Trigonometry in the Real World – Measuring Heights With Shadows

Trigonometry is often introduced with triangles drawn on paper, but its real power comes when students take it outside. One of the simplest and most satisfying applications is using shadows to measure the height of tall objects — from trees and lampposts to buildings.

---

The Principle

When sunlight hits an object, it forms a right-angled triangle between the object, its shadow, and the line of sight to the top.
If students measure:

• The length of the shadow, and

• The angle of elevation from the tip of the shadow to the top of the object,
  they can use trigonometry to find the object’s height.

Using:

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ $$\text{height} = \text{shadow length} \times \tan(\theta)$$

---

Example Calculation

Object	Shadow Length (m)	Angle of Elevation (°)	Calculated Height (m)
Tree	5.0	35	3.5
Lamp Post	7.5	40	6.3
Building	10.0	45	10.0

Students can verify results by comparing with a tape measure or building data, reinforcing accuracy and proportional reasoning.

---

Skills Highlight

• Applying trigonometric ratios (tan, sin, cos) to real situations.

• Measuring and estimating angles with a clinometer or phone app.

• Recognising sources of error such as uneven ground or moving shadows.

• Relating maths to everyday objects and outdoor measurement.

---

Why It Works in Teaching

Taking trigonometry outdoors turns numbers into meaning. Students see how angles and ratios describe the real world. It transforms abstract formulas into a practical tool for problem-solving — and shows that maths genuinely measures the world around us.

Measuring Height Using Shadows – Trigonometry in Action

---

Layout Description

1. Scene Illustration (Top Section)

• A simple side-on diagram showing:

  • A tree or lamppost standing vertically on the left.

  • A horizontal ground line extending to the right.

  • A shadow cast along the ground (labelled shadow length = adjacent side).

  • A ray of sunlight coming down diagonally to the tip of the shadow, forming a right-angled triangle.

  • The angle of elevation (θ) marked between the shadow and the line to the top of the object.

Labels:

• Height of object (opposite side)

• Shadow length (adjacent side)

• Angle θ (angle of elevation)

• Sunlight direction

• Right angle marked between the height and the ground.

---

2. Formula Section (Middle)
Clearly display the key equation underneath the diagram:

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \quad \Rightarrow \quad \text{Height} = \text{Shadow Length} \times \tan(\theta)$$

Below this, show a quick worked example:

Example: Shadow = 7.5 m, Angle = 40°

Height = 7.5 × tan(40°) = 6.3 m

---

3. Quick Tips (Bottom Section)
Three short side boxes:

• Tip 1: Measure the shadow quickly before the sun moves.

• Tip 2: Use a clinometer or phone app to measure the angle.

• Tip 3: Keep the ground level for accurate results.