A tower on a building and a flag on a tower

Two triangles that share a base

A flagstaff stands on the roof of a building. From a point $P$ on the ground you look at the bottom of the flagstaff, which is the roof, and then at its top.

Both lines of sight start at $P$. Both end on the same vertical line. The two right triangles therefore share the same base, the distance $x$ from $P$ to the building.

Let the building be $b$ metres tall and the flagstaff $t$ metres long. The angle to the bottom is $\alpha$ and the angle to the top is $\beta$.

• Bottom of the flagstaff. $\tan\alpha = \dfrac{b}{x}$.
• Top of the flagstaff. $\tan\beta = \dfrac{b + t}{x}$.

The top is higher than the roof, so $\beta$ is larger than $\alpha$. The flagstaff is the difference of the two heights.

Example 1: a flagstaff on a building

A flagstaff stands on a building $24$ m tall. From a point on the ground, the angle of elevation of the bottom of the flagstaff is $30°$ and of its top is $45°$. How long is the flagstaff?

1. The distance. $\tan 30° = \dfrac{24}{x}$, so $x = 24\sqrt{3} \approx 41.57$ m.
2. The height of the top. $\tan 45° = \dfrac{24 + t}{x}$, so $24 + t = x = 41.57$ m.
3. Subtract the building. $t = 41.57 - 24 = 17.57$ m.

Check. $t = 24(\sqrt{3} - 1) = 24 \times 0.732 = 17.57$ m. It agrees.

Example 2: a statue on a pedestal

A statue $2.4$ m tall stands on a pedestal. From a point on the ground, the angle of elevation of the top of the pedestal is $45°$ and of the top of the statue is $60°$. How tall is the pedestal?

1. Name the height. Let the pedestal be $p$ metres. The distance to it is $x$.
2. The pedestal. $\tan 45° = \dfrac{p}{x}$, so $x = p$.
3. The statue's top. $\tan 60° = \dfrac{p + 2.4}{x}$, so $p + 2.4 = p\sqrt{3}$.
4. Solve. $p(\sqrt{3} - 1) = 2.4$, so $p = \dfrac{2.4}{0.732} \approx 3.28$ m.

The top of the statue is $3.28 + 2.4 = 5.68$ m above the ground.

Example 3: looking down and looking up

From the roof of a building $15$ m tall, the angle of depression of the foot of a tower is $60°$. The angle of elevation of the top of the tower is $30°$. How tall is the tower?

1. Depression becomes elevation. From the foot of the tower, the angle of elevation of the roof is also $60°$. These are alternate angles between parallel lines.
2. The distance. $\tan 60° = \dfrac{15}{x}$, so $x = \dfrac{15}{\sqrt{3}} \approx 8.66$ m.
3. The rise above the roof. $\tan 30° = \dfrac{r}{x}$, so $r = 8.66 \times \dfrac{1}{\sqrt{3}} = 5$ m.
4. The tower. It is $15 + 5 = 20$ m tall.

Here the horizontal line through the roof is the starting line for both angles. The tower stands at distance $x$ from the building.

Practise with new numbers

Try it in Python

The program finds the distance to a building, the height of the top of the flagstaff and the length of the flagstaff.

Common mistakes

• Giving the height of the top as the flagstaff. Subtract the building.
• Using the roof as the ground. The angle $\alpha$ and the angle $\beta$ are both measured at $P$ on the ground.
• Mixing up the two tangents. The larger angle belongs to the top, which is higher.
• Forgetting that depression equals elevation. Alternate angles are equal, so the angle of depression from the top is the angle of elevation from the bottom.

Remember these

• Both lines of sight end on one vertical line, so the two triangles share the base $x$.
• Bottom: $\tan\alpha = \dfrac{b}{x}$. Top: $\tan\beta = \dfrac{b + t}{x}$.
• The flagstaff is the top height minus the building height: $t = x(\tan\beta - \tan\alpha)$.
• Check by putting $x$ back into both tangents.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why do the two triangles in a flagstaff problem share the same base?

2. From a point $30$ m from a tower the angle of elevation of the top is $45°$. What is the height of the tower?

3. The top of a building is at $20$ m, and the top of a mast on it is at $32$ m above the ground. How long is the mast?

4. From a window the angle of depression of a car is $25°$. What is the angle of elevation of the window from the car?

5. In example 1, the top of the flagstaff is at $41.57$ m and the bottom at $24$ m. Why is the flagstaff not $41.57$ m long?