Pyramids: volume and surface area
A temple roof, a tent and the pyramids of Egypt are square pyramids. Change the side of the square base and the height, and find the slant height, the volume and the area of the faces.
Core concept
A pyramid holds a third of the prism on the same base and height, V = ⅓a²h; its four triangular faces need the slant height √(h² + (a/2)²)
Topic: Area and volume
A square pyramid has a square base of side a and four triangles that meet at the top (the apex). Its height h is measured straight up from the centre of the base to the apex. In Class 9 a prism's volume: How much space something takes up, measured in millilitres or cubic centimetres. All key words was the area: How much flat space a shape covers, measured in square units like cm². All key words of its end × its length; a pyramid on the same base and height holds exactly one third of that: V = ⅓a²h. Each triangular face has base a and a height of its own, the slant height l. Half the base, the height and the slant height make a right-angled triangle, so l = √(h² + (a/2)²) by Pythagoras. The four faces together (the lateral surface) are 4 × ½ × a × l = 2al, and the total surface adds the base: 2al + a². Combined solids, like a tent with a cone on a cylinder, are worked out part by part.
What you will learn
- Find the slant height of a square pyramid with Pythagoras' theorem.
- Find the volume of a square pyramid as ⅓ × base area × height.
- Find the lateral and total surface area of a square pyramid.
Before you start
- Know the area of a triangle, Pythagoras' theorem and the volume of a prism.
Materials
- A sheet of card, a ruler and scissors to make a pyramid
Equipment
- None
Procedure
- Record a pyramid with a = 6 m and h = 4 m.
- Record a pyramid with a = 8 m and h = 3 m, and compare.
- Record a = 10 m, h = 12 m.
- Record a = 12 m, h = 8 m.
How this simulation is simplified
- The apex is straight above the centre of the square base.