Geometry and measures
Heights and distances
Use angles of elevation and depression with the standard angles to find a height or a distance you cannot measure directly.
You cannot hold a tape measure against the top of a tower, but you can stand a known distance away and measure the angle you have to look up through. That angle, measured up from the horizontal, is the angle of elevation. Measured down from the horizontal — from a cliff top to a boat, say — it is the angle of depression. Everything in this skill is one right-angled triangle and one of the three ratios.
Which ratio to reach for
Label the triangle first: the height you want is usually the opposite side, the distance along the ground is the adjacent, and a rope, ladder or line of sight is the hypotenuse. Opposite and adjacent together mean tangent. A hypotenuse in the pair means sine or cosine.
Depression equals elevation
The horizontal at the cliff top is parallel to the ground, and the line of sight crosses both. So the angle of depression from the top down to the boat is exactly the angle of elevation from the boat up to the top. Whenever a question gives you a depression, redraw it at the bottom as an elevation and the triangle stops looking strange.
Worked example
A kite is on a straight string 100 units long, at 30° to the ground. How high is the kite?
The string is the hypotenuse and the height is the opposite side.
Naming the sides before choosing a ratio is what stops the whole topic being guesswork.
Try it together
A tower stands on level ground. From a point 45 m from its base, the angle of elevation of the top is 30°. Take √3 = 1.73. Let us find the height.
The height is opposite the angle and the 45 is adjacent to it, so this is a tangent question.
1.tan 30° is 1 over √3, so the height is 45 ÷ √3. Multiplying top and bottom by √3 turns that into 45√3 ÷ 3. What is 45 ÷ 3?
Have a go
Have a go on your own: from a point 30 m from the base of a tower, the angle of elevation of its top is 45°. How tall is the tower, in metres?
Hint: tan 45° is 1.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.