# 3D Shapes

After studying this section, you will be able to:

• recognise a prism as a solid with a uniform cross-section
• find the volume of a prism, a pyramid, a cone and a sphere

This video and images below explain the faces, vertices and edges of common three-dimensional shapes.

You should know and be able to use the formulae for the areas of the following shapes: parallelogram, rhombus, trapezium. These may be needed when calculating the volumes of prisms.

If a solid has a uniform cross-section; that is, the cross-sectional area is the same throughout its length then the solid is a prism.

You should also know the formulae for the volumes of the cone and the pyramid (although you would be given them in an examination).

Cones and Pyramids

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• A cone can be treated as a pyramid with a circular base.
• The formula for the volume of a cone is the same as that for a pyramid:

Volume = 1/3 x base area x vertical height

Volume = 1/3πr2h

where r is the radius of the base and h is the vertical height of the cone.

• The curved surface area of a cone is given by:

S = πrl

where l is the slant height of the cone

• The total surface area of a cone consists of the curved surface area plus the area of its circular base.
• The total surface area is given by:

A = πrl + πr2

KEY POINT - The formula for the volume and curved surface area of a cone are given on the formula sheet that is included with the examination. However, it is much better to learn them.

Spheres

• A sphere is a ball shape.
• The Earth and other planets are approximately spherical.
• The volume of a sphere is given by the formula:

V = 4/3πr3

• The surface area of a sphere is given by the formula:

A = 4πr2

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Practice Question

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