Heights & Distances

At a glance 11 formulas

1. 1Height h, distance d$h = d\tan\theta,\ \theta = \text{elevation}$$d = h\cot\theta,\ \text{also the shadow}$
2. 2Depression equals elevationThe angle of depression from A to B equals the angle of elevation from B to A (alternate angles), so a look down uses the same right triangle.
3. 3Handy angle factsAt $45^\circ$ the height equals the distance. Since $\tan 60^\circ = 3\tan 30^\circ$, an object is 3 times as far away when seen at $30^\circ$ as at $60^\circ$.
4. 4Walking towards it$h = \dfrac{d}{\cot\alpha - \cot\beta}$$d = \text{walk in};\ \text{elevation } \alpha \text{ rises to } \beta$
5. 5Opposite sides$h = \dfrac{d}{\cot\alpha + \cot\beta}$$d = \text{gap};\ \text{elevations } \alpha, \beta \text{ from either side}$
6. 6Complementary elevations$h = \sqrt{xy}$$\text{elevations } \theta,\ 90^\circ - \theta \text{ at distances } x, y$
7. 7Split at eye level$H = h\left(1 + \dfrac{\tan\alpha}{\tan\beta}\right)$$H = \text{tower},\ h = \text{eye level};\ \text{top } \alpha,\ \text{foot } \beta$
8. 8Pole on a roof$\text{pole} = d(\tan\beta - \tan\alpha)$$d = \text{distance};\ \text{roof at } \alpha,\ \text{tip at } \beta$
9. 9Steady approachFor an object moving straight at the foot at a steady speed, the distance left is $h\cot\theta$, so times are in the ratio of the drops in $\cot\theta$; $h$ cancels.
10. 10Cloud and its reflection$H = h\,\dfrac{\tan\beta + \tan\alpha}{\tan\beta - \tan\alpha}$$H = \text{cloud},\ h = \text{eye};\ \text{cloud } \alpha,\ \text{image } \beta$
11. 11South and eastPoints due south and due east of a tower are $h\cot\alpha$ and $h\cot\beta$ from its foot, so they are $h\sqrt{\cot^2\alpha + \cot^2\beta}$ apart (Pythagoras on the ground).