Area Bounded by a Circle, Surface Area and Volume of Some Solids
Primary 6 · Mathematics
Area Bounded by a Circle, Surface Area and Volume of Some Solids
Lesson 4 of 5
16 min
Let's begin
At break time Ishimwe's team plays football with a leather ball. At home, his grandmother weaves an agaseke, the traditional Rwandan basket with a tall, pointed lid.
The ball is a sphere and the lid is a cone. Neither has flat faces to unfold into rectangles. So how do you find the area of a surface that curves all over?
A sphere is a perfectly round ball. Every point on its surface is the same distance, r, from its centre.
The surface area of a sphere is exactly four times the area of a circle with the same radius:
surface area of a sphere = 4 × π × r × r
Worked example 1
a football with a radius of 11 cm.
Worked example 2
a sphere with a radius of 5 cm. S = 4 × 3.14 × 5 × 5 = 4 × 3.14 × 25 = 314 cm².
A cone has a circular base and a curved surface that rises to a point, the apex.
If you cut the curved surface along a slant line and flatten it, you get a fan shape. Its area is:
curved surface area of a cone = π × r × l
A closed cone also has its circular base, so: total surface area = π × r × r + π × r × l.
Worked example 3
an agaseke lid with a base radius of 7 cm and a slant height of 10 cm. The lid is open underneath, so we need the curved surface only: 3.14 × 7 × 10 = 219.8 cm².
Worked example 4
a solid cone with r = 3 cm and l = 5 cm.
Tip
A sphere is worth four circles: 4 × π × r × r.
A cone's curve is π × r × slant. Add the base (π × r × r) only if the cone is closed underneath.
Use π = 3.14.
Answers
What you learned today
Sources and credits
The sphere formula is adapted from OpenStax. Written by Ganzaa: the agaseke and football examples and the whole section on cones, because the open source used here does not cover the surface area of a cone.
Quick check
Use π = 3.14. What is the surface area of a ball with a radius of 3 cm?
