Circumference and area of a circle

Diameter of the trampoline

\(\displaystyle 3.6\text{ m}\)

A circular trampoline has diameter 3.6 m and a safety pad runs around its edge. What is the length of the pad, to 2 decimal places?

C = 2πr is circumference in cm; A = πr² is area in cm². The dots mark the selected radius on the two curves. Move the radius from 1 to 2 to 4 and compare the readouts. The vertical scale compares numerical values, not equal units of length and area

What happened when the radius was doubled from 1 to 2, and again from 2 to 4?

Circumference and area of a circle

Circumference is the distance around a circle and area is the space inside it. Circumference uses the radius once, while area uses the radius twice

1. 1Find the radius Halve a stated diameter before using either circle formula \(\displaystyle \begin{gathered}d=3.6\text{ m}\\r=\frac{3.6}{2}=1.8\text{ m}\end{gathered}\)
2. 2Use one radius for circumference The distance around grows in proportion to the radius \(\displaystyle \begin{gathered}C=2\pi r=2\pi(1.8)\\C=3.6\pi\text{ m}\\C\approx11.31\text{ m}\end{gathered}\)
3. 3Square the radius for area The space inside grows with the square of the radius \(\displaystyle \begin{gathered}A=\pi r^2=\pi(1.8)^2\\A=3.24\pi\text{ m}^2\\A\approx10.18\text{ m}^2\end{gathered}\)
4. 4Round once at the end Keep pi exact in the working and round only the final answer \(\displaystyle \begin{gathered}3.6\pi\text{ m}\approx11.31\text{ m}\\3.24\pi\text{ m}^2\approx10.18\text{ m}^2\end{gathered}\)

Diameter of the rug

\(\displaystyle 2.4\text{ m}\)

A round rug has diameter 2.4 m. It needs edge binding around its outside. Find the length of the binding, to 2 decimal places

Halve the diameter to find the radius first, then use the circumference formula. Write exactly 2 decimal places

Diameter of the rug

\(\displaystyle 2.4\text{ m}\)

The same rug needs a non-slip underlay covering its whole inside. Find the area of the underlay, to 2 decimal places

Square the radius, not the diameter. Write exactly 2 decimal places

Work backwards from an exact measure

Cancel the common factor π before recovering the positive radius. Then calculate the other measure

1. 1Circumference given If C = 16π cm, then 2r = 16 and r = 8 cm. The area is 64π cm² \(\displaystyle \begin{gathered}16\pi\text{ cm}=2\pi r\\r=8\text{ cm}\\A=\pi(8)^2=64\pi\text{ cm}^2\end{gathered}\)
2. 2Area given If A = 36π cm², then r² = 36 and r = 6 cm. The circumference is 12π cm \(\displaystyle \begin{gathered}36\pi\text{ cm}^2=\pi r^2\\r=6\text{ cm}\\C=2\pi(6)=12\pi\text{ cm}\end{gathered}\)

Diameter

\(\displaystyle d=9\text{ cm}\)

A student uses a diameter of 9 cm to calculate circumference. Which is the first incorrect line?