Measure without climbing
Draw the right triangle, then pick the ratio with the side you know and the side you want.
How tall is the Minar?
You cannot climb the Minar with a tape. But the student can find its height from the ground.
She knows three things: how far she stands from the Minar, the angle of elevation of the top, and her own height. Her line of sight AC and the horizontal AB make the right triangle ABC.
Which part of the Minar's height can the right triangle ABC give her?
What this lesson covers
The idea
A height or distance is found from a right triangle formed with the line of sight: choose the ratio of the known angle involving the known side and the required side, solve it, and add the observer's height if needed.
How tall is the Minar?
You cannot climb the Minar with a tape. But the student can find its height from the ground.
She knows three things: how far she stands from the Minar, the angle of elevation of the top, and her own height. Her line of sight AC and the horizontal AB make the right triangle ABC.
Which part of the Minar's height can the right triangle ABC give her?
- BC, the part above her eye level
- CD, the whole Minar
- AB, her distance from the Minar
Solve four situations
Each situation is drawn as a right triangle, to scale. Tap the ratio that contains the side you know and the side you want. Read each step, and watch for the moment you add the observer's height.
The method
CD = BC + BD, where BC comes from tan A = BC/AB (or cot A = AB/BC) and BD = AE is the observer's height.
To find a height or distance: 1. Draw the right triangle formed by the line of sight. 2. Mark the known angle, the known side and the required side. 3. Choose the ratio of the known angle that has the known side and the required side. 4. Solve it. 5. If the angle was measured from the observer's eye, add the observer's height.
Notes
Find a height or distance from the right triangle formed by the line of sight. Choose the ratio of the known angle that has the known side and the required side, solve it, and add the observer's height if needed.
Check yourself
A student knows the distance AB and the angle of elevation A. She wants the height BC. Which ratio should she use?
From a point P on the ground, the angle of elevation of the top of a 10 m tall building is 30°. How far is P from the foot of the building, in metres? (Take √3 = 1.732.)
Answer: 17.32
tan 30° = AB / AP, so 1/√3 = 10 / AP and AP = 10√3 = 10 × 1.732 = 17.32 m.
A girl 1.5 m tall stands 20 m from a pole. The angle of elevation of the top of the pole from her eyes is 45°. How tall is the pole, in metres?
Answer: 21.5
Above her eyes: tan 45° = 1 = h / 20, so h = 20 m. The pole is 20 + 1.5 = 21.5 m tall.
Why do we add 1.5 m to 20 m in the last question?
- sin A = BC/AC. sin A needs the line of sight AC, which she does not know.
- cos A = AB/AC. cos A does not have BC in it.
- tan A = BC/AB — correct. Yes! tan A has the side she knows (AB) and the side she wants (BC).
- The angle is measured from the ground.. The angle is measured from her eyes, which are 1.5 m above the ground.
- The right triangle starts at her eyes, so it gives only the part of the pole above them. — correct. Yes! The 20 m is the part above her eye level. Her eyes are 1.5 m above the ground, so we add 1.5 m.
- Every height problem needs 1.5 m added.. We add the observer's height only when the angle was measured from her eyes. Her height changes from person to person.