medium difficulty6 min

Shadow-based reasoning

Shadow-based reasoning uses the length of a shadow and the angle of elevation of the sun or light source to find a height or distance.

In one line: Shadow sits on the ground, so it is the adjacent side.

Practice This Concept

Understand it

When sunlight falls on a pole, tree, or building, the object, its shadow, and the sun ray form a right triangle.
The shadow becomes the horizontal side, and the height becomes the vertical side.
This makes tan the most useful ratio in many shadow questions.
These problems are very common because they connect trigonometry with everyday observation.

Example

A tree casts a 12 m shadow when the angle of elevation of the sun is 30 degrees. Then tan 30 = height/12, so the tree height can be found.

Formula and rule

On level ground, tan θ = height / shadow length when the light ray and ground form a right triangle.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1A flagpole casts a 9 m shadow and the sun's angle of elevation is 45 degrees.
2Since tan 45 = height/9, the height is 9 m.

Method you can reuse

1Draw the object and its shadow.
2Mark the sun angle.
3Identify height and shadow as opposite and adjacent.
4Use tan θ.
5Solve.
6Write the answer with units.

Sketch it in your answer

Draw the object vertical, the shadow horizontal on the ground, and the sun ray as the slant side.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In a shadow problem on level ground, which side usually represents the shadow length?

Think first, then show answer

The shadow length usually represents the horizontal side on the ground. That is why tan is often used to relate the shadow and the height.

Common wrong answer

A common wrong answer is to use the shadow as the hypotenuse. The hypotenuse is the sun ray, not the ground shadow.

Don't mix this up

Students sometimes use the shadow as the slant side instead of the horizontal side.

Exam tip

In shadow questions, always treat the shadow length as the ground side unless the question says the ground is sloping.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Find height from shadow length and sun angle, or find shadow length from height and sun angle.

1
Enough for 1 mark
Shadow-based reasoning uses the length of a shadow and the angle of elevation of the sun or light source to find a height or distance.
2
Add for 2 marks
Example: A tree casts a 12 m shadow when the angle of elevation of the sun is 30 degrees. Then tan 30 = height/12, so the tree height can be found.
3
Add for 3 marks
When sunlight falls on a pole, tree, or building, the object, its shadow, and the sun ray form a right triangle.

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