3D Pythagoras & Trigonometry Flashcards

All 6 cards in this deck

What is the formula for the space diagonal of a cuboid with edges $a$, $b$ and $c$?

e.g. a cuboid 2 cm by 3 cm by 6 cm

$d = \sqrt{a^2 + b^2 + c^2}$
$d = \sqrt{4 + 9 + 36} = 7$ cm

What is meant by the angle between a line and a plane?

The angle between the line and its projection (its 'shadow') on the plane.

What are the steps to find the angle between a space diagonal of a cuboid and the base?

e.g. base diagonal 8 cm, vertical height 6 cm

1. Draw the right-angled triangle made by the base diagonal, the vertical edge and the space diagonal.
2. Use tan with opposite = height, adjacent = base diagonal: $\tan\theta = \frac{6}{8}$.
3. $\theta = \tan^{-1}(0.75) = 36.9°$ (1 d.p.).

What is meant by the angle between two planes?

The angle between two lines, one drawn in each plane, that both meet the line of intersection at right angles.

True or false? In a square-based pyramid, the angle between a sloping face and the base equals the angle between a sloping edge and the base.

False. The face angle is measured at the midpoint of a base edge using the slant height, so it is a different (larger) angle.

In a 3D problem, what do you do when the triangle you need has no right angle?

Sketch that triangle on its own with its true lengths, then use the sine rule or cosine rule.