Circumference and Area of a Circle and a Sector
Circumference of a circle is the distance around its edge, found with C = 2πr, where r is the radius. A circle with radius 5 has C = 10π. The area of a circle is the space inside it, found with A = πr².
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π and Circumference
- is a number used in circle formulas and is usually rounded to 3.14 in calculations.
- The formula for circumference is or .
Finding the Circumference of a Circle
- If the radius is , substitute into .
- This gives , which is approximately .
Finding the Area of a Circle
- The formula for circle area is .
- If the radius is , the area is , about .
Circle Sectors
- A sector is a fraction of a circle based on the central angle.
- Use for arc length and for area.
Practice Questions
Test your understanding
A circle has a radius of . What is the circumference of the circle? Give your answer in terms of .
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The formula for the circumference of a circle is .
The radius is , so .
'In terms of ' means leaving in the answer as a factor rather than working it out as a decimal, so the answer is , not .
A circle has a diameter of . What is the circumference of the circle? Give your answer in terms of .
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The formula for the circumference of a circle in terms of the diameter is .
The diameter is , so .
You can check this with the radius form: the radius is , and , the same answer.
The radius of a circle is . What is the area of the circle in terms of ?
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The formula for the area of a circle is .
The radius is , so .
Note that means , not , and that 'in terms of ' means keeping as a factor in the answer, so the area is , not .
The diameter of a circle is . What is the area of the circle in terms of ?
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The formula for the area of a circle is , which needs the radius, not the diameter.
The diameter is , so the radius is .
Then .
'In terms of ' means leaving in the answer as a factor, so the area is .
A sector of a circle has a central angle of and a radius of . What is the area of the sector in terms of ?
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The area of a sector is , where is the central angle.
Here and .
Whole circle: .
Fraction of the circle: , so the sector is a quarter of the circle.
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A sector of a circle has a central angle of and a radius of . What is the arc length of the sector in terms of ?
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The arc length of a sector is , where is the central angle.
Here and .
Whole circumference: .
Fraction of the circle: .
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Interactive Activity
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Frequently asked questions
The questions students bump into most on this topic
The circumference is the distance around the edge of a circle, its perimeter. Roll a wheel 1 metre across through one full turn and it travels about 3.14 metres: the circumference of any circle is π times its diameter. Written as a formula, C = πd, or C = 2πr using the radius.
Multiply the diameter by π, or the radius by 2π. A circle with a radius of 5 cm has circumference 2 × π × 5 = 10π cm, about 31.4 cm; with a diameter of 8 cm it is 8π cm. To go backwards, divide the circumference by π to get the diameter, and halve that for the radius.
No. π is the number you get when you divide any circle’s circumference by its diameter, and it is the same for every circle: 3.14159…, with infinite decimal places and no repeating pattern. 3.14 is a rounded value for quick working. That is why the formula is C = πd: the circumference is always π times the diameter.
Square the radius, then multiply by π: A = πr². A circle with a diameter of 6 cm has a radius of 3 cm, so its area is π × 3² = 9π cm², about 28.26 cm². If you are given the diameter, halve it first: a diameter of 10 cm gives a radius of 5 cm and an area of 25π cm².
A sector is a slice of a circle, so take the same slice of the circle’s area. The fraction is the central angle over 360: a 60° sector is 60/360 = 1/6 of the circle, so its area is 1/6 × πr². A 90° sector of a circle with radius 8 cm is 1/4 × π × 8² = 16π cm². No new formula is needed.
Take the same fraction of the circumference: arc length = angle/360 × 2πr. For a 60° sector of a circle with radius 6 cm, that is 1/6 × 2π × 6 = 2π cm. For the perimeter of the sector, add the two radii to the arc: 2π + 6 + 6, about 18.3 cm.
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