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Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

Lesson generation model

System default

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Tutoring abilities

What happens when you cut a rectangular sandwich diagonally from corner to corner?

You get two matching triangles with a neat, square corner called a **right angle**.

<!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">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }
    .stepper-nav {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 420px;
      justify-content: center;
    }
    .step-btn {
      flex: 1;
      padding: 8px 12px;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      background: #ffffff;
      color: #64748b;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: all 0.25s ease;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
    }
    .step-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: 420px;
      height: 220px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }
    .tri2-group {
      transition: transform 0.6s cubic-bezier(0.4, 0, 0.2, 1), opacity 0.6s ease;
      transform-origin: 210px 110px;
    }
    .tri1-path {
      transition: fill-opacity 0.4s ease, stroke 0.4s ease;
    }
    .measure-label {
      font-size: 14px;
      font-weight: 700;
      fill: #475569;
      text-anchor: middle;
      dominant-baseline: central;
    }
    .formula-card {
      width: 100%;
      max-width: 420px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 16px;
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 48px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
    }
    .katex {
      font-size: 1.15em;
    }
  </style>
</head>
<body>

  
  <div class="stepper-nav">
    <button class="step-btn active" id="btn-0" onclick="setStep(0)">1. Triangle</button>
    <button class="step-btn" id="btn-1" onclick="setStep(1)">2. Rectangle</button>
    <button class="step-btn" id="btn-2" onclick="setStep(2)">3. Half Area</button>
  </div>

  
  <div class="stage">
    <svg viewBox="0 0 420 220">
      <defs>
        
        <pattern id="grid" width="30" height="30" patternUnits="userSpaceOnUse">
          <path d="M 30 0 L 0 0 0 30" fill="none" stroke="#f1f5f9" stroke-width="1.5" />
        </pattern>
      </defs>

      
      <rect x="90" y="30" width="240" height="160" fill="url(#grid)" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="4 4" rx="4" />

      
      
      <polygon id="tri1" class="tri1-path" points="90,190 330,190 90,30" fill="#22b7ff" fill-opacity="0.35" stroke="#0284c7" stroke-width="2.5" stroke-linejoin="round" />

      
      <path d="M 90,172 L 108,172 L 108,190" fill="none" stroke="#0284c7" stroke-width="2" />
      <circle cx="99" cy="181" r="2" fill="#0284c7" />

      
      
      <g id="tri2-wrap" class="tri2-group" style="opacity: 0; transform: translate(50px, -40px) rotate(15deg);">
        <polygon points="330,30 90,30 330,190" fill="#38bdf8" fill-opacity="0.25" stroke="#0284c7" stroke-width="2.5" stroke-dasharray="6 4" stroke-linejoin="round" />
        
        <path d="M 330,48 L 312,48 L 312,30" fill="none" stroke="#0284c7" stroke-width="1.5" />
      </g>

      
      
      <line x1="90" y1="205" x2="330" y2="205" stroke="#64748b" stroke-width="1.5" marker-end="url(#arrow)" />
      <text x="210" y="215" class="measure-label">base (b)</text>

      
      <line x1="72" y1="30" x2="72" y2="190" stroke="#64748b" stroke-width="1.5" />
      <text x="50" y="110" class="measure-label" transform="rotate(-90 50 110)">height (h)</text>
    </svg>
  </div>

  
  <div class="formula-card">
    <div id="formula-target"></div>
  </div>

  <script>
    const formulas = [
      "\\text{Area} = \\,\\text{?}",
      "\\text{Rectangle Area} = b \\times h",
      "\\text{Triangle Area} = \\frac{1}{2} \\times b \\times h"
    ];

    let currentStep = 0;

    function renderFormula(index) {
      const el = document.getElementById('formula-target');
      if (window.katex) {
        katex.render(formulas[index], el, { throwOnError: false });
      } else {
        el.textContent = index === 2 ? "Area = 1/2 × base × height" : index === 1 ? "Rectangle = base × height" : "Area = ?";
      }
    }

    function setStep(index) {
      currentStep = index;
      
      // Update buttons
      for (let i = 0; i < 3; i++) {
        const btn = document.getElementById('btn-' + i);
        if (btn) btn.classList.toggle('active', i === index);
      }

      const tri2 = document.getElementById('tri2-wrap');
      const tri1 = document.getElementById('tri1');

      if (index === 0) {
        // Step 1: Only primary triangle
        tri2.style.opacity = '0';
        tri2.style.transform = 'translate(60px, -40px) rotate(20deg)';
        tri1.setAttribute('fill', '#22b7ff');
        tri1.setAttribute('fill-opacity', '0.45');
      } else if (index === 1) {
        // Step 2: Both triangles combined to rectangle
        tri2.style.opacity = '1';
        tri2.style.transform = 'translate(0px, 0px) rotate(0deg)';
        tri1.setAttribute('fill', '#22b7ff');
        tri1.setAttribute('fill-opacity', '0.35');
      } else if (index === 2) {
        // Step 3: Triangle highlighted as exactly half
        tri2.style.opacity = '0.35';
        tri2.style.transform = 'translate(0px, 0px) rotate(0deg)';
        tri1.setAttribute('fill', '#22b7ff');
        tri1.setAttribute('fill-opacity', '0.65');
      }

      renderFormula(index);
    }

    // Initialize
    window.addEventListener('DOMContentLoaded', () => {
      setStep(0);
    });
  </script>
</body>
</html>

How do we find out how much space is inside one of those triangles?

**Area** is the total number of square units that fit inside a flat shape.

For any rectangle, we find the area by multiplying its bottom edge (**base**) by its straight up-and-down edge (**height**).

<!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">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      gap: 12px;
      user-select: none;
    }

    .nav-tabs {
      display: flex;
      gap: 8px;
      background: #f1f5f9;
      padding: 4px;
      border-radius: 20px;
    }

    .tab-btn {
      border: none;
      background: transparent;
      padding: 6px 14px;
      border-radius: 16px;
      font-size: 14px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      transition: all 0.25s ease;
    }

    .tab-btn.active {
      background: #ffffff;
      color: #0284c7;
      box-shadow: 0 2px 6px rgba(0, 0, 0, 0.08);
    }

    .stage-container {
      width: 100%;
      max-width: 420px;
      height: 230px;
      position: relative;
      display: flex;
      align-items: center;
      justify-content: center;
    }

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

    .grid-line {
      stroke: #e2e8f0;
      stroke-width: 1;
      stroke-dasharray: 2 2;
    }

    .tri-a {
      fill: #38bdf8;
      fill-opacity: 0.85;
      stroke: #0284c7;
      stroke-width: 2;
      transition: transform 0.6s cubic-bezier(0.34, 1.3, 0.64, 1), opacity 0.6s ease;
      cursor: pointer;
    }

    .tri-b {
      fill: #a78bfa;
      fill-opacity: 0.85;
      stroke: #7c3aed;
      stroke-width: 2;
      transition: transform 0.6s cubic-bezier(0.34, 1.3, 0.64, 1), opacity 0.6s ease;
      cursor: pointer;
    }

    .diag-line {
      stroke: #0f172a;
      stroke-width: 2;
      stroke-dasharray: 4 4;
      transition: opacity 0.3s ease;
    }

    .dim-line {
      stroke: #64748b;
      stroke-width: 1.5;
    }

    .dim-text {
      font-size: 14px;
      fill: #475569;
      font-weight: 600;
      text-anchor: middle;
      dominant-baseline: middle;
    }

    .right-angle {
      fill: none;
      stroke: #0284c7;
      stroke-width: 1.5;
    }

    .formula-panel {
      min-height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 18px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      padding: 6px 20px;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.04);
    }
  </style>
</head>
<body>

  
  <div class="nav-tabs">
    <button class="tab-btn active" id="btn-rect" onclick="setStep(0)">Rectangle</button>
    <button class="tab-btn" id="btn-split" onclick="setStep(1)">2 Halves</button>
    <button class="tab-btn" id="btn-tri" onclick="setStep(2)">1 Triangle</button>
  </div>

  
  <div class="stage-container">
    <svg viewBox="0 0 320 200">
      <defs>
        <pattern id="grid" width="20" height="20" patternUnits="userSpaceOnUse">
          <path d="M 20 0 L 0 0 0 20" fill="none" stroke="#f1f5f9" stroke-width="1"/>
        </pattern>
      </defs>

      
      <rect x="40" y="20" width="240" height="140" fill="url(#grid)"/>

      
      <g id="grp-b" transform="translate(0, 0)">
        <polygon class="tri-b" points="40,20 280,20 280,160"/>
        <path d="M 266,20 L 266,34 L 280,34" fill="none" stroke="#7c3aed" stroke-width="1.5"/>
      </g>

      
      <g id="grp-a" transform="translate(0, 0)">
        <polygon class="tri-a" points="40,20 40,160 280,160"/>
        
        <path class="right-angle" d="M 40,146 L 54,146 L 54,160"/>
      </g>

      
      <line id="diag" class="diag-line" x1="40" y1="20" x2="280" y2="160" opacity="0"/>

      
      <g id="dim-base">
        <line class="dim-line" x1="40" y1="175" x2="280" y2="175"/>
        <path d="M 40,171 L 40,179 M 280,171 L 280,179" stroke="#64748b" stroke-width="1.5"/>
        <text class="dim-text" x="160" y="190">base (b)</text>
      </g>

      
      <g id="dim-height">
        <line class="dim-line" x1="25" y1="20" x2="25" y2="160"/>
        <path d="M 21,20 L 29,20 M 21,160 L 29,160" stroke="#64748b" stroke-width="1.5"/>
        <text class="dim-text" x="12" y="90" transform="rotate(-90 12 90)">height (h)</text>
      </g>
    </svg>
  </div>

  
  <div class="formula-panel" id="formula"></div>

  <script>
    const formulas = [
      "\\text{Area} = b \\times h",
      "\\text{2 Triangles} = b \\times h",
      "\\text{Area} = \\frac{1}{2} \\times b \\times h"
    ];

    let currentStep = 0;

    function renderMath(index) {
      const el = document.getElementById('formula');
      katex.render(formulas[index], el, {
        throwOnError: false,
        displayMode: false
      });
    }

    function setStep(step) {
      currentStep = step;

      // Update button styles
      document.getElementById('btn-rect').classList.toggle('active', step === 0);
      document.getElementById('btn-split').classList.toggle('active', step === 1);
      document.getElementById('btn-tri').classList.toggle('active', step === 2);

      const grpA = document.getElementById('grp-a');
      const grpB = document.getElementById('grp-b');
      const diag = document.getElementById('diag');

      if (step === 0) {
        // Whole Rectangle
        grpA.style.transform = "translate(0px, 0px)";
        grpB.style.transform = "translate(0px, 0px)";
        grpB.style.opacity = "0.85";
        diag.style.opacity = "0";
      } else if (step === 1) {
        // Separated / 2 Halves
        grpA.style.transform = "translate(-14px, 12px)";
        grpB.style.transform = "translate(14px, -12px)";
        grpB.style.opacity = "0.85";
        diag.style.opacity = "1";
      } else if (step === 2) {
        // Just 1 Triangle (half)
        grpA.style.transform = "translate(0px, 0px)";
        grpB.style.transform = "translate(24px, -20px)";
        grpB.style.opacity = "0.08";
        diag.style.opacity = "0";
      }

      renderMath(step);
    }

    // Auto-cycle through steps calmly
    let timer = setInterval(() => {
      setStep((currentStep + 1) % 3);
    }, 3800);

    // Pause cycling on user interaction
    ['btn-rect', 'btn-split', 'btn-tri', 'grp-a', 'grp-b'].forEach(id => {
      const el = document.getElementById(id);
      if (el) {
        el.addEventListener('click', () => clearInterval(timer));
        el.addEventListener('touchstart', () => clearInterval(timer), { passive: true });
      }
    });

    // Triangle click toggle
    document.getElementById('grp-a').addEventListener('click', () => {
      setStep((currentStep + 1) % 3);
    });
    document.getElementById('grp-b').addEventListener('click', () => {
      setStep((currentStep + 1) % 3);
    });

    // Initial render
    setStep(0);
  </script>
</body>
</html>

Since two identical right triangles make the whole rectangle, one triangle must be exactly **half** of that rectangle.

That is why the area formula for a right triangle is $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.

<!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">
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      gap: 10px;
      user-select: none;
    }

    .stepper {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 420px;
    }
    .step-btn {
      flex: 1;
      padding: 8px 4px;
      font-size: 13px;
      font-weight: 600;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      border-radius: 8px;
      cursor: pointer;
      text-align: center;
      transition: all 0.25s ease;
    }
    .step-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.3);
    }

    .canvas-container {
      position: relative;
      width: 100%;
      max-width: 420px;
      height: 240px;
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 12px;
      overflow: hidden;
      display: flex;
      align-items: center;
      justify-content: center;
    }

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

    .grid-line {
      stroke: #f1f5f9;
      stroke-width: 1.5;
    }

    .rect-ghost {
      fill: rgba(34, 183, 255, 0.08);
      stroke: #22b7ff;
      stroke-width: 2;
      stroke-dasharray: 4 4;
      transition: all 0.5s ease;
    }

    .tri-primary {
      fill: rgba(34, 183, 255, 0.35);
      stroke: #0284c7;
      stroke-width: 2.5;
      transition: all 0.5s ease;
    }

    .tri-secondary {
      fill: rgba(245, 158, 11, 0.2);
      stroke: #d97706;
      stroke-width: 2;
      stroke-dasharray: 4 4;
      transition: all 0.5s cubic-bezier(0.4, 0, 0.2, 1);
      transform-origin: center;
    }

    .dimension-label {
      font-size: 13px;
      font-weight: 700;
      fill: #1e2945;
      text-anchor: middle;
      dominant-baseline: middle;
    }

    .right-angle {
      stroke: #0284c7;
      stroke-width: 1.5;
      fill: none;
    }

    .formula-card {
      width: 100%;
      max-width: 420px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px 12px;
      display: flex;
      align-items: center;
      justify-content: space-around;
      min-height: 48px;
    }

    .formula-math {
      font-size: 16px;
      color: #0f172a;
    }

    .area-badge {
      font-size: 14px;
      font-weight: 700;
      color: #0284c7;
      background: #f0f9ff;
      padding: 4px 10px;
      border-radius: 6px;
      border: 1px solid #bae6fd;
    }
  </style>
</head>
<body>

  <div class="stepper">
    <button class="step-btn active" id="btn-0" onclick="setStep(0)">1. Rectangle</button>
    <button class="step-btn" id="btn-1" onclick="setStep(1)">2. Halve</button>
    <button class="step-btn" id="btn-2" onclick="setStep(2)">3. Triangle</button>
  </div>

  <div class="canvas-container">
    <svg id="scene" viewBox="0 0 360 220">
      
      <g id="grid"></g>

      
      <g id="shapes">
        
        <polygon id="tri-comp" class="tri-secondary" points="50,40 290,40 290,180" />
        
        
        <polygon id="tri-main" class="tri-primary" points="50,180 290,180 290,40" />

        
        <path class="right-angle" d="M 276,180 L 276,166 L 290,166" />
      </g>

      
      <text x="170" y="202" class="dimension-label">base (b)</text>
      <text x="316" y="110" class="dimension-label">height (h)</text>
    </svg>
  </div>

  <div class="formula-card">
    <div id="math-exp" class="formula-math"></div>
    <div id="badge-val" class="area-badge">Area = b × h</div>
  </div>

  <script>
    // Draw background grid
    const gridG = document.getElementById('grid');
    for (let x = 50; x <= 290; x += 40) {
      gridG.innerHTML += `<line x1="${x}" y1="40" x2="${x}" y2="180" class="grid-line"/>`;
    }
    for (let y = 40; y <= 180; y += 35) {
      gridG.innerHTML += `<line x1="50" y1="${y}" x2="290" y2="${y}" class="grid-line"/>`;
    }

    let currentStep = 0;
    const formulas = [
      "\\text{Area} = b \\times h",
      "\\text{Area} = \\frac{b \\times h}{2}",
      "\\text{Area} = \\frac{1}{2} \\times b \\times h"
    ];
    const badges = [
      "Full Rectangle",
      "2 Equal Halves",
      "1 Right Triangle"
    ];

    function updateView() {
      // Button states
      for (let i = 0; i < 3; i++) {
        document.getElementById(`btn-${i}`).classList.toggle('active', i === currentStep);
      }

      // Render math
      katex.render(formulas[currentStep], document.getElementById('math-exp'), {
        throwOnError: false
      });
      document.getElementById('badge-val').textContent = badges[currentStep];

      // Shapes animation
      const triComp = document.getElementById('tri-comp');
      const triMain = document.getElementById('tri-main');

      if (currentStep === 0) {
        triComp.style.opacity = "1";
        triComp.style.transform = "translate(0px, 0px)";
        triComp.style.fill = "rgba(34, 183, 255, 0.35)";
        triComp.style.stroke = "#0284c7";
        triComp.style.strokeDasharray = "none";
        triMain.style.fill = "rgba(34, 183, 255, 0.35)";
      } else if (currentStep === 1) {
        triComp.style.opacity = "0.75";
        triComp.style.transform = "translate(-12px, -10px)";
        triComp.style.fill = "rgba(245, 158, 11, 0.25)";
        triComp.style.stroke = "#d97706";
        triComp.style.strokeDasharray = "4 4";
        triMain.style.fill = "rgba(34, 183, 255, 0.35)";
      } else {
        triComp.style.opacity = "0";
        triComp.style.transform = "translate(-24px, -20px)";
        triMain.style.fill = "rgba(34, 183, 255, 0.5)";
      }
    }

    function setStep(step) {
      currentStep = step;
      updateView();
    }

    // Auto-cycle gently
    let autoInterval = setInterval(() => {
      currentStep = (currentStep + 1) % 3;
      updateView();
    }, 4000);

    // Pause auto on user touch/click
    document.body.addEventListener('pointerdown', () => {
      clearInterval(autoInterval);
    }, { once: true });

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

What does this look like with real measurements?

Imagine a triangle with a **base** of $6\text{ inches}$ and a **height** of $4\text{ inches}$.

First, find the full rectangle area: $6 \times 4 = 24\text{ square inches}$.

Next, take half of $24$: $\frac{1}{2} \times 24 = 12\text{ square inches}$.

<!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">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <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;
      padding: 12px 16px;
      min-height: 100%;
      user-select: none;
    }

    .controls {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 380px;
      justify-content: center;
    }

    .step-btn {
      flex: 1;
      padding: 8px 10px;
      font-size: 14px;
      font-weight: 600;
      border: 1px solid #e6e6e6;
      border-radius: 8px;
      background: #ffffff;
      color: #6b7280;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
      transition: all 0.25s ease;
      text-align: center;
    }

    .step-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }

    .canvas-box {
      width: 100%;
      max-width: 360px;
      display: flex;
      flex-direction: column;
      align-items: center;
      position: relative;
    }

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

    .grid-line {
      stroke: #e2e8f0;
      stroke-width: 1;
    }

    .rect-ghost {
      stroke: #cbd5e1;
      stroke-width: 2;
      stroke-dasharray: 4 4;
      fill: none;
      transition: opacity 0.4s ease;
    }

    .tri-top {
      fill: #e0f2fe;
      stroke: #38bdf8;
      stroke-width: 2;
      transition: transform 0.5s ease, opacity 0.5s ease;
      transform-origin: 200px 140px;
    }

    .tri-main {
      fill: #22b7ff;
      fill-opacity: 0.25;
      stroke: #0284c7;
      stroke-width: 2.5;
      transition: fill 0.3s ease;
    }

    .dim-label {
      font-size: 14px;
      font-weight: 600;
      fill: #475569;
    }

    .math-bar {
      min-height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
      padding: 6px 16px;
      border-radius: 10px;
      background: #ffffff;
      border: 1px solid #e6e6e6;
      box-shadow: 0 1px 3px rgba(0,0,0,0.04);
      width: 100%;
      max-width: 360px;
      font-size: 17px;
    }
  </style>
</head>
<body>

  
  <div class="controls">
    <button class="step-btn active" onclick="setStep(0)">1. Rectangle</button>
    <button class="step-btn" onclick="setStep(1)">2. Halve</button>
    <button class="step-btn" onclick="setStep(2)">3. Triangle</button>
  </div>

  
  <div class="canvas-box">
    <svg viewBox="0 0 320 200">
      <defs>
        
        <pattern id="grid" width="40" height="40" patternUnits="userSpaceOnUse">
          <path d="M 40 0 L 0 0 0 40" fill="none" stroke="#f1f5f9" stroke-width="1.5"/>
        </pattern>
      </defs>

      
      <g transform="translate(45, 15)">
        <rect width="240" height="160" fill="url(#grid)" />
        <rect width="240" height="160" class="rect-ghost" id="rectGhost" opacity="0"/>

        
        <polygon id="triTop" class="tri-top" points="0,0 240,0 240,160" />

        
        <polygon id="triMain" class="tri-main" points="0,0 0,160 240,160" />

        
        <path d="M 0 144 L 16 144 L 16 160" fill="none" stroke="#0284c7" stroke-width="1.5" />

        
        <text x="120" y="182" text-anchor="middle" class="dim-label">6 in (base)</text>
        <text x="-12" y="85" text-anchor="middle" class="dim-label" transform="rotate(-90 -12 85)">4 in (height)</text>
      </g>
    </svg>
  </div>

  
  <div class="math-bar" id="mathOut"></div>

  <script>
    const steps = [
      {
        formula: "\\text{Area} = 6 \\times 4 = 24\\text{ in}^2",
        triTopTransform: "translate(0, 0)",
        triTopOpacity: 1,
        triTopFill: "rgba(34, 183, 255, 0.25)",
        ghostOpacity: 0
      },
      {
        formula: "\\frac{1}{2} \\times 24 = 12\\text{ in}^2",
        triTopTransform: "translate(22px, -18px)",
        triTopOpacity: 0.6,
        triTopFill: "#e0f2fe",
        ghostOpacity: 1
      },
      {
        formula: "\\text{Area} = \\frac{1}{2} \\times 6 \\times 4 = 12\\text{ in}^2",
        triTopTransform: "translate(50px, -40px)",
        triTopOpacity: 0,
        triTopFill: "#e0f2fe",
        ghostOpacity: 1
      }
    ];

    let currentStep = 0;
    const btns = document.querySelectorAll('.step-btn');
    const triTop = document.getElementById('triTop');
    const rectGhost = document.getElementById('rectGhost');
    const mathOut = document.getElementById('mathOut');

    function renderMath(latex) {
      if (window.katex) {
        katex.render(latex, mathOut, { throwOnError: false });
      } else {
        mathOut.textContent = latex;
      }
    }

    function setStep(index) {
      currentStep = index;
      btns.forEach((btn, idx) => {
        btn.classList.toggle('active', idx === index);
      });

      const s = steps[index];
      triTop.style.transform = s.triTopTransform;
      triTop.style.opacity = s.triTopOpacity;
      triTop.style.fill = s.triTopFill;
      rectGhost.style.opacity = s.ghostOpacity;
      renderMath(s.formula);
    }

    // Auto cycle
    let autoPlayTimer = setInterval(() => {
      setStep((currentStep + 1) % steps.length);
    }, 3800);

    btns.forEach(btn => {
      btn.addEventListener('click', () => {
        clearInterval(autoPlayTimer);
      });
    });

    // Initial render
    setStep(0);
  </script>
</body>
</html>

A common mistake is forgetting to take **half** at the very end.

If you only multiply base times height, you have found the area of the entire rectangle, not just the single triangle!

Next time you see any right triangle, picture its missing matching half to see the whole rectangle right away.

math

Discovering the Area of Right-Angled Triangles. Explain why the area of a right-angled triangle is half the area of its surrounding rectangle (A = 1/2 × b × h).

Intuition for the Area of a Triangle Formula

How to Calculate the Area of a Triangle

Area of Right Triangle Proof

To master the conceptual understanding of the area of a right-angled triangle under standard MNU 2-11c, instruction should follow a Concrete-Representational-Abstract (CRA) progression. Begin with physical paper folding and cutting activities where learners fold a rectangle along its diagonal to see that two identical triangles form one whole rectangle. Transition to drawing on squared dot grid paper (representational), counting whole and half squares to verify that area equals half the base times height. Finally, introduce the algebraic expression $A = \frac{1}{2} \times b \times h$ (abstract). Explicitly emphasize identifying the two sides framing the right angle to prevent errors with the hypotenuse.

Discovering the Area of Right-Angled Triangles

The Rectangle Secret

Step-by-Step Example