About Height and Distance
These questions are about a right angled triangle you cannot measure directly. The height of a tower and the distance along the ground are the two shorter sides, and the angle of elevation links them through the tangent. Almost every question is one tangent ratio away.
What you need to understand
- The angle of elevation is measured upward from the horizontal; the angle of depression is measured downward.
- The angle of elevation equals the angle of depression between the same two points, because they are alternate angles.
- Tangent of the angle equals the height divided by the distance along the ground.
- tan 45 is 1, tan 30 is 1 divided by the square root of 3, and tan 60 is the square root of 3.
- At the same time of day the sun makes the same angle with every upright object, so a pole and a tower are similar triangles and their heights are in the ratio of their shadows.
- Two observations at different distances from the same object give two equations, which can be subtracted to remove the unknown distance.
Formulas to remember
How to work through these questions
- Draw the triangle and mark which side is the height and which is the distance along the ground.
- Choose the ratio that involves the two sides you have and the side you want.
- If the question gives a distance as a multiple of the square root of 3, expect the root to cancel and the answer to be whole.
- For two observations, write one tangent equation per observation and subtract one from the other.
- Check the answer is realistic, because a tower is usually taller than the distance you stand from it only when the angle exceeds 45 degrees.
Mistakes that cost marks
- Using the sine or cosine where the tangent is needed.
- Dividing where the ratio calls for multiplying, which swaps the height and the distance.
- Mixing up the two complementary angles, 30 and 60, which changes the answer by a factor of three.
- Forgetting that the angle of elevation is measured from the horizontal and not from the vertical.
- Applying the shadow ratio upside down, so that the taller object appears to have the shorter shadow.
Worked example
From a point 30 metres from the foot of a tower the angle of elevation of the top is 45 degrees. Find the height of the tower.
- The height and the distance along the ground are the two shorter sides, so use the tangent.
- tan of the angle of elevation = height divided by distance.
- tan 45 = 1, so height = distance x 1 = 30 metres.
- Check: an angle of 45 degrees means the height equals the distance, so both must be 30 metres.
Answer: 30 metres
Practice questions with answers
A few Height and Distance questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
At a certain time of day a pole of height 22 m has a shadow of 11 m. What is the height of a pole whose shadow at that same time is 66 m?
Answer: Option A — with explanation
The sun is at the same angle for both objects, so the two triangles are similar and the heights are in the same ratio as the shadows.
Height / shadow is 22/11 = 2 for the first pole.
So the second height is 2 x 66 = 132 m.
Common mistakes
- repeated the height already given: 22
- inverted the proportion: 3.67
- added the difference in the shadow lengths to the first height: 77
Question 2
The angle of elevation of the top of a tower from a point on the ground is 45 degrees. If the point is 89 m from the foot of the tower, find the height of the tower.
-
A
178
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B
133.5
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C
44.5
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D
89
Answer: Option D — with explanation
The height and the distance along the ground are the two shorter sides of a right angled triangle, so the tangent of the elevation angle links them.
tan 45 degrees = 1, and tan = height / distance, so height = distance x 1 = 89 m.
Common mistakes
- halved the distance when the tangent is one: 44.5
- doubled the distance when the tangent is one: 178
- used three halves in place of the tangent of 45 degrees: 133.5
Question 3
A surveyor stands $28\sqrt{3}$ m from the foot of a tower and measures the angle of elevation of its top as 30 degrees. Find the height of the tower.
Answer: Option D — with explanation
The tangent of the angle of elevation equals the height divided by the distance along the ground.
$\tan 30^\circ = \dfrac{1}{\sqrt{3}}$, so the height is $28\sqrt{3}$ divided by $\sqrt{3}$, which is 28 m.
Writing the distance as a multiple of the square root of three is what keeps the answer whole.
Common mistakes
- used the tangent of 60 degrees instead of 30 degrees: 84
- divided by three instead of by the square root of three: 9.33
- took the height as one and a half times the factor in the distance: 42
Question 4
The angle of elevation of the top of a tower from a point A on the ground is 30 degrees. On walking $26\sqrt{3}$ m towards the tower the angle of elevation becomes 60 degrees. Find the height of the tower.
Answer: Option D — with explanation
Let the height be h and the shorter distance along the ground be d.
$\tan 60^\circ = \dfrac{h}{d}$, so d is h divided by $\sqrt{3}$, and $\tan 30^\circ$ gives d + $26\sqrt{3}$ = h$\sqrt{3}$.
Subtracting, the distance walked equals $\dfrac{2h}{\sqrt{3}}$, so h is 39 m.
Common mistakes
- doubled the height: 78
- took the factor in the distance walked as the height: 26
- applied the halving a second time: 19.5
Question 5
A pole 17 m high casts a shadow 4 m long. At the same moment another pole casts a shadow 16 m long. Find the height of the second pole.
Answer: Option C — with explanation
The sun is at the same angle for both objects, so the two triangles are similar and the heights are in the same ratio as the shadows.
Height / shadow is 17/4 = 17/4 for the first pole.
So the second height is 17/4 x 16 = 68 m.
Common mistakes
- added the difference in the shadow lengths to the first height: 29
- inverted the proportion: 4.25
- repeated the height already given: 17
Frequently asked questions
Why is the tangent the ratio to use?
Because the tangent compares the side opposite the angle with the side next to it, and those are exactly the height and the ground distance in these problems.
What is the difference between the angle of elevation and the angle of depression?
The angle of elevation is measured upward from the horizontal when you look at something above you. The angle of depression is measured downward when you look at something below you. Between the same two points the two angles are equal.
Why does the answer sometimes come out as a whole number when a root appears?
Because the distance is usually given as a multiple of the square root of three. Dividing that by the square root of three removes the root entirely.
How do two angle observations help?
Each observation gives one equation linking the height and the horizontal distance. Subtracting one from the other removes the horizontal distance and leaves the height alone.
