Surface Area of Solids
Measurement · Surface Area of Solids
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 13 Surface Area of Solids
Lesson objectives
By the end of this topic, you should be able to:
- Find the surface area of prisms and pyramids.
- Find the surface area of a cone, including the curved surface.
- Find the surface area of a frustum.
- Find the surface area of a sphere and hemisphere.
Surface Area of Solids
The surface area of a solid is the total area of every face, flat or curved.
a) The general method
Imagine the solid unfolded into its net. Every face appears flat and at true size.
Work out each face, then add.
Marking each face on a sketch as you use it stops you missing one or counting it twice.
Decide first whether the solid is closed or open. An open pipe has no circular ends. A container open at the top has just one end.
b) Prisms
For any prism, surface area = 2 × cross-section + perimeter of cross-section × length.
A cuboid gives 2(lw + lh + wh).
A triangular prism gives two triangles plus three rectangles.
c) Cylinders
A cylinder unrolls into two circles and one rectangle.
The rectangle's length is the circumference, 2πr, so its area is 2πrh.
Closed cylinder: 2πr² + 2πrh.
d) Cones
A cone unrolls into a circle and a sector.
The curved surface area is πrl, where l is the slant height, not the vertical height.
Total surface area of a closed cone: πr² + πrl.
The slant height comes from Pythagoras: l² = r² + h².
Using h in place of l is the commonest error in cone questions. They are different lengths.
e) Frustum
A frustum is a cone with the top cut off parallel to the base.
Its curved surface is the large cone's curved surface minus the small cone's.
Add the two circular ends if the frustum is closed.
f) Spheres and hemispheres
A sphere has surface area 4πr². It has no flat faces at all.
A hemisphere has a curved surface of 2πr², which is half the sphere.
A closed hemisphere also has a flat circular face of πr², giving 3πr² in total.
Read whether the hemisphere is solid or a bowl. That decides whether the flat circle counts.
g) The formulas together
h) Where this is used
Painting or plating a tank. Costing sheet metal. Calculating heat loss, which depends on surface area. Packaging design.
Words to know
- Surface area -- the total area of all faces of a solid, equal to the area of its net.
- Slant height -- the distance from the apex to the base edge measured along a face.
- Curved surface area -- the area of the curved part only, excluding flat faces.
- Frustum -- the solid remaining when the top of a cone or pyramid is cut off parallel to the base.
- Hemisphere -- half a sphere.
:::checkpoint Check yourself
- Find the surface area of a closed cylinder with radius 7 cm and height 10 cm.
- A cone has radius 5 cm and vertical height 12 cm. Find its slant height.
- Find the total surface area of that cone.
- Why does a closed hemisphere have surface area 3πr²? :::
Bridge to practice
The exercises begin with nets and prisms, move through pyramids, cones and the slant height, and finish with spheres, frustums and composite solids. Before using any formula, sketch the net and mark which faces the question actually requires.