Advanced

Tutor model

i62 💰:9.4

i52 💰:7.9

🏆 i67 💰:62.5

i70 💰:125.0

i68 💰:125.0

🏆 i66 💰:75.0

i65 💰:125.0

i63 💰:62.5

i61 💰:75.0

i60 💰:9.4

i60 💰:37.5

i58 💰:62.5

i57 💰:22.5

i57 💰:37.5

i56 💰:45.0

⚡ i55 💰:11.25

i55 💰:37.5

i54 💰:62.5

i52 💰:35.0

i51 💰:37.5

i49 💰:15.0

i49 💰:62.5

i48 💰:11.3

i47 💰:7.5

i44 💰:25.0

i43 💰:37.5

i42 💰:187.5

i40 💰:5.0

i39 💰:100.0

i38 💰:37.5

i37 💰:11.0

i36 💰:187.5

i34 💰:12.5

i33 💰:37.5

i30 💰:1.0

i29 💰:1.5

i28 💰:150.0

i23 💰:20.0

i20 💰:4.0

i18 💰:1.0

i16 💰:25.0

i13 💰:1.5

Keep session private

What would you like to learn?

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

i58 💰:18.8

i31 💰:11.0

i25 💰:31.3

Tutoring abilities

How can you measure the height of a giant skyscraper without climbing to the very top with a tape measure?

You can stand safely on the ground and use a special upward angle to find the height with math.

When you look straight ahead, your eyes make a flat line called the **horizontal line**.

The **angle of elevation** is the angle formed when you tilt your eyes upward from that flat line to look at a high object.

<!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;
      user-select: none;
      -webkit-user-select: none;
    }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      width: 100%;
      min-height: 100%;
      padding: 12px;
    }
    .diagram-wrap {
      width: 100%;
      max-width: 540px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }
    svg {
      width: 100%;
      height: auto;
      overflow: visible;
    }
    .controls {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      max-width: 480px;
      background: #ffffff;
      padding: 10px 16px;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.06);
      border: 1px solid #e6e6e6;
      gap: 12px;
    }
    .slider-label {
      font-size: 14px;
      font-weight: 600;
      color: #1e2945;
      white-space: nowrap;
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .badge {
      background: #eef8ff;
      color: #0284c7;
      padding: 2px 8px;
      border-radius: 6px;
      font-weight: 700;
      font-size: 15px;
      min-width: 46px;
      text-align: center;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #22b7ff;
      cursor: pointer;
      height: 6px;
    }
    .preset-btns {
      display: flex;
      gap: 6px;
    }
    .preset-btn {
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      border-radius: 6px;
      padding: 4px 10px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      transition: all 0.2s ease;
    }
    .preset-btn:hover, .preset-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
    }
  </style>
</head>
<body>
  <div class="diagram-wrap">
    <svg viewBox="0 0 500 320">
      
      <line x1="20" y1="280" x2="480" y2="280" stroke="#cbd5e1" stroke-width="3" stroke-linecap="round" />

      
      <polygon id="triangle-poly" points="80,240 420,240 420,100" fill="#22b7ff" fill-opacity="0.08" />

      
      <line x1="80" y1="240" x2="420" y2="240" stroke="#0284c7" stroke-width="2.5" stroke-dasharray="6,4" />
      <text x="250" y="258" text-anchor="middle" font-size="13" font-weight="600" fill="#0284c7">Horizontal Line</text>

      
      <line id="height-line" x1="420" y1="240" x2="420" y2="100" stroke="#64748b" stroke-width="2.5" stroke-dasharray="4,4" />
      
      
      <path id="right-angle" d="M 405 240 L 405 225 L 420 225" fill="none" stroke="#64748b" stroke-width="2" />

      
      <line id="sight-line" x1="80" y1="240" x2="420" y2="100" stroke="#ff9800" stroke-width="3.5" stroke-linecap="round" />
      
      
      <text id="sight-label" x="235" y="160" transform="rotate(-22, 235, 160)" text-anchor="middle" font-size="13" font-weight="700" fill="#d97706">Line of Sight</text>

      
      <path id="angle-arc" d="M 140 240 A 60 60 0 0 0 135 217" fill="none" stroke="#ff9800" stroke-width="3" />
      <text id="angle-text" x="155" y="232" font-size="15" font-weight="800" fill="#ea580c">22°</text>
      <text id="angle-name" x="120" y="272" font-size="12" font-weight="700" fill="#ea580c">Angle of Elevation (θ)</text>

      
      <g transform="translate(48, 205)">
        <text font-size="42" x="0" y="38">🧍</text>
        <circle cx="27" cy="10" r="4" fill="#0284c7" />
      </g>

      
      <g id="target-obj" transform="translate(400, 70)">
        <text font-size="38" x="-19" y="12" text-anchor="middle">🎈</text>
        <circle cx="0" cy="0" r="4" fill="#ff9800" />
      </g>
    </svg>

    <div class="controls">
      <div class="slider-label">
        <span>Angle θ:</span>
        <span id="angle-val" class="badge">22°</span>
      </div>
      <input type="range" id="angle-slider" min="15" max="50" value="22" step="1">
      <div class="preset-btns">
        <button class="preset-btn" onclick="setAngle(20)">20°</button>
        <button class="preset-btn" onclick="setAngle(35)">35°</button>
        <button class="preset-btn" onclick="setAngle(48)">48°</button>
      </div>
    </div>
  </div>

  <script>
    const slider = document.getElementById('angle-slider');
    const angleVal = document.getElementById('angle-val');
    const targetObj = document.getElementById('target-obj');
    const sightLine = document.getElementById('sight-line');
    const heightLine = document.getElementById('height-line');
    const trianglePoly = document.getElementById('triangle-poly');
    const sightLabel = document.getElementById('sight-label');
    const angleArc = document.getElementById('angle-arc');
    const angleText = document.getElementById('angle-text');
    const rightAngle = document.getElementById('right-angle');
    const presetBtns = document.querySelectorAll('.preset-btn');

    const ox = 80;  // Observer eye X
    const oy = 240; // Observer eye Y
    const tx = 420; // Target X

    function update(deg) {
      angleVal.textContent = deg + '°';
      slider.value = deg;

      const rad = (deg * Math.PI) / 180;
      const dx = tx - ox; // 340
      const dy = dx * Math.tan(rad);
      const ty = oy - dy;

      // Update balloon position
      targetObj.setAttribute('transform', `translate(${tx}, ${ty})`);

      // Update lines
      sightLine.setAttribute('x2', tx);
      sightLine.setAttribute('y2', ty);
      heightLine.setAttribute('y2', ty);
      trianglePoly.setAttribute('points', `${ox},${oy} ${tx},${oy} ${tx},${ty}`);

      // Update line of sight label
      const midX = (ox + tx) / 2;
      const midY = (oy + ty) / 2 - 12;
      sightLabel.setAttribute('x', midX);
      sightLabel.setAttribute('y', midY);
      sightLabel.setAttribute('transform', `rotate(${-deg}, ${midX}, ${midY})`);

      // Update Arc
      const arcR = 65;
      const arcEndX = ox + arcR * Math.cos(rad);
      const arcEndY = oy - arcR * Math.sin(rad);
      angleArc.setAttribute('d', `M ${ox + arcR} ${oy} A ${arcR} ${arcR} 0 0 0 ${arcEndX.toFixed(1)} ${arcEndY.toFixed(1)}`);

      // Position angle text in middle of arc
      const textR = 85;
      const midAngleRad = rad / 2;
      const textX = ox + textR * Math.cos(midAngleRad);
      const textY = oy - textR * Math.sin(midAngleRad) + 5;
      angleText.setAttribute('x', textX.toFixed(1));
      angleText.setAttribute('y', textY.toFixed(1));
      angleText.textContent = deg + '°';

      // Update active preset button highlight
      presetBtns.forEach(btn => {
        btn.classList.toggle('active', parseInt(btn.textContent) === parseInt(deg));
      });
    }

    function setAngle(deg) {
      update(deg);
    }

    slider.addEventListener('input', (e) => update(parseInt(e.target.value)));

    // Initial render
    update(22);
  </script>
</body>
</html>

How do we connect this upward angle to the height of the object?

Looking up creates a **right triangle**, which is a triangle with one square $90^\circ$ corner at the base of the object.

The **tangent ratio** compares the side opposite your angle to the side adjacent (next to) your angle: $$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$$

<!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>Angle of Elevation</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; }
    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: space-between;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }
    .diagram-area {
      width: 100%;
      max-width: 520px;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.06);
      padding: 8px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
    }
    svg {
      width: 100%;
      height: auto;
      overflow: visible;
    }
    .math-bar {
      margin-top: 6px;
      padding: 6px 12px;
      background: #f1f5f9;
      border-radius: 8px;
      font-size: 15px;
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 38px;
      width: 100%;
    }
    .controls {
      width: 100%;
      max-width: 520px;
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      margin-top: 10px;
    }
    .control-card {
      background: #ffffff;
      padding: 8px 12px;
      border-radius: 10px;
      box-shadow: 0 1px 4px rgba(0,0,0,0.05);
      display: flex;
      flex-direction: column;
      gap: 4px;
    }
    .control-label {
      font-size: 13px;
      font-weight: 600;
      color: #4b5563;
      display: flex;
      justify-content: space-between;
    }
    input[type="range"] {
      width: 100%;
      accent-color: #22b7ff;
      cursor: pointer;
    }
  </style>
</head>
<body>

  <div class="diagram-area">
    <svg viewBox="0 0 460 210" id="svgCanvas">
      
      <line x1="30" y1="185" x2="430" y2="185" stroke="#cbd5e1" stroke-width="2" stroke-dasharray="4,4" />

      
      <line id="adjLine" x1="70" y1="150" x2="370" y2="150" stroke="#64748b" stroke-width="2.5" stroke-dasharray="5,4" />

      
      <line id="oppLine" x1="370" y1="150" x2="370" y2="40" stroke="#10b981" stroke-width="3.5" stroke-linecap="round" />

      
      <line id="hypLine" x1="70" y1="150" x2="370" y2="40" stroke="#22b7ff" stroke-width="3" stroke-linecap="round" />

      
      <path id="rightAngle" d="M 352 150 L 352 132 L 370 132" fill="none" stroke="#94a3b8" stroke-width="2" />

      
      <path id="angleArc" fill="rgba(34, 183, 255, 0.18)" stroke="#22b7ff" stroke-width="2" />

      
      <text id="treeIcon" x="370" y="150" font-size="34" text-anchor="middle" dominant-baseline="auto">🌲</text>

      
      <text x="60" y="156" font-size="24" text-anchor="middle" dominant-baseline="middle">👁️</text>

      
      <text id="lblAdj" x="220" y="172" font-size="13" font-weight="600" fill="#475569" text-anchor="middle">Adjacent = 15 m</text>
      <text id="lblOpp" x="382" y="95" font-size="13" font-weight="600" fill="#059669" text-anchor="start">Opposite = 11 m</text>
      <text id="lblTheta" x="125" y="142" font-size="14" font-weight="700" fill="#0284c7" text-anchor="start">θ</text>
    </svg>

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

  <div class="controls">
    <div class="control-card">
      <div class="control-label">
        <span>Adjacent (d)</span>
        <span id="distVal">15 m</span>
      </div>
      <input type="range" id="distRange" min="8" max="22" value="15" step="1">
    </div>
    <div class="control-card">
      <div class="control-label">
        <span>Opposite (h)</span>
        <span id="heightVal">11 m</span>
      </div>
      <input type="range" id="heightRange" min="4" max="16" value="11" step="1">
    </div>
  </div>

  <script>
    const distRange = document.getElementById('distRange');
    const heightRange = document.getElementById('heightRange');
    const distVal = document.getElementById('distVal');
    const heightVal = document.getElementById('heightVal');
    const mathDisplay = document.getElementById('mathDisplay');

    const adjLine = document.getElementById('adjLine');
    const oppLine = document.getElementById('oppLine');
    const hypLine = document.getElementById('hypLine');
    const rightAngle = document.getElementById('rightAngle');
    const angleArc = document.getElementById('angleArc');
    const treeIcon = document.getElementById('treeIcon');
    const lblAdj = document.getElementById('lblAdj');
    const lblOpp = document.getElementById('lblOpp');
    const lblTheta = document.getElementById('lblTheta');

    const x0 = 70;
    const y0 = 150;
    const pxPerMeter = 14;

    function update() {
      const d = parseFloat(distRange.value);
      const h = parseFloat(heightRange.value);

      distVal.textContent = d + ' m';
      heightVal.textContent = h + ' m';

      const xTree = x0 + (d * pxPerMeter);
      const yTreeTop = y0 - (h * pxPerMeter);

      // Update lines
      adjLine.setAttribute('x2', xTree);
      oppLine.setAttribute('x1', xTree);
      oppLine.setAttribute('x2', xTree);
      oppLine.setAttribute('y1', y0);
      oppLine.setAttribute('y2', yTreeTop);

      hypLine.setAttribute('x2', xTree);
      hypLine.setAttribute('y2', yTreeTop);

      // Right angle marker
      rightAngle.setAttribute('d', `M ${xTree - 16} ${y0} L ${xTree - 16} ${y0 - 16} L ${xTree} ${y0 - 16}`);

      // Angle arc
      const rad = Math.atan2(h, d);
      const deg = (rad * 180 / Math.PI).toFixed(1);
      const arcR = Math.min(48, d * pxPerMeter * 0.4);
      const arcX = x0 + arcR * Math.cos(rad);
      const arcY = y0 - arcR * Math.sin(rad);
      angleArc.setAttribute('d', `M ${x0} ${y0} L ${x0 + arcR} ${y0} A ${arcR} ${arcR} 0 0 0 ${arcX} ${arcY} Z`);

      // Tree icon position and size
      treeIcon.setAttribute('x', xTree);
      treeIcon.setAttribute('y', y0 + 6);
      const scale = Math.max(0.7, Math.min(1.4, h / 10));
      treeIcon.setAttribute('transform', `scale(1, ${scale}) translate(0, ${y0 * (1 - scale) / scale})`);

      // Label positions
      lblAdj.setAttribute('x', (x0 + xTree) / 2);
      lblAdj.textContent = `Adjacent = ${d} m`;

      lblOpp.setAttribute('x', xTree + 10);
      lblOpp.setAttribute('y', (y0 + yTreeTop) / 2 + 4);
      lblOpp.textContent = `Opposite = ${h} m`;

      const thetaOffset = arcR + 12;
      lblTheta.setAttribute('x', x0 + thetaOffset * Math.cos(rad / 2));
      lblTheta.setAttribute('y', y0 - thetaOffset * Math.sin(rad / 2) + 4);
      lblTheta.textContent = `θ = ${deg}°`;

      // KaTeX math rendering
      const tex = `\\tan(\\theta) = \\frac{\\text{Opp}}{\\text{Adj}} = \\frac{${h}}{${d}} \\implies \\theta = \\tan^{-1}\\left(\\frac{${h}}{${d}}\\right) = ${deg}^\\circ`;
      if (window.katex) {
        katex.render(tex, mathDisplay, { throwOnError: false });
      } else {
        mathDisplay.textContent = `tan(θ) = ${h}/${d} → θ = ${deg}°`;
      }
    }

    distRange.addEventListener('input', update);
    heightRange.addEventListener('input', update);
    update();
  </script>
</body>
</html>

What happens when we measure a real tree in a park?

Imagine you stand $20\text{ m}$ away from a tree, and your eyes look up at an angle of $30^\circ$ to see the top.

First, find the height above your eye level by multiplying distance by tangent: $$h = 20 \times \tan(30^\circ) \approx 20 \times 0.577 = 11.54\text{ m}$$

If your eye height is $1.5\text{ m}$ above the ground, add it to find the **total height**: $$11.54\text{ m} + 1.5\text{ m} = 13.04\text{ m}$$

<!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>Angle of Elevation - Height Calculation</title>
  <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css">
  <script src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
    }
    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      padding: 12px;
      min-height: 100%;
    }
    .diagram-wrap {
      width: 100%;
      max-width: 500px;
      position: relative;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    /* Controls */
    .controls {
      display: flex;
      gap: 8px;
      margin-top: 10px;
      width: 100%;
      max-width: 500px;
      justify-content: center;
    }
    .step-btn {
      flex: 1;
      padding: 8px 6px;
      font-size: 13px;
      font-weight: 600;
      border: 1px solid #e6e6e6;
      background: #ffffff;
      color: #6b7280;
      border-radius: 8px;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
    }
    .step-btn.active {
      border-color: #22b7ff;
      background: #eef9ff;
      color: #0284c7;
      box-shadow: 0 0 0 2px rgba(34, 183, 255, 0.25);
    }
    /* Formula Card */
    .formula-bar {
      margin-top: 8px;
      width: 100%;
      max-width: 500px;
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 8px;
      padding: 8px 12px;
      text-align: center;
      font-size: 15px;
      min-height: 42px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    /* Dynamic Highlights */
    .hl-blue { stroke: #22b7ff; fill: rgba(34, 183, 255, 0.12); }
    .hl-orange { stroke: #f59e0b; fill: rgba(245, 158, 11, 0.15); }
    .hl-green { stroke: #10b981; fill: rgba(16, 185, 129, 0.15); }
    .dim-line { stroke-width: 2; stroke-linecap: round; }
    .guide-line { stroke-dasharray: 4 3; stroke-width: 1.5; }
    .faded { opacity: 0.25; transition: opacity 0.3s ease; }
    .active-part { opacity: 1; transition: opacity 0.3s ease; }
  </style>
</head>
<body>

  <div class="diagram-wrap">
    <svg viewBox="0 0 460 250">
      
      <line x1="10" y1="230" x2="450" y2="230" stroke="#94a3b8" stroke-width="2.5" stroke-linecap="round" />
      
      
      <text x="35" y="215" font-size="28" text-anchor="middle">🧍</text>
      <circle cx="35" cy="182" r="3" fill="#1e2945" />

      
      <line x1="35" y1="182" x2="330" y2="182" stroke="#64748b" class="guide-line" />
      
      
      <polygon id="triangle-poly" points="35,182 330,182 330,45" fill="rgba(34,183,255,0.08)" />

      
      <line x1="35" y1="182" x2="330" y2="45" stroke="#22b7ff" stroke-width="2.5" />
      
      
      <path d="M 75 182 A 40 40 0 0 0 71 165" fill="none" stroke="#22b7ff" stroke-width="2" />
      <text x="82" y="176" font-size="13" font-weight="bold" fill="#0284c7">30°</text>

      
      <path d="M 318 182 L 318 170 L 330 170" fill="none" stroke="#64748b" stroke-width="1.5" />

      
      <line x1="35" y1="238" x2="330" y2="238" stroke="#64748b" stroke-width="1.5" marker-start="url(#dot)" marker-end="url(#dot)" />
      <text x="182" y="248" font-size="12" font-weight="600" fill="#64748b" text-anchor="middle">20 m</text>

      
      <rect x="330" y="45" width="45" height="185" rx="3" fill="#cbd5e1" stroke="#64748b" stroke-width="2" />
      
      <rect x="340" y="60" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="356" y="60" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="340" y="85" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="356" y="85" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="340" y="110" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="356" y="110" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="340" y="135" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="356" y="135" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="340" y="160" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="356" y="160" width="10" height="12" rx="1" fill="#f8fafc" />
      <rect x="348" y="208" width="12" height="22" rx="1" fill="#475569" />

      
      <g id="grp-opp" class="active-part">
        <line x1="384" y1="45" x2="384" y2="182" stroke="#22b7ff" stroke-width="2.5" />
        <line x1="379" y1="45" x2="389" y2="45" stroke="#22b7ff" stroke-width="2" />
        <line x1="379" y1="182" x2="389" y2="182" stroke="#22b7ff" stroke-width="2" />
        <text x="393" y="118" font-size="12" font-weight="bold" fill="#0284c7">11.54 m</text>
      </g>

      
      <g id="grp-eye" class="active-part">
        <line x1="384" y1="182" x2="384" y2="230" stroke="#f59e0b" stroke-width="2.5" />
        <line x1="379" y1="230" x2="389" y2="230" stroke="#f59e0b" stroke-width="2" />
        <text x="393" y="210" font-size="12" font-weight="bold" fill="#d97706">1.5 m</text>
      </g>

      
      <g id="grp-total" class="faded">
        <line x1="440" y1="45" x2="440" y2="230" stroke="#10b981" stroke-width="2.5" />
        <line x1="435" y1="45" x2="445" y2="45" stroke="#10b981" stroke-width="2" />
        <line x1="435" y1="230" x2="445" y2="230" stroke="#10b981" stroke-width="2" />
        <text x="445" y="142" font-size="12" font-weight="bold" fill="#059669" transform="rotate(90, 445, 142)" text-anchor="middle">13.04 m</text>
      </g>
    </svg>
  </div>

  
  <div class="controls">
    <button class="step-btn active" onclick="setStep(0)">1. Trig Height</button>
    <button class="step-btn" onclick="setStep(1)">2. Eye Height</button>
    <button class="step-btn" onclick="setStep(2)">3. Total Height</button>
  </div>

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

  <script>
    const formulas = [
      "\\text{opp} = 20 \\times \\tan(30^\\circ) = 11.54\\text{ m}",
      "\\text{eye height} = 1.5\\text{ m}",
      "\\text{total} = 11.54\\text{ m} + 1.5\\text{ m} = 13.04\\text{ m}"
    ];

    const grpOpp = document.getElementById('grp-opp');
    const grpEye = document.getElementById('grp-eye');
    const grpTotal = document.getElementById('grp-total');
    const formulaEl = document.getElementById('formula');
    const buttons = document.querySelectorAll('.step-btn');

    function renderFormula(idx) {
      if (window.katex) {
        katex.render(formulas[idx], formulaEl, { throwOnError: false });
      } else {
        formulaEl.textContent = formulas[idx];
      }
    }

    function setStep(idx) {
      buttons.forEach((btn, i) => {
        btn.classList.toggle('active', i === idx);
      });

      if (idx === 0) {
        grpOpp.className.baseVal = 'active-part';
        grpEye.className.baseVal = 'faded';
        grpTotal.className.baseVal = 'faded';
      } else if (idx === 1) {
        grpOpp.className.baseVal = 'faded';
        grpEye.className.baseVal = 'active-part';
        grpTotal.className.baseVal = 'faded';
      } else {
        grpOpp.className.baseVal = 'active-part';
        grpEye.className.baseVal = 'active-part';
        grpTotal.className.baseVal = 'active-part';
      }

      renderFormula(idx);
    }

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

What is the most common mistake students make on height problems?

A common mistake is forgetting to add the observer's eye height at the very end.

The triangle only calculates the height starting from your eyes, so always add your eye height to get the true height from the ground.

What if you already know the building height, but need to find the angle instead?

If a tower is $50\text{ m}$ tall above eye level and you stand $50\text{ m}$ away, then $\tan(\theta) = \frac{50}{50} = 1$.

We use the **inverse tangent** (written $\tan^{-1}$) to find the angle: $\theta = \tan^{-1}(1) = 45^\circ$.

math

Calculating Angles of Elevation in Practical Contexts. Measure and calculate angles of elevation to find building, tree, or mountain heights.

National 5 / Fourth Level Mathematics: Trigonometry with Right-Angled Triangles

Angles of Elevation and Depression - Modeling with Right Triangles

Making and Using a Clinometer: Practical Applications of Trigonometry

Although the prompt header references Early Level (P1), the explicit curriculum code (MTH 4-16a) and trigonometry content correspond to Scottish CfE Fourth Level / National 4/5 (secondary stage). Effective pedagogy bridges practical fieldwork (constructing simple straw-and-protractor clinometers) with formal geometric modelling. Emphasize a three-stage routine: (1) draw and label the physical situation into a mathematical right-angled triangle, (2) select and compute the correct trigonometric ratio (primarily $\tan$), and (3) contextualize the result by accounting for observer height and rounding sensibly for real-world measurements.

Calculating Angles of Elevation in Practical Contexts

What Is an Angle of Elevation?

The Triangle Formula

Example: Measuring a Tall Tree