Separate a radius from a diameter and retain exact multiples of π.
Core & Extended · Syllabus C5.3, E5.3
Notes and key methods
A circle of radius r has circumference 2πr and area \(\displaystyle \pi r^{2}\). Its diameter is 2r. Circumference uses length units, while area uses square units.
When an answer is requested as kπ, enter the coefficient k rather than a decimal approximation to the whole expression. A semicircle perimeter includes its straight diameter as well as its curved boundary. Only the curved part is half a circumference.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Circumference and circle area
A circle has radius 8 cm. A semicircle is formed by cutting along a diameter.
Find the diameter in cm.
Write the full circumference as kπ cm. Find k.
Write the full circle area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
The semicircle perimeter is aπ+b cm. Find b.
Show the reasoning
(a)
Convert radius to diameter.
Double the radius to obtain 16 cm.
Answer: \(16\).
(b)
Use the circumference formula with the radius.
Using 2πr gives coefficient 16.
Answer: \(16\).
(c)
Square the radius for the area coefficient.
Using \(\displaystyle \pi r^{2}\) gives coefficient 64.
Answer: \(64\).
(d)
Include the straight cut edge in the perimeter.
The straight diameter contributes 16 cm; the curved term is 8π cm.
Answer: \(16\).
Example 2 · Circumference and circle area
A circle has radius 9 cm. A semicircle is formed by cutting along a diameter.
Find the diameter in cm.
Write the full circumference as kπ cm. Find k.
Write the full circle area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
The semicircle perimeter is aπ+b cm. Find b.
Show the reasoning
(a)
Convert radius to diameter.
Double the radius to obtain 18 cm.
Answer: \(18\).
(b)
Use the circumference formula with the radius.
Using 2πr gives coefficient 18.
Answer: \(18\).
(c)
Square the radius for the area coefficient.
Using \(\displaystyle \pi r^{2}\) gives coefficient 81.
Answer: \(81\).
(d)
Include the straight cut edge in the perimeter.
The straight diameter contributes 18 cm; the curved term is 9π cm.
Answer: \(18\).
Example 3 · Circumference and circle area
A circle has radius 10 cm. A semicircle is formed by cutting along a diameter.
Find the diameter in cm.
Write the full circumference as kπ cm. Find k.
Write the full circle area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
The semicircle perimeter is aπ+b cm. Find b.
Show the reasoning
(a)
Convert radius to diameter.
Double the radius to obtain 20 cm.
Answer: \(20\).
(b)
Use the circumference formula with the radius.
Using 2πr gives coefficient 20.
Answer: \(20\).
(c)
Square the radius for the area coefficient.
Using \(\displaystyle \pi r^{2}\) gives coefficient 100.
Answer: \(100\).
(d)
Include the straight cut edge in the perimeter.
The straight diameter contributes 20 cm; the curved term is 10π cm.
Answer: \(20\).
Common mistakes
Keep intermediate values exact until the final step.