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Tutoring abilities

Imagine walking along the outer curved crust of a single giant pizza slice instead of walking all the way around the whole pizza.

The entire distance around a full circle is the **circumference**, while any curved portion of that circle's edge is called an **arc**.

<!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="1:1">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <script src="{{VENDOR_CDN}}/classace-diagram.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: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      gap: 10px;
    }
    .display-area {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      width: 100%;
      max-width: 380px;
      position: relative;
    }
    svg {
      width: 100%;
      max-width: 260px;
      height: auto;
      overflow: visible;
    }
    .math-panel {
      background: #ffffff;
      padding: 8px 16px;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      font-size: 17px;
      min-height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
      margin-top: 4px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.04);
    }
    .controls {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
      width: 100%;
      max-width: 380px;
    }
    .slider-row {
      display: flex;
      align-items: center;
      gap: 12px;
      width: 100%;
      padding: 0 8px;
    }
    input[type=range] {
      flex: 1;
      height: 6px;
      border-radius: 3px;
      background: #e2e8f0;
      outline: none;
      -webkit-appearance: none;
      accent-color: #22b7ff;
      cursor: pointer;
    }
    .preset-pills {
      display: flex;
      gap: 6px;
      flex-wrap: wrap;
      justify-content: center;
      width: 100%;
    }
    .pill {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      color: #1e2945;
      padding: 5px 12px;
      border-radius: 20px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.03);
      transition: background-color 0.2s, color 0.2s, border-color 0.2s;
    }
    .pill.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
    }
    .legend {
      display: flex;
      gap: 16px;
      font-size: 12px;
      font-weight: 600;
      margin-top: 2px;
    }
    .legend-item {
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .legend-color {
      width: 12px;
      height: 4px;
      border-radius: 2px;
    }
  </style>
</head>
<body>

  <div class="display-area">
    <svg viewBox="0 0 300 300" id="circleSvg">
      
      <circle cx="150" cy="150" r="100" fill="#f1f5f9" stroke="#cbd5e1" stroke-width="4" stroke-dasharray="4 4" />

      
      <path id="sectorPath" fill="rgba(34, 183, 255, 0.15)" stroke="none" />

      
      <path id="angleArc" fill="none" stroke="#22b7ff" stroke-width="2" />
      <text id="angleText" x="150" y="150" text-anchor="middle" font-size="14" font-weight="700" fill="#0284c7"></text>

      
      <line x1="150" y1="150" x2="250" y2="150" stroke="#94a3b8" stroke-width="2" />
      <line id="radius2" x1="150" y1="150" x2="250" y2="150" stroke="#94a3b8" stroke-width="2" />

      
      <circle cx="150" cy="150" r="4" fill="#64748b" />

      
      <path id="arcPath" fill="none" stroke="#22b7ff" stroke-width="6" stroke-linecap="round" />
    </svg>

    <div class="legend">
      <div class="legend-item">
        <div class="legend-color" style="background: #22b7ff; height: 6px;"></div>
        <span>Arc</span>
      </div>
      <div class="legend-item">
        <div class="legend-color" style="background: #cbd5e1; border-top: 2px dashed #94a3b8; height: 0;"></div>
        <span>Circumference (<span id="cKatex"></span>)</span>
      </div>
    </div>
  </div>

  <div class="math-panel" id="mathDisplay"></div>

  <div class="controls">
    <div class="slider-row">
      <input type="range" id="angleSlider" min="15" max="360" step="15" value="90">
    </div>
    <div class="preset-pills">
      <button class="pill" data-angle="60">60° (⅙)</button>
      <button class="pill active" data-angle="90">90° (¼)</button>
      <button class="pill" data-angle="180">180° (½)</button>
      <button class="pill" data-angle="270">270° (¾)</button>
      <button class="pill" data-angle="360">360° (1)</button>
    </div>
  </div>

  <script>
    const CX = 150;
    const CY = 150;
    const R = 100;

    const slider = document.getElementById('angleSlider');
    const pills = document.querySelectorAll('.pill');
    const mathDisplay = document.getElementById('mathDisplay');
    const cKatex = document.getElementById('cKatex');

    if (window.katex) {
      katex.render('C', cKatex);
    }

    function gcd(a, b) {
      return b === 0 ? a : gcd(b, a % b);
    }

    function updateDiagram(deg) {
      const rad = (deg * Math.PI) / 180;
      const endX = CX + R * Math.cos(-rad);
      const endY = CY + R * Math.sin(-rad);

      const largeArc = deg > 180 ? 1 : 0;

      // Sector Path
      let sectorD = '';
      if (deg >= 360) {
        sectorD = `M ${CX} ${CY - R} A ${R} ${R} 0 1 1 ${CX} ${CY + R} A ${R} ${R} 0 1 1 ${CX} ${CY - R} Z`;
      } else {
        sectorD = `M ${CX} ${CY} L ${CX + R} ${CY} A ${R} ${R} 0 ${largeArc} 0 ${endX} ${endY} Z`;
      }
      document.getElementById('sectorPath').setAttribute('d', sectorD);

      // Arc Path
      let arcD = '';
      if (deg >= 360) {
        arcD = `M ${CX + R} ${CY} A ${R} ${R} 0 1 0 ${CX - R} ${CY} A ${R} ${R} 0 1 0 ${CX + R} ${CY}`;
      } else {
        arcD = `M ${CX + R} ${CY} A ${R} ${R} 0 ${largeArc} 0 ${endX} ${endY}`;
      }
      document.getElementById('arcPath').setAttribute('d', arcD);

      // Radius 2
      const radius2 = document.getElementById('radius2');
      radius2.setAttribute('x2', endX);
      radius2.setAttribute('y2', endY);

      // Angle indicator
      const angleArc = document.getElementById('angleArc');
      const angleText = document.getElementById('angleText');
      const rAngle = 30;

      if (deg > 10 && deg < 360) {
        const ax = CX + rAngle * Math.cos(-rad);
        const ay = CY + rAngle * Math.sin(-rad);
        angleArc.setAttribute('d', `M ${CX + rAngle} ${CY} A ${rAngle} ${rAngle} 0 ${largeArc} 0 ${ax} ${ay}`);
        angleArc.style.display = 'block';

        const midRad = -rad / 2;
        const tx = CX + (rAngle + 18) * Math.cos(midRad);
        const ty = CY + (rAngle + 18) * Math.sin(midRad) + 5;
        angleText.setAttribute('x', tx);
        angleText.setAttribute('y', ty);
        angleText.textContent = `${deg}°`;
        angleText.style.display = 'block';
      } else if (deg === 360) {
        angleArc.setAttribute('d', `M ${CX + rAngle} ${CY} A ${rAngle} ${rAngle} 0 1 0 ${CX - rAngle} ${CY} A ${rAngle} ${rAngle} 0 1 0 ${CX + rAngle} ${CY}`);
        angleArc.style.display = 'block';
        angleText.setAttribute('x', CX);
        angleText.setAttribute('y', CY - rAngle - 8);
        angleText.textContent = '360°';
        angleText.style.display = 'block';
      } else {
        angleArc.style.display = 'none';
        angleText.style.display = 'none';
      }

      // Simplify fraction
      const g = gcd(deg, 360);
      const num = deg / g;
      const den = 360 / g;

      let fracLatex = '';
      if (den === 1) {
        fracLatex = 'C';
      } else if (num === 1) {
        fracLatex = `\\frac{1}{${den}} C`;
      } else {
        fracLatex = `\\frac{${num}}{${den}} C`;
      }

      const formulaLatex = `\\text{Arc} = \\frac{${deg}^\\circ}{360^\\circ} \\times C = ${fracLatex}`;
      
      if (window.katex) {
        katex.render(formulaLatex, mathDisplay, { throwOnError: false });
      } else {
        mathDisplay.textContent = `Arc = (${deg}° / 360°) × C = ${num}/${den} C`;
      }

      // Update pills
      pills.forEach(p => {
        if (parseInt(p.dataset.angle, 10) === deg) {
          p.classList.add('active');
        } else {
          p.classList.remove('active');
        }
      });
    }

    slider.addEventListener('input', (e) => {
      updateDiagram(parseInt(e.target.value, 10));
    });

    pills.forEach(p => {
      p.addEventListener('click', () => {
        const val = parseInt(p.dataset.angle, 10);
        slider.value = val;
        updateDiagram(val);
      });
    });

    // Initial render
    updateDiagram(90);
  </script>
</body>
</html>

How do we find the exact length of that single curved piece?

A full spin around the center of any circle makes a complete turn of **360 degrees** ($360^\circ$).

If our slice opens up by a **central angle** of $\theta$, that slice takes up the fraction $\frac{\theta}{360^\circ}$ of the entire circle.

Over 2,200 years ago, the Greek astronomer Eratosthenes used this exact ratio of angles to calculate the circumference of planet Earth.

Since the whole perimeter is $C = 2\pi r$, multiplying that total distance by our fraction gives the **arc length formula**:

$$\text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r$$

<!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, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
    }
    .diagram-grid {
      display: grid;
      grid-template-columns: 1fr 1.1fr;
      gap: 16px;
      width: 100%;
      max-width: 580px;
      align-items: center;
    }
    @media (max-width: 480px) {
      .diagram-grid {
        grid-template-columns: 1fr;
        gap: 12px;
      }
    }
    .viz-container {
      display: flex;
      justify-content: center;
      align-items: center;
      position: relative;
    }
    svg {
      width: 100%;
      max-width: 220px;
      height: auto;
      overflow: visible;
    }
    .math-panel {
      display: flex;
      flex-direction: column;
      gap: 10px;
      background: #ffffff;
      padding: 14px;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      border: 1px solid #e6e6e6;
    }
    .preset-row {
      display: flex;
      flex-wrap: wrap;
      gap: 6px;
    }
    .preset-btn {
      flex: 1;
      min-width: 44px;
      padding: 6px 4px;
      font-size: 13px;
      font-weight: 600;
      border: 1px solid #d1d5db;
      border-radius: 8px;
      background: #ffffff;
      color: #374151;
      cursor: pointer;
      text-align: center;
      transition: background 0.2s, border-color 0.2s, color 0.2s;
    }
    .preset-btn.active {
      background: #22b7ff;
      border-color: #0284c7;
      color: #ffffff;
    }
    .slider-wrap {
      display: flex;
      align-items: center;
      gap: 10px;
      font-size: 13px;
      color: #6b7280;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #0284c7;
      cursor: pointer;
    }
    .calc-box {
      background: #f0f9ff;
      border: 1px solid #bae6fd;
      border-radius: 8px;
      padding: 10px;
      display: flex;
      flex-direction: column;
      gap: 6px;
      text-align: center;
    }
    .formula-row {
      font-size: 16px;
      min-height: 28px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    .sub-formula {
      font-size: 14px;
      color: #0369a1;
      min-height: 24px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
  </style>
</head>
<body>

<div class="diagram-grid">
  
  <div class="viz-container">
    <svg viewBox="0 0 220 220" id="circleSvg">
      
      <circle cx="110" cy="110" r="85" fill="none" stroke="#e2e8f0" stroke-width="6" stroke-dasharray="4,4" />
      
      
      <path id="sectorPath" fill="rgba(34, 183, 255, 0.15)" stroke="none" />
      
      
      <line x1="110" y1="110" x2="110" y2="25" stroke="#94a3b8" stroke-width="2" />
      
      
      <line id="radiusArm" x1="110" y1="110" x2="110" y2="25" stroke="#94a3b8" stroke-width="2" />
      
      
      <path id="anglePath" fill="none" stroke="#0284c7" stroke-width="2" />
      
      
      <path id="arcPath" fill="none" stroke="#0284c7" stroke-width="6" stroke-linecap="round" />
      
      
      <circle cx="110" cy="110" r="3.5" fill="#1e2945" />

      
      <text x="116" y="70" font-size="13" font-style="italic" fill="#64748b" font-weight="600">r</text>

      
      <text id="arcLabel" font-size="12" font-weight="700" fill="#0284c7" text-anchor="middle">Arc</text>
    </svg>
  </div>

  
  <div class="math-panel">
    <div class="preset-row">
      <button class="preset-btn" data-angle="60">60°</button>
      <button class="preset-btn active" data-angle="90">90°</button>
      <button class="preset-btn" data-angle="120">120°</button>
      <button class="preset-btn" data-angle="180">180°</button>
      <button class="preset-btn" data-angle="270">270°</button>
    </div>

    <div class="slider-wrap">
      <span>θ:</span>
      <input type="range" id="angleSlider" min="15" max="345" step="15" value="90">
      <span id="angleValue" style="min-width: 35px; font-weight:600; color:#1e2945;">90°</span>
    </div>

    <div class="calc-box">
      <div class="formula-row" id="katexFraction"></div>
      <div class="sub-formula" id="katexResult"></div>
    </div>
  </div>
</div>

<script>
  const CX = 110, CY = 110, R = 85;

  function gcd(a, b) {
    return b === 0 ? a : gcd(b, a % b);
  }

  function getFraction(angle) {
    const divisor = gcd(angle, 360);
    return {
      num: angle / divisor,
      den: 360 / divisor
    };
  }

  function updateDiagram(angle) {
    const rad = (angle - 90) * (Math.PI / 180);
    const x2 = CX + R * Math.cos(rad);
    const y2 = CY + R * Math.sin(rad);
    const largeArc = angle > 180 ? 1 : 0;

    // 1. Sector fill
    const sectorD = `M ${CX} ${CY} L ${CX} ${CY - R} A ${R} ${R} 0 ${largeArc} 1 ${x2} ${y2} Z`;
    document.getElementById('sectorPath').setAttribute('d', sectorD);

    // 2. Radius Arm
    const radiusArm = document.getElementById('radiusArm');
    radiusArm.setAttribute('x2', x2);
    radiusArm.setAttribute('y2', y2);

    // 3. Highlighted Arc
    const arcD = `M ${CX} ${CY - R} A ${R} ${R} 0 ${largeArc} 1 ${x2} ${y2}`;
    document.getElementById('arcPath').setAttribute('d', arcD);

    // 4. Angle Arc (radius 24)
    const ar = 24;
    const ax2 = CX + ar * Math.cos(rad);
    const ay2 = CY + ar * Math.sin(rad);
    const angleD = `M ${CX} ${CY - ar} A ${ar} ${ar} 0 ${largeArc} 1 ${ax2} ${ay2}`;
    document.getElementById('anglePath').setAttribute('d', angleD);

    // 5. Arc text label position (at midpoint of arc, slightly outside)
    const midRad = ((angle / 2) - 90) * (Math.PI / 180);
    const labelR = R + 18;
    const lx = CX + labelR * Math.cos(midRad);
    const ly = CY + labelR * Math.sin(midRad) + 4;
    const arcLabel = document.getElementById('arcLabel');
    arcLabel.setAttribute('x', lx);
    arcLabel.setAttribute('y', ly);

    // 6. Update Math / KaTeX
    const frac = getFraction(angle);
    let fracStr = `\\frac{${frac.num}}{${frac.den}}`;
    if (frac.den === 1) fracStr = `${frac.num}`;

    if (window.katex) {
      katex.render(`\\text{Fraction} = \\frac{${angle}^\\circ}{360^\\circ} = ${fracStr}`, document.getElementById('katexFraction'), { throwOnError: false });
      katex.render(`\\text{Arc} = ${fracStr} \\times 2\\pi r`, document.getElementById('katexResult'), { throwOnError: false });
    } else {
      document.getElementById('katexFraction').textContent = `Fraction = ${angle}° / 360° = ${frac.num}/${frac.den}`;
      document.getElementById('katexResult').textContent = `Arc = ${frac.num}/${frac.den} × 2πr`;
    }

    // 7. Update slider & text
    document.getElementById('angleSlider').value = angle;
    document.getElementById('angleValue').textContent = angle + '°';

    // 8. Update active button
    document.querySelectorAll('.preset-btn').forEach(btn => {
      btn.classList.toggle('active', parseInt(btn.dataset.angle) === angle);
    });
  }

  // Event Listeners
  const slider = document.getElementById('angleSlider');
  slider.addEventListener('input', (e) => {
    updateDiagram(parseInt(e.target.value));
  });

  document.querySelectorAll('.preset-btn').forEach(btn => {
    btn.addEventListener('click', () => {
      const ang = parseInt(btn.dataset.angle);
      updateDiagram(ang);
    });
  });

  // Init
  updateDiagram(90);
</script>

</body>
</html>

Let's test this formula on a real problem with actual numbers.

Suppose a circle has a radius of $r = 6\text{ cm}$ and a central angle of $\theta = 60^\circ$.

First, find the angle fraction: $\frac{60^\circ}{360^\circ} = \frac{1}{6}$ of the full circle.

Next, multiply this fraction by the full circumference $2\pi(6) = 12\pi\text{ cm}$.

$$\text{Arc Length} = \frac{1}{6} \times 12\pi = 2\pi \approx 6.28\text{ cm}$$

<!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>Arc Length as a Fraction of Circumference</title>
  <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;
      -webkit-tap-highlight-color: transparent;
      user-select: none;
    }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
    }
    .diagram-grid {
      display: flex;
      flex-direction: row;
      align-items: center;
      justify-content: center;
      gap: 16px;
      width: 100%;
      max-width: 620px;
    }
    @media (max-width: 480px) {
      .diagram-grid {
        flex-direction: column;
        gap: 10px;
      }
    }
    .svg-container {
      position: relative;
      width: 220px;
      height: 220px;
      flex-shrink: 0;
    }
    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }
    .math-panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 12px 16px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      flex: 1;
      min-width: 200px;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.04);
    }
    .math-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
      font-size: 15px;
    }
    .highlight-box {
      background: #f0f9ff;
      border-left: 3px solid #0284c7;
      padding: 6px 10px;
      border-radius: 4px 8px 8px 4px;
    }
    .btn-group {
      display: flex;
      flex-wrap: wrap;
      gap: 6px;
      justify-content: center;
      width: 100%;
      max-width: 620px;
      margin-top: 12px;
    }
    .pill-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 6px 12px;
      border-radius: 20px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .pill-btn:hover {
      border-color: #0284c7;
      color: #0284c7;
    }
    .pill-btn.active {
      background: #0284c7;
      border-color: #0284c7;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.35);
    }
    .slider-wrap {
      width: 100%;
      max-width: 620px;
      margin-top: 10px;
      display: flex;
      align-items: center;
      gap: 12px;
      padding: 0 4px;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #0284c7;
      cursor: pointer;
      height: 6px;
    }
    .slider-label {
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
      min-width: 38px;
      text-align: right;
    }
  </style>
</head>
<body>

  <div class="diagram-grid">
    
    <div class="svg-container">
      <svg viewBox="0 0 260 260">
        
        <circle cx="130" cy="130" r="95" fill="none" stroke="#cbd5e1" stroke-width="4" stroke-dasharray="4 4" />
        
        
        <path id="sectorPath" fill="rgba(2, 132, 199, 0.12)" />

        
        <path id="angleArc" fill="none" stroke="#0284c7" stroke-width="2" />
        <text id="angleText" font-size="12" font-weight="700" fill="#0369a1" text-anchor="middle" dominant-baseline="central"></text>

        
        <line x1="130" y1="130" x2="225" y2="130" stroke="#64748b" stroke-width="2" />
        <line id="radiusEndLine" x1="130" y1="130" x2="225" y2="130" stroke="#64748b" stroke-width="2" />

        
        <path id="arcPath" fill="none" stroke="#0284c7" stroke-width="7" stroke-linecap="round" />

        
        <circle cx="130" cy="130" r="4.5" fill="#1e2945" />

        
        <text x="175" y="122" font-size="12" font-weight="600" fill="#475569" text-anchor="middle">r = 6 cm</text>
      </svg>
    </div>

    
    <div class="math-panel">
      <div class="math-row">
        <span id="texCircumference"></span>
      </div>
      <div class="math-row">
        <span id="texFraction"></span>
      </div>
      <div class="math-row highlight-box">
        <span id="texArc"></span>
      </div>
    </div>
  </div>

  
  <div class="slider-wrap">
    <span class="slider-label">Angle:</span>
    <input type="range" id="angleSlider" min="15" max="360" step="15" value="60">
    <span class="slider-label" id="angleValueDisplay">60°</span>
  </div>

  <div class="btn-group">
    <button class="pill-btn active" data-angle="60">60° (⅙)</button>
    <button class="pill-btn" data-angle="90">90° (¼)</button>
    <button class="pill-btn" data-angle="120">120° (⅓)</button>
    <button class="pill-btn" data-angle="180">180° (½)</button>
    <button class="pill-btn" data-angle="360">360° (1)</button>
  </div>

  <script>
    const CX = 130;
    const CY = 130;
    const R = 95;
    const r_val = 6;
    const circumference = 2 * Math.PI * r_val; // 12 * pi

    const slider = document.getElementById('angleSlider');
    const angleDisplay = document.getElementById('angleValueDisplay');
    const arcPath = document.getElementById('arcPath');
    const sectorPath = document.getElementById('sectorPath');
    const angleArc = document.getElementById('angleArc');
    const angleText = document.getElementById('angleText');
    const radiusEndLine = document.getElementById('radiusEndLine');
    const buttons = document.querySelectorAll('.pill-btn');

    function gcd(a, b) {
      return b === 0 ? a : gcd(b, a % b);
    }

    function renderMath(angle) {
      const g = gcd(angle, 360);
      const num = angle / g;
      const den = 360 / g;

      const fracTex = den === 1 ? '1' : `\\frac{${num}}{${den}}`;
      const arcExactCoeff = (num * 12) / den;
      const arcLengthVal = (angle / 360) * 12 * Math.PI;

      let exactArcTex = '';
      if (Number.isInteger(arcExactCoeff)) {
        exactArcTex = arcExactCoeff === 1 ? '\\pi' : (arcExactCoeff === 0 ? '0' : `${arcExactCoeff}\\pi`);
      } else {
        const agcd = gcd(num * 12, den);
        exactArcTex = `\\frac{${(num * 12) / agcd}\\pi}{${den / agcd}}`;
      }

      if (window.katex) {
        katex.render(`C = 2\\pi(6) = 12\\pi \\approx 37.70\\text{ cm}`, document.getElementById('texCircumference'), { throwOnError: false });
        katex.render(`\\text{Fraction} = \\frac{${angle}^\\circ}{360^\\circ} = ${fracTex}`, document.getElementById('texFraction'), { throwOnError: false });
        katex.render(`\\text{Arc} = ${fracTex} \\times 12\\pi = ${exactArcTex} \\approx ${arcLengthVal.toFixed(2)}\\text{ cm}`, document.getElementById('texArc'), { throwOnError: false });
      } else {
        document.getElementById('texCircumference').innerText = `C = 12π ≈ 37.7 cm`;
        document.getElementById('texFraction').innerText = `Fraction = ${angle}° / 360° = ${num}/${den}`;
        document.getElementById('texArc').innerText = `Arc = ${num}/${den} × 12π ≈ ${arcLengthVal.toFixed(2)} cm`;
      }
    }

    function updateVisual(angle) {
      const rad = (angle * Math.PI) / 180;
      const endX = CX + R * Math.cos(-rad);
      const endY = CY + R * Math.sin(-rad);

      radiusEndLine.setAttribute('x2', endX);
      radiusEndLine.setAttribute('y2', endY);

      if (angle >= 360) {
        arcPath.setAttribute('d', `M ${CX + R} ${CY} A ${R} ${R} 0 1 0 ${CX + R - 0.01} ${CY}`);
        sectorPath.setAttribute('d', `M ${CX} ${CY} L ${CX + R} ${CY} A ${R} ${R} 0 1 0 ${CX + R - 0.01} ${CY} Z`);
      } else {
        const largeArc = angle > 180 ? 1 : 0;
        const dArc = `M ${CX + R} ${CY} A ${R} ${R} 0 ${largeArc} 0 ${endX} ${endY}`;
        arcPath.setAttribute('d', dArc);
        sectorPath.setAttribute('d', `M ${CX} ${CY} L ${CX + R} ${CY} A ${R} ${R} 0 ${largeArc} 0 ${endX} ${endY} Z`);
      }

      // Small angle indicator arc
      const rAngle = 32;
      const ax = CX + rAngle * Math.cos(-rad);
      const ay = CY + rAngle * Math.sin(-rad);
      const largeAngle = angle > 180 ? 1 : 0;

      if (angle >= 360) {
        angleArc.setAttribute('d', `M ${CX + rAngle} ${CY} A ${rAngle} ${rAngle} 0 1 0 ${CX + rAngle - 0.01} ${CY}`);
      } else {
        angleArc.setAttribute('d', `M ${CX + rAngle} ${CY} A ${rAngle} ${rAngle} 0 ${largeAngle} 0 ${ax} ${ay}`);
      }

      // Angle label position
      const midRad = rad / 2;
      const labelDist = Math.max(rAngle + 16, 46);
      angleText.setAttribute('x', CX + labelDist * Math.cos(-midRad));
      angleText.setAttribute('y', CY + labelDist * Math.sin(-midRad));
      angleText.textContent = `${angle}°`;

      angleDisplay.textContent = `${angle}°`;
      slider.value = angle;

      buttons.forEach(b => {
        if (parseInt(b.dataset.angle) === angle) {
          b.classList.add('active');
        } else {
          b.classList.remove('active');
        }
      });

      renderMath(angle);
    }

    slider.addEventListener('input', (e) => {
      updateVisual(parseInt(e.target.value));
    });

    buttons.forEach(b => {
      b.addEventListener('click', () => {
        const val = parseInt(b.dataset.angle);
        updateVisual(val);
      });
    });

    // Initialize at 60°
    updateVisual(60);
  </script>
</body>
</html>

What simple mistake trips up most people when calculating arc lengths?

A common mistake is using the area formula $\pi r^2$ instead of the circumference formula $2\pi r$.

Remember: arc length is a **one-dimensional distance** along the border, so you always multiply your angle fraction by the outer boundary length $2\pi r$.

Whenever you see a circular slice, always ask yourself first: what fraction of $360^\circ$ is this angle?

math

Understanding Arc Length as a Fraction of Circumference. Calculate arc length using the ratio of the central angle to 360 degrees.

Arc Length from Subtended Angle

Arcs and Sectors - Fourth Level & National 5 Maths

Arc Length Video & Practice Questions

Curricular Level Clarification: Standard MTH 4-16b corresponds to Fourth Level Mathematics in Scotland's Curriculum for Excellence (typically secondary S1–S3, ages 11–14). To build conceptual understanding before relying purely on algebraic formulas, begin with intuitive fraction benchmarks (e.g., $180^\circ = \frac{1}{2}$, $90^\circ = \frac{1}{4}$, $120^\circ = \frac{1}{3}$) before generalizing to arbitrary angle ratios $\frac{\theta}{360^\circ}$. Emphasize the distinction between linear distance (length in cm/m) and angular measure (degrees).

Understanding Arc Length as a Fraction of Circumference

The Fraction of a Full Turn

Step-by-Step Example