Volume of pyramids, cones, spheres and hemispheres, curved surface areas, slant height by Pythagoras, and frustums and composite solids.
What you will learn
- Find the volume of a pyramid and a cone
- Find the volume and surface area of a sphere and hemisphere
- Find a cone's slant height and curved surface area
- Find the volume of a frustum or a composite solid (Higher)
The formulas you are given, and how to use them
These formulas are usually given to you, but you must know which length goes where.
pyramid: V = ⅓ × base area × height
cone: V = ⅓πr²h, curved surface = πrl
sphere: V = 4/3 πr³, surface area = 4πr²
h is the perpendicular height. l is the slant height, along the sloping side. A hemisphere is half a sphere: V = ⅔πr³.
A pyramid with a 6 cm by 5 cm base and height 4 cm has V = ⅓ × 30 × 4 = 40 cm³. Working backwards, multiply the volume by 3 first: a pyramid of volume 50 cm³ on a 5 cm by 6 cm base has 30 × h = 150, so h = 5 cm.
For a sphere's radius from its volume, undo each step: 4/3 × π × r³ = 288π gives r³ = 288 × 3 ÷ 4 = 216, so r = ∛216 = 6 cm. Take the cube root, because r is cubed.
The rest of this lesson and its 8 practice questions are locked.
In this lesson
- The formulas you are given, and how to use them
- Slant height by Pythagoras
- Frustums and composite solids (Higher)
- Try it: An easy one first
8 practice questions, marked as you go; about 25 minutes in all.
Key terms
Perpendicular, Slant, Cube, Solid, Frustum
In this unit: Area, perimeter and volume
Perimeter and area of rectangles, triangles, parallelograms, trapeziums and compound shapes, circles, arcs and sectors, and the volume and surface area of prisms, cylinders, pyramids, cones and spheres.
- Perimeter and area
- Circles
- Arcs and sectors
- Volume and surface area of prisms
- Cones, pyramids and spheres
- Solids, nets, plans and elevations