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Tutoring abilities

Look around your room — can you find a shape made only of straight lines that closes all the way up like a fence?

A **polygon** is any flat, closed shape made only of straight sides.

Every polygon has straight **sides** and pointy corners called **vertices** (one corner is called a **vertex**).

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      background: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }
    .shape-nav {
      display: flex;
      gap: 6px;
      flex-wrap: wrap;
      justify-content: center;
      width: 100%;
      max-width: 440px;
    }
    .shape-btn {
      background: #ffffff;
      border: 2px solid #e5e7eb;
      border-radius: 10px;
      padding: 6px 10px;
      font-size: 13px;
      font-weight: 600;
      color: #4b5563;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 5px;
      transition: background 0.2s, border-color 0.2s, color 0.2s;
    }
    .shape-btn.active {
      border-color: #22b7ff;
      background: #e0f2fe;
      color: #0369a1;
    }
    .display-area {
      position: relative;
      width: 100%;
      max-width: 320px;
      aspect-ratio: 1 / 1;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }
    .polygon-fill {
      fill: #f0f9ff;
      stroke: #bae6fd;
      stroke-width: 2;
      transition: all 0.3s ease;
    }
    .side-line {
      stroke: #0284c7;
      stroke-width: 5;
      stroke-linecap: round;
      transition: stroke-dasharray 0.4s ease, opacity 0.3s ease;
    }
    .side-label {
      font-size: 13px;
      font-weight: 700;
      fill: #0369a1;
      text-anchor: middle;
      dominant-baseline: central;
    }
    .side-badge {
      fill: #ffffff;
      stroke: #0284c7;
      stroke-width: 1.5;
    }
    .vertex-dot {
      fill: #f59e0b;
      stroke: #ffffff;
      stroke-width: 2.5;
      cursor: pointer;
      transition: transform 0.2s ease;
    }
    .vertex-label {
      font-size: 11px;
      font-weight: 800;
      fill: #ffffff;
      text-anchor: middle;
      dominant-baseline: central;
      pointer-events: none;
    }
    .stats-row {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      width: 100%;
      max-width: 380px;
    }
    .stat-pill {
      background: #ffffff;
      border: 1.5px solid #e5e7eb;
      border-radius: 20px;
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 700;
      display: flex;
      align-items: center;
      gap: 6px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.04);
    }
    .stat-pill.sides {
      color: #0284c7;
      border-color: #bae6fd;
      background: #f0f9ff;
    }
    .stat-pill.vertices {
      color: #d97706;
      border-color: #fde68a;
      background: #fffbeb;
    }
    .equals-sign {
      font-size: 18px;
      font-weight: 800;
      color: #9ca3af;
    }
  </style>
</head>
<body>

  
  <div class="shape-nav" id="shapeNav">
    <button class="shape-btn active" data-sides="3">Triangle</button>
    <button class="shape-btn" data-sides="4">Quadrilateral</button>
    <button class="shape-btn" data-sides="5">Pentagon</button>
    <button class="shape-btn" data-sides="6">Hexagon</button>
    <button class="shape-btn" data-sides="8">Octagon</button>
  </div>

  
  <div class="display-area">
    <svg viewBox="0 0 300 300" id="polySvg">
      <polygon id="polyBackground" class="polygon-fill" />
      <g id="sidesGroup"></g>
      <g id="verticesGroup"></g>
    </svg>
  </div>

  
  <div class="stats-row">
    <div class="stat-pill sides">
      <span style="display:inline-block;width:12px;height:3px;background:#0284c7;border-radius:2px;"></span>
      <span id="sideCount">3 Sides</span>
    </div>
    <span class="equals-sign">=</span>
    <div class="stat-pill vertices">
      <span style="display:inline-block;width:8px;height:8px;background:#f59e0b;border-radius:50%;"></span>
      <span id="vertexCount">3 Vertices</span>
    </div>
  </div>

  <script>
    const shapes = [
      { name: 'Triangle', sides: 3, rot: -90 },
      { name: 'Quadrilateral', sides: 4, rot: -45 },
      { name: 'Pentagon', sides: 5, rot: -90 },
      { name: 'Hexagon', sides: 6, rot: 0 },
      { name: 'Octagon', sides: 8, rot: 22.5 }
    ];

    const cx = 150, cy = 150, radius = 95;
    let currentSides = 3;

    const nav = document.getElementById('shapeNav');
    const svgPoly = document.getElementById('polyBackground');
    const sidesGroup = document.getElementById('sidesGroup');
    const verticesGroup = document.getElementById('verticesGroup');
    const sideCountEl = document.getElementById('sideCount');
    const vertexCountEl = document.getElementById('vertexCount');

    function getPolygonPoints(n, rotDeg) {
      const points = [];
      const rotRad = (rotDeg * Math.PI) / 180;
      for (let i = 0; i < n; i++) {
        const angle = rotRad + (i * 2 * Math.PI) / n;
        const x = cx + radius * Math.cos(angle);
        const y = cy + radius * Math.sin(angle);
        points.push({ x, y });
      }
      return points;
    }

    function renderPolygon(n) {
      const shapeConfig = shapes.find(s => s.sides === n) || shapes[0];
      const points = getPolygonPoints(n, shapeConfig.rot);

      // 1. Update background polygon
      const pointsAttr = points.map(p => `${p.x.toFixed(1)},${p.y.toFixed(1)}`).join(' ');
      svgPoly.setAttribute('points', pointsAttr);

      // 2. Draw Sides & Numbers
      sidesGroup.innerHTML = '';
      for (let i = 0; i < n; i++) {
        const p1 = points[i];
        const p2 = points[(i + 1) % n];

        // Side line
        const line = document.createElementNS('http://www.w3.org/2000/svg', 'line');
        line.setAttribute('x1', p1.x);
        line.setAttribute('y1', p1.y);
        line.setAttribute('x2', p2.x);
        line.setAttribute('y2', p2.y);
        line.setAttribute('class', 'side-line');
        sidesGroup.appendChild(line);

        // Side label position (midpoint pushed outward slightly)
        const midX = (p1.x + p2.x) / 2;
        const midY = (p1.y + p2.y) / 2;
        const offset = 18;
        const dx = midX - cx;
        const dy = midY - cy;
        const dist = Math.sqrt(dx * dx + dy * dy);
        const lx = midX + (dx / dist) * offset;
        const ly = midY + (dy / dist) * offset;

        // Side badge background circle
        const badge = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        badge.setAttribute('cx', lx);
        badge.setAttribute('cy', ly);
        badge.setAttribute('r', '9');
        badge.setAttribute('class', 'side-badge');
        sidesGroup.appendChild(badge);

        // Side number text
        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', lx);
        text.setAttribute('y', ly);
        text.setAttribute('class', 'side-label');
        text.textContent = i + 1;
        sidesGroup.appendChild(text);
      }

      // 3. Draw Vertices (corner dots)
      verticesGroup.innerHTML = '';
      points.forEach((p, i) => {
        const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        circle.setAttribute('cx', p.x);
        circle.setAttribute('cy', p.y);
        circle.setAttribute('r', '11');
        circle.setAttribute('class', 'vertex-dot');
        verticesGroup.appendChild(circle);

        const vText = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        vText.setAttribute('x', p.x);
        vText.setAttribute('y', p.y);
        vText.setAttribute('class', 'vertex-label');
        vText.textContent = i + 1;
        verticesGroup.appendChild(vText);
      });

      // 4. Update texts
      sideCountEl.textContent = `${n} Sides`;
      vertexCountEl.textContent = `${n} Vertices`;
    }

    // Button event handling
    nav.addEventListener('click', (e) => {
      const btn = e.target.closest('.shape-btn');
      if (!btn) return;

      document.querySelectorAll('.shape-btn').forEach(b => b.classList.remove('active'));
      btn.classList.add('active');

      currentSides = parseInt(btn.dataset.sides, 10);
      renderPolygon(currentSides);
    });

    // Initial render
    renderPolygon(3);
  </script>
</body>
</html>

How do you know how many vertices a polygon will have if you already know its sides?

Every closed polygon always has the **exact same number** of sides and vertices.

Because every time two straight lines meet to close the shape, they create exactly one new corner.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
    }
    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }
    .main-layout {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }
    .shape-selector {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 6px;
      width: 100%;
    }
    .tab-btn {
      background: #ffffff;
      border: 1px solid #dbe2ea;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 14px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
      transition: background 0.2s ease, color 0.2s ease, border-color 0.2s ease, box-shadow 0.2s ease;
    }
    .tab-btn:hover {
      background: #f1f5f9;
      color: #1e2945;
    }
    .tab-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }
    .display-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      width: 100%;
      padding: 14px;
      display: flex;
      flex-direction: column;
      align-items: center;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
    }
    .svg-container {
      width: 220px;
      height: 220px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }
    .poly-fill {
      fill: rgba(34, 183, 255, 0.08);
      stroke: #22b7ff;
      stroke-width: 4;
      stroke-linejoin: round;
      transition: all 0.35s ease;
    }
    .vertex-dot {
      fill: #ff9800;
      stroke: #ffffff;
      stroke-width: 2.5;
      cursor: pointer;
      transition: transform 0.3s ease;
    }
    .side-label {
      font-size: 11px;
      font-weight: 700;
      fill: #0284c7;
      text-anchor: middle;
      dominant-baseline: central;
    }
    .side-tag {
      fill: #e0f2fe;
      stroke: #bae6fd;
      stroke-width: 1;
    }
    .metrics {
      display: flex;
      justify-content: center;
      gap: 16px;
      width: 100%;
      margin-top: 10px;
    }
    .metric-badge {
      display: flex;
      align-items: center;
      gap: 6px;
      padding: 6px 14px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 700;
    }
    .badge-sides {
      background: #e0f2fe;
      color: #0369a1;
      border: 1px solid #bae6fd;
    }
    .badge-vertices {
      background: #fef3c7;
      color: #b45309;
      border: 1px solid #fde68a;
    }
    .indicator-icon {
      width: 10px;
      height: 10px;
      border-radius: 50%;
      display: inline-block;
    }
    .indicator-side {
      background: #0284c7;
      border-radius: 2px;
      width: 12px;
      height: 3px;
    }
    .indicator-vertex {
      background: #ff9800;
      border-radius: 50%;
      width: 9px;
      height: 9px;
    }
    .equal-sign {
      display: flex;
      align-items: center;
      font-size: 18px;
      font-weight: 800;
      color: #94a3b8;
    }
  </style>
</head>
<body>

  <div class="main-layout">
    
    <div class="shape-selector">
      <button class="tab-btn active" onclick="setSides(3)">3</button>
      <button class="tab-btn" onclick="setSides(4)">4</button>
      <button class="tab-btn" onclick="setSides(5)">5</button>
      <button class="tab-btn" onclick="setSides(6)">6</button>
      <button class="tab-btn" onclick="setSides(8)">8</button>
    </div>

    
    <div class="display-card">
      <div class="svg-container">
        <svg id="polySvg" viewBox="0 0 200 200">
          <polygon id="polyShape" class="poly-fill" points=""></polygon>
          <g id="sideTagsGroup"></g>
          <g id="vertexGroup"></g>
        </svg>
      </div>

      
      <div class="metrics">
        <div class="metric-badge badge-sides">
          <span class="indicator-side"></span>
          <span id="sideCount">3 Sides</span>
        </div>
        <div class="equal-sign">=</div>
        <div class="metric-badge badge-vertices">
          <span class="indicator-vertex"></span>
          <span id="vertexCount">3 Vertices</span>
        </div>
      </div>
    </div>
  </div>

  <script>
    let currentN = 3;

    function getPoints(n) {
      const cx = 100, cy = 100;
      const r = 74;
      const pts = [];
      const offset = (n % 2 === 1) ? -Math.PI / 2 : -Math.PI / 2 + Math.PI / n;
      for (let i = 0; i < n; i++) {
        const angle = offset + (i * 2 * Math.PI / n);
        pts.push({
          x: cx + r * Math.cos(angle),
          y: cy + r * Math.sin(angle)
        });
      }
      return pts;
    }

    function setSides(n) {
      currentN = n;
      
      // Update buttons
      document.querySelectorAll('.tab-btn').forEach(btn => {
        btn.classList.toggle('active', parseInt(btn.textContent) === n);
      });

      const pts = getPoints(n);
      
      // Draw Polygon
      const pointsAttr = pts.map(p => `${p.x.toFixed(1)},${p.y.toFixed(1)}`).join(' ');
      document.getElementById('polyShape').setAttribute('points', pointsAttr);

      // Draw Side Labels
      const sideTagsGroup = document.getElementById('sideTagsGroup');
      sideTagsGroup.innerHTML = '';
      for (let i = 0; i < n; i++) {
        const p1 = pts[i];
        const p2 = pts[(i + 1) % n];
        const mx = (p1.x + p2.x) / 2;
        const my = (p1.y + p2.y) / 2;

        // Position slightly outward from center
        const vx = mx - 100;
        const vy = my - 100;
        const dist = Math.hypot(vx, vy);
        const lx = mx + (vx / dist) * 14;
        const ly = my + (vy / dist) * 14;

        // Tag background
        const rect = document.createElementNS('http://www.w3.org/2000/svg', 'rect');
        rect.setAttribute('x', lx - 8);
        rect.setAttribute('y', ly - 8);
        rect.setAttribute('width', 16);
        rect.setAttribute('height', 16);
        rect.setAttribute('rx', 4);
        rect.setAttribute('class', 'side-tag');
        sideTagsGroup.appendChild(rect);

        // Tag number text
        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', lx);
        text.setAttribute('y', ly);
        text.setAttribute('class', 'side-label');
        text.textContent = (i + 1);
        sideTagsGroup.appendChild(text);
      }

      // Draw Vertex Points
      const vertexGroup = document.getElementById('vertexGroup');
      vertexGroup.innerHTML = '';
      pts.forEach(p => {
        const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        circle.setAttribute('cx', p.x);
        circle.setAttribute('cy', p.y);
        circle.setAttribute('r', 6);
        circle.setAttribute('class', 'vertex-dot');
        vertexGroup.appendChild(circle);
      });

      // Update Count Labels
      document.getElementById('sideCount').textContent = `${n} Sides`;
      document.getElementById('vertexCount').textContent = `${n} Vertices`;
    }

    // Initial render
    setSides(3);
  </script>
</body>
</html>

What names do we give these shapes as we add more and more sides?

We name each polygon by counting its **sides** and **vertices**.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      gap: 12px;
    }

    .nav-tabs {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 6px;
      width: 100%;
      max-width: 480px;
    }

    .tab-btn {
      border: 1.5px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      padding: 6px 12px;
      border-radius: 20px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
      transition: all 0.2s ease;
    }

    .tab-btn:hover {
      border-color: #22b7ff;
      color: #1e2945;
    }

    .tab-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.35);
    }

    .stage {
      position: relative;
      width: 100%;
      max-width: 320px;
      aspect-ratio: 1 / 1;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }

    .stat-bar {
      display: flex;
      justify-content: center;
      align-items: center;
      gap: 12px;
      width: 100%;
      max-width: 420px;
    }

    .pill {
      display: flex;
      align-items: center;
      gap: 6px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      padding: 6px 14px;
      border-radius: 12px;
      font-size: 14px;
      font-weight: 600;
    }

    .pill-dot {
      width: 10px;
      height: 10px;
      border-radius: 50%;
    }

    .dot-edge {
      background: #22b7ff;
    }

    .dot-vert {
      background: #f59e0b;
    }

    .pill-count {
      color: #1e2945;
      font-size: 15px;
      font-weight: 700;
    }

    .equal-sign {
      color: #94a3b8;
      font-size: 18px;
      font-weight: 700;
    }

    .poly-edge {
      stroke: #22b7ff;
      stroke-width: 4;
      stroke-linecap: round;
      stroke-linejoin: round;
      fill: rgba(34, 183, 255, 0.08);
      transition: all 0.3s ease;
    }

    .vert-node {
      fill: #f59e0b;
      stroke: #ffffff;
      stroke-width: 2.5;
      transition: all 0.3s ease;
    }

    .edge-badge {
      fill: #0284c7;
      font-size: 11px;
      font-weight: 700;
      text-anchor: middle;
      dominant-baseline: central;
    }

    .badge-bg {
      fill: #ffffff;
      stroke: #bae6fd;
      stroke-width: 1.5;
    }
  </style>
</head>
<body>

  <div class="nav-tabs" id="tabs"></div>

  <div class="stage">
    <svg viewBox="0 0 300 300" id="polySvg">
      <polygon id="polyShape" class="poly-edge" points="" />
      <g id="edgeBadges"></g>
      <g id="vertGroup"></g>
    </svg>
  </div>

  <div class="stat-bar">
    <div class="pill">
      <span class="pill-dot dot-edge"></span>
      <span id="sideText">Sides: <span class="pill-count" id="sideNum">3</span></span>
    </div>
    <span class="equal-sign">=</span>
    <div class="pill">
      <span class="pill-dot dot-vert"></span>
      <span id="vertText">Vertices: <span class="pill-count" id="vertNum">3</span></span>
    </div>
  </div>

  <script>
    const shapes = [
      { name: 'Triangle', n: 3 },
      { name: 'Quadrilateral', n: 4 },
      { name: 'Pentagon', n: 5 },
      { name: 'Hexagon', n: 6 },
      { name: 'Octagon', n: 8 }
    ];

    let activeIndex = 0;
    const cx = 150;
    const cy = 150;
    const r = 100;

    const tabsContainer = document.getElementById('tabs');
    const polyShape = document.getElementById('polyShape');
    const edgeBadges = document.getElementById('edgeBadges');
    const vertGroup = document.getElementById('vertGroup');
    const sideNum = document.getElementById('sideNum');
    const vertNum = document.getElementById('vertNum');

    shapes.forEach((s, idx) => {
      const btn = document.createElement('button');
      btn.className = 'tab-btn' + (idx === 0 ? ' active' : '');
      btn.innerHTML = `${s.name} <span style="opacity:0.7">(${s.n})</span>`;
      btn.addEventListener('click', () => setShape(idx));
      tabsContainer.appendChild(btn);
    });

    function getPoints(n) {
      const pts = [];
      const offset = (n % 2 === 1) ? -Math.PI / 2 : -Math.PI / 2 + Math.PI / n;
      for (let i = 0; i < n; i++) {
        const angle = offset + (i * 2 * Math.PI / n);
        pts.push({
          x: Math.round((cx + r * Math.cos(angle)) * 10) / 10,
          y: Math.round((cy + r * Math.sin(angle)) * 10) / 10
        });
      }
      return pts;
    }

    function setShape(idx) {
      activeIndex = idx;
      const data = shapes[idx];
      const n = data.n;

      Array.from(tabsContainer.children).forEach((btn, i) => {
        btn.classList.toggle('active', i === idx);
      });

      const pts = getPoints(n);
      polyShape.setAttribute('points', pts.map(p => `${p.x},${p.y}`).join(' '));

      vertGroup.innerHTML = '';
      pts.forEach(p => {
        const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        circle.setAttribute('cx', p.x);
        circle.setAttribute('cy', p.y);
        circle.setAttribute('r', '7');
        circle.setAttribute('class', 'vert-node');
        vertGroup.appendChild(circle);
      });

      edgeBadges.innerHTML = '';
      for (let i = 0; i < n; i++) {
        const p1 = pts[i];
        const p2 = pts[(i + 1) % n];
        const mx = (p1.x + p2.x) / 2;
        const my = (p1.y + p2.y) / 2;

        const vx = mx - cx;
        const vy = my - cy;
        const len = Math.hypot(vx, vy);
        const bx = mx + (len ? (vx / len) * 16 : 0);
        const by = my + (len ? (vy / len) * 16 : 0);

        const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        circle.setAttribute('cx', bx);
        circle.setAttribute('cy', by);
        circle.setAttribute('r', '9');
        circle.setAttribute('class', 'badge-bg');

        const txt = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        txt.setAttribute('x', bx);
        txt.setAttribute('y', by);
        txt.setAttribute('class', 'edge-badge');
        txt.textContent = i + 1;

        edgeBadges.appendChild(circle);
        edgeBadges.appendChild(txt);
      }

      sideNum.textContent = n;
      vertNum.textContent = n;
    }

    setShape(0);
  </script>
</body>
</html>

Does a shape stop being a hexagon if its sides look tilted or stretched out?

A **regular polygon** has all sides the same length and all corners the same size.

An **irregular polygon** has sides or corners of different sizes, but it still has the same total count.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Classifying Polygons</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: #f8f9fa;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 12px;
      min-height: 100%;
    }
    .shape-nav {
      display: flex;
      gap: 6px;
      width: 100%;
      max-width: 440px;
      justify-content: center;
    }
    .nav-btn {
      flex: 1;
      padding: 6px 4px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      text-align: center;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }
    .nav-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }
    .stage {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      padding: 12px;
      position: relative;
    }
    .toggle-bar {
      display: flex;
      background: #f1f5f9;
      border-radius: 8px;
      padding: 2px;
      gap: 2px;
      margin-bottom: 8px;
    }
    .toggle-btn {
      padding: 4px 12px;
      font-size: 13px;
      font-weight: 600;
      border: none;
      background: transparent;
      border-radius: 6px;
      color: #64748b;
      cursor: pointer;
      transition: background 0.2s, color 0.2s;
    }
    .toggle-btn.active {
      background: #ffffff;
      color: #1e2945;
      box-shadow: 0 1px 3px rgba(0, 0, 0, 0.1);
    }
    svg {
      width: 100%;
      max-width: 240px;
      height: 190px;
      overflow: visible;
    }
    polygon {
      fill: rgba(34, 183, 255, 0.12);
      stroke: #22b7ff;
      stroke-width: 3;
      stroke-linejoin: round;
      transition: all 0.3s ease;
    }
    .vertex-dot {
      fill: #ff9800;
      stroke: #ffffff;
      stroke-width: 2;
      transition: cx 0.3s ease, cy 0.3s ease;
    }
    .vertex-text {
      font-size: 10px;
      font-weight: 700;
      fill: #ffffff;
      text-anchor: middle;
      dominant-baseline: central;
      pointer-events: none;
    }
    .side-label {
      font-size: 11px;
      font-weight: 700;
      fill: #0284c7;
      text-anchor: middle;
      dominant-baseline: central;
      background: white;
    }
    .counts-bar {
      display: flex;
      gap: 12px;
      width: 100%;
      max-width: 440px;
      justify-content: center;
    }
    .badge {
      flex: 1;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 12px;
      font-size: 14px;
      font-weight: 600;
    }
    .badge-icon {
      width: 10px;
      height: 10px;
      border-radius: 50%;
    }
    .badge-sides .badge-icon {
      border-radius: 2px;
      background: #22b7ff;
    }
    .badge-vertices .badge-icon {
      background: #ff9800;
    }
    .badge span {
      color: #1e2945;
      font-weight: 700;
    }
  </style>
</head>
<body>

  <div class="shape-nav" role="tablist">
    <button class="nav-btn active" data-sides="3" onclick="setSides(3)">3 Sides</button>
    <button class="nav-btn" data-sides="4" onclick="setSides(4)">4 Sides</button>
    <button class="nav-btn" data-sides="5" onclick="setSides(5)">5 Sides</button>
    <button class="nav-btn" data-sides="6" onclick="setSides(6)">6 Sides</button>
  </div>

  <div class="stage">
    <div class="toggle-bar">
      <button class="toggle-btn active" id="btn-reg" onclick="setType('reg')">Regular</button>
      <button class="toggle-btn" id="btn-irreg" onclick="setType('irreg')">Irregular</button>
    </div>

    <svg viewBox="0 0 240 190" id="polySvg">
      <polygon id="polyShape" points="" />
      <g id="sideLabels"></g>
      <g id="vertexMarkers"></g>
    </svg>
  </div>

  <div class="counts-bar">
    <div class="badge badge-sides">
      <div class="badge-icon"></div>
      Sides: <span id="sidesCount">3</span>
    </div>
    <div class="badge badge-vertices">
      <div class="badge-icon"></div>
      Vertices: <span id="verticesCount">3</span>
    </div>
  </div>

  <script>
    const shapes = {
      3: {
        reg: [[120, 25], [35, 165], [205, 165]],
        irreg: [[80, 30], [30, 165], [210, 125]]
      },
      4: {
        reg: [[50, 30], [190, 30], [190, 170], [50, 170]],
        irreg: [[40, 55], [195, 30], [165, 165], [60, 130]]
      },
      5: {
        reg: [[120, 22], [205, 80], [175, 168], [65, 168], [35, 80]],
        irreg: [[70, 30], [195, 45], [205, 155], [105, 170], [35, 110]]
      },
      6: {
        reg: [[120, 20], [195, 60], [195, 135], [120, 175], [45, 135], [45, 60]],
        irreg: [[60, 25], [180, 35], [210, 105], [165, 170], [80, 145], [30, 95]]
      }
    };

    let currentSides = 3;
    let currentType = 'reg';

    function setSides(n) {
      currentSides = n;
      document.querySelectorAll('.nav-btn').forEach(btn => {
        btn.classList.toggle('active', parseInt(btn.dataset.sides) === n);
      });
      render();
    }

    function setType(type) {
      currentType = type;
      document.getElementById('btn-reg').classList.toggle('active', type === 'reg');
      document.getElementById('btn-irreg').classList.toggle('active', type === 'irreg');
      render();
    }

    function render() {
      const pts = shapes[currentSides][currentType];
      const ptsString = pts.map(p => p.join(',')).join(' ');
      document.getElementById('polyShape').setAttribute('points', ptsString);

      // Render vertex dots
      const vertexGroup = document.getElementById('vertexMarkers');
      vertexGroup.innerHTML = '';
      pts.forEach((p, i) => {
        const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        circle.setAttribute('cx', p[0]);
        circle.setAttribute('cy', p[1]);
        circle.setAttribute('r', '8');
        circle.setAttribute('class', 'vertex-dot');
        vertexGroup.appendChild(circle);

        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', p[0]);
        text.setAttribute('y', p[1]);
        text.setAttribute('class', 'vertex-text');
        text.textContent = (i + 1);
        vertexGroup.appendChild(text);
      });

      // Render side labels
      const sideGroup = document.getElementById('sideLabels');
      sideGroup.innerHTML = '';
      for (let i = 0; i < pts.length; i++) {
        const p1 = pts[i];
        const p2 = pts[(i + 1) % pts.length];
        const mx = (p1[0] + p2[0]) / 2;
        const my = (p1[1] + p2[1]) / 2;

        // Vector away from center (120, 100)
        let dx = mx - 120;
        let dy = my - 100;
        const len = Math.hypot(dx, dy) || 1;
        const ox = mx + (dx / len) * 14;
        const oy = my + (dy / len) * 14;

        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', ox);
        text.setAttribute('y', oy);
        text.setAttribute('class', 'side-label');
        text.textContent = 'S' + (i + 1);
        sideGroup.appendChild(text);
      }

      document.getElementById('sidesCount').textContent = currentSides;
      document.getElementById('verticesCount').textContent = currentSides;
    }

    render();
  </script>
</body>
</html>

Let's put your detective skills to work on a mystery shape!

Let's count the sides of a stop sign: it has 1, 2, 3, 4, 5, 6, 7, **8 straight sides**.

Because it has 8 sides and 8 vertices, this shape is an **octagon**.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif; }
    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }
    .diagram-wrap {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }
    .nav-bar {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 6px;
      width: 100%;
    }
    .nav-btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 10px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
      transition: background 0.2s, color 0.2s, border-color 0.2s, box-shadow 0.2s;
    }
    .nav-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }
    .canvas-card {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 12px;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
    }
    svg {
      width: 100%;
      max-width: 250px;
      height: 220px;
      overflow: visible;
    }
    .poly-fill {
      fill: rgba(34, 183, 255, 0.08);
      stroke: #22b7ff;
      stroke-width: 3.5;
      stroke-linejoin: round;
      transition: all 0.3s ease;
    }
    .vertex-group circle {
      fill: #ffffff;
      stroke: #0284c7;
      stroke-width: 2.5;
      transition: transform 0.2s ease;
    }
    .vertex-group text {
      fill: #0284c7;
      font-size: 12px;
      font-weight: 700;
      text-anchor: middle;
      dominant-baseline: central;
    }
    .side-label {
      fill: #64748b;
      font-size: 11px;
      font-weight: 600;
      text-anchor: middle;
      dominant-baseline: central;
    }
    .info-row {
      display: flex;
      justify-content: center;
      gap: 10px;
      width: 100%;
      margin-top: 4px;
    }
    .badge {
      background: #f1f5f9;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #334155;
      display: flex;
      align-items: center;
      gap: 5px;
    }
    .badge-name {
      background: rgba(34, 183, 255, 0.12);
      color: #0284c7;
      font-weight: 700;
      font-size: 15px;
    }
    .badge-count {
      color: #0284c7;
    }
  </style>
</head>
<body>
  <div class="diagram-wrap">
    <div class="nav-bar" id="navBar"></div>

    <div class="canvas-card">
      <svg id="polySvg" viewBox="0 0 240 220"></svg>
      <div class="info-row">
        <div class="badge badge-name" id="polyName">Octagon</div>
        <div class="badge"><span class="badge-count" id="polySides">8</span> sides</div>
        <div class="badge"><span class="badge-count" id="polyVertices">8</span> vertices</div>
      </div>
    </div>
  </div>

  <script>
    const shapes = [
      { n: 3, name: "Triangle" },
      { n: 4, name: "Quadrilateral" },
      { n: 5, name: "Pentagon" },
      { n: 6, name: "Hexagon" },
      { n: 7, name: "Heptagon" },
      { n: 8, name: "Octagon" }
    ];

    let currentIndex = 5; // Default Octagon (matches preceding text)

    const navBar = document.getElementById('navBar');
    const polySvg = document.getElementById('polySvg');
    const polyName = document.getElementById('polyName');
    const polySides = document.getElementById('polySides');
    const polyVertices = document.getElementById('polyVertices');

    function renderNav() {
      navBar.innerHTML = '';
      shapes.forEach((s, idx) => {
        const btn = document.createElement('button');
        btn.className = `nav-btn ${idx === currentIndex ? 'active' : ''}`;
        btn.textContent = `${s.n} · ${s.name}`;
        btn.onclick = () => {
          currentIndex = idx;
          updateShape();
        };
        navBar.appendChild(btn);
      });
    }

    function getPolygonPoints(n, cx = 120, cy = 110, r = 82) {
      const points = [];
      const angleOffset = -Math.PI / 2;
      for (let i = 0; i < n; i++) {
        const angle = angleOffset + (i * 2 * Math.PI / n);
        const x = cx + r * Math.cos(angle);
        const y = cy + r * Math.sin(angle);
        points.push({ x, y, angle });
      }
      return points;
    }

    function updateShape() {
      renderNav();
      const current = shapes[currentIndex];
      polyName.textContent = current.name;
      polySides.textContent = current.n;
      polyVertices.textContent = current.n;

      const pts = getPolygonPoints(current.n);
      const pointsStr = pts.map(p => `${p.x.toFixed(1)},${p.y.toFixed(1)}`).join(' ');

      let svgHtml = `<polygon class="poly-fill" points="${pointsStr}" />`;

      // Side markers
      for (let i = 0; i < pts.length; i++) {
        const next = pts[(i + 1) % pts.length];
        const midX = (pts[i].x + next.x) / 2;
        const midY = (pts[i].y + next.y) / 2;
        
        // Push label slightly outward
        const dx = midX - 120;
        const dy = midY - 110;
        const dist = Math.sqrt(dx * dx + dy * dy) || 1;
        const labelX = midX + (dx / dist) * 12;
        const labelY = midY + (dy / dist) * 12;

        svgHtml += `<text class="side-label" x="${labelX.toFixed(1)}" y="${labelY.toFixed(1)}">s${i+1}</text>`;
      }

      // Vertices with numbered dots
      pts.forEach((p, idx) => {
        svgHtml += `
          <g class="vertex-group">
            <circle cx="${p.x.toFixed(1)}" cy="${p.y.toFixed(1)}" r="10" />
            <text x="${p.x.toFixed(1)}" y="${p.y.toFixed(1)}">${idx + 1}</text>
          </g>
        `;
      });

      polySvg.innerHTML = svgHtml;
    }

    updateShape();
  </script>
</body>
</html>

Whenever you spot an unfamiliar shape, just tap and count its corners — the number will instantly tell you its name!

math

Classifying Polygons by Number of Sides and Vertices. Classify regular and irregular polygons with 3 to 10 sides by counting sides and vertices.

What are 2D Shapes and Polygons? (Properties of 2D Shapes)

Classify Polygons: Up to 10 and 12 Sides by Counting Sides and Vertices

Polygon Explorations: Making, Classifying, and Sorting 2D Shapes

To achieve mastery aligned with Curriculum for Excellence (CfE MTH 2-16a / Early-to-Primary progressions), implement a Concrete-Pictorial-Abstract (CPA) pedagogical framework. Begin with concrete multisensory exploration: have learners construct regular and irregular polygons using geoboards, craft sticks, and playdough connectors (vertices). Teach explicit tracking strategies (e.g., placing a start peg or dot-marking vertices) to prevent skip- or double-counting. Connect learning to the local Scottish built and natural environment through 'shape scavenger hunts' (identifying octagonal stop signs, hexagonal honeycombs, triangular roof trusses, and irregular quadrilaterals in paving stones), actively discussing why specific polygons provide structural strength or visibility.

Classifying Polygons by Number of Sides and Vertices

What is a Polygon?

The Matching Rule

Counting Shapes from 3 to 10