Pythagoras' Theorem
Senior 2 · Mathematics
Pythagoras' Theorem
Lesson 4 of 4
22 min
The reconstructed royal palace at the King's Palace Museum in Nyanza has a tall, pointed thatched roof. Treat such a roof as a cone: its rafters run from the edge of the roof to the tip. How long must they be? That length is the slant height of a cone, and Pythagoras' theorem finds it from the radius and the height. Slant heights are also needed for the surface area of pyramids and cones, and for turning a slant height back into the vertical height you need for volume (Unit 4).
In a right cone, the vertical height h, the radius r and the slant height l form a right-angled triangle, with the right angle at the centre of the base and l as the hypotenuse.
Cone: l2 = r2 + h2
Worked example 1
A cone has radius 5 cm and height 12 cm. l = √(25 + 144) = √169 = 13 cm.
Worked example 2
from slant height to volume. A cone has slant height 10 cm and radius 6 cm. Find its volume (π = 3.14).
h2 = 100 − 36 = 64, so h = 8 cm. Then V = ⅓ × 3.14 × 36 × 8 = 301.44 cm3.
Worked example 3
a conical roof. A round house has a roof of radius 4 m rising 3 m to its tip. Each rafter is √(16 + 9) = √25 = 5 m long.
For a pyramid with a square base of side s and vertical height h:
Worked example 4
A square pyramid has base side 10 m and height 12 m.
Slant height = √(122 + 52) = √169 = 13 m.
Half of the base diagonal = ½ × 10√2 = 5√2 m, so the lateral edge = √(144 + 50) = √194 ≈ 13.93 m.
Tip
Memory hook. The slant height is always the hypotenuse, so it is longer than both the height and the radius (or half-side). For a pyramid, draw the little right-angled triangle inside it: apex, centre of the base, midpoint of an edge. Use half the base side, not the whole side.
A cone has radius 9 cm and slant height 15 cm. Find its vertical height.
Solution: h2 = 225 − 81 = 144, so h = 12 cm.
A square pyramid has base side 8 cm and slant height 5 cm. Find its vertical height.
Solution: Half the side is 4. h2 = 25 − 16 = 9, so h = 3 cm.
A cone has radius 5 cm and height 12 cm. Find its slant height, then explain why it must be longer than 12 cm.
Solution: l = 13 cm. It is the hypotenuse of a right-angled triangle with legs 5 and 12, and a hypotenuse is always the longest side.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
