Circles — Challenge
Every circle has the same ratio of circumference to diameter. That ratio is π, and it drives both formulas.
What these problems cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Circumference — The perimeter of a circle: C = 2π r.
- π — The circumference of any circle divided by its diameter — about 3.14159.
- Radius — Centre to edge. Half the diameter.
- Diameter — All the way across through the centre. d = 2r.
- Arc — Part of the edge; a "pizza slice" bounded by two radii.
- Sector — Part of the edge; a "pizza slice" bounded by two radii.
Problem 1
What is the circumference of a circle of radius 5 centimetres?
- 5π
- 10π
- 20π
- 25π
- 50π
Worked answer
10π
The circumference of a circle is 2π r.
With r=5 that is 2π· 5 = 10π centimetres.
The expression 25π would be the area, which uses r2 instead.
Problem 2
What is the area of a circle of radius 6 centimetres?
- 6π
- 12π
- 36π
- 72π
- 144π
Worked answer
36π
The area of a circle is π r2.
With r=6 that is π· 36 = 36π square centimetres.
The value 12π is the circumference, which uses 2r rather than r2.
Problem 3
A circle has a circumference of 16π centimetres. What is its area in square centimetres?
- 16π
- 32π
- 64π
- 128π
- 256π
Worked answer
64π
From 2π r = 16π we get r = 8 centimetres.
Then the area is π r2 = π· 64 = 64π square centimetres.
The radius has to be found first; the circumference cannot be squared directly.
Problem 4
What is the circumference of a circle of diameter 14 centimetres?
- 7π
- 14π
- 28π
- 49π
- 196π
Worked answer
14π
The circumference is π d when the diameter is given, which is the same as 2π r.
π· 14 = 14π centimetres.
Using 14 as the radius by mistake would double this to 28π.
Problem 5
A circle has an area of 25π square centimetres. What is its radius in centimetres?
- 5
- 10
- 12.5
- 25
- 50
Worked answer
5
From π r2 = 25π we get r2 = 25.
Taking the positive square root gives r = 5 centimetres.
Halving the 25 to get 12.5 confuses area with circumference.
Problem 6
What is the area of a semicircle of radius 4 centimetres?
- 2π
- 4π
- 8π
- 16π
- 32π
Worked answer
8π
The full circle has area π· 42 = 16π square centimetres.
A semicircle is half of that: 8π square centimetres.
Problem 7
What is the arc length of a 90-degree sector of a circle of radius 8 centimetres?
- π
- 2π
- 4π
- 8π
- 16π
Worked answer
4π
A 90-degree sector is 90360 = 14 of the circle.
The full circumference is 2π· 8 = 16π centimetres.
A quarter of that is 4π centimetres.
Problem 8
What is the area of a 60-degree sector of a circle of radius 6 centimetres?
- π
- 3π
- 6π
- 12π
- 36π
Worked answer
6π
A 60-degree sector is 60360 = 16 of the circle.
The full area is π· 36 = 36π square centimetres.
One sixth of that is 6π square centimetres.
Problem 9
What is the circumference of a circle of radius 3.5 centimetres?
- 3.5π
- 7π
- 12.25π
- 14π
- 49π
Worked answer
7π
The circumference is 2π r.
2· 3.5 = 7, so the circumference is 7π centimetres.
The value 12.25π is the area, since 3.52 = 12.25.
Problem 10
Two circles share the same centre, with radii 3 and 5 centimetres. What is the area of the ring between them, in square centimetres?
- 2π
- 4π
- 8π
- 16π
- 34π
Worked answer
16π
The larger circle has area 25π and the smaller has area 9π square centimetres.
The ring is the difference: 25π-9π = 16π square centimetres.
Subtracting the radii first and squaring would give 4π, which is not the same thing.