Area of a circle
Use the radius to calculate area. A diameter is twice the radius
\[A=\pi r^2\]
- Find the radius Halve a given diameter \(\displaystyle r=\frac d2\)
- Square before multiplying by π Keep π exact until the final rounding step \(\displaystyle \begin{gathered}r=5\text{ cm}\\A=25\pi\text{ cm}^2\end{gathered}\)
- Compare scaled circles If the radius is multiplied by a factor, area is multiplied by the square of that factor \(\displaystyle \pi(kr)^2=k^2\pi r^2\)
- Work backwards Use the positive square root of area divided by π \(\displaystyle r=\sqrt{\frac A\pi}\)
Worked example
A circular garden bed has diameter 18 m. Find its area to 2 decimal places
\[d=18\text{ m}\]
- \(\displaystyle r=\frac{18}{2}=9\text{ m}\)The radius is half the stated diameter
- \(\displaystyle \begin{gathered}A=\pi(9)^2=81\pi\text{ m}^2\\A\approx254.47\text{ m}^2\end{gathered}\)Square the radius for area and round the exact value once
Common mistakes
Using the diameter as the radius
✕\(\displaystyle \begin{gathered}d=18\text{ m}\\A=\pi(18)^2=324\pi\text{ m}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}r=9\text{ m}\\A=81\pi\text{ m}^2\end{gathered}\)
Halve the diameter before squaring. Using the diameter makes the area four times too largeUsing the wrong measure or unit
✕\(\displaystyle A=18\pi\text{ m}\)✓\(\displaystyle A=81\pi\text{ m}^2\)
Area uses the squared radius and is measured in square units, not ordinary length unitsScaling area as a length
✕\(\displaystyle A_{\mathrm{new}}=2A_{\mathrm{old}}\)✓\(\displaystyle A_{\mathrm{new}}=4A_{\mathrm{old}}\)
When a radius doubles, its square is multiplied by four. The area therefore quadruplesDifficulty levels
introFind area from a radius
standardFind area from a diameter
challengeRecover a positive radius from an exact area
examRecognise an area rounded to the stated precision
Questions about this topic
Why must a diameter be halved?+
The formulas are written using the radius, and a diameter is twice the radius. Therefore, halve the diameter to find the radius
Why is area measured in square units?+
Area describes a two-dimensional region, so its unit is a length unit multiplied by itself, such as cm² or m²
When should π be rounded?+
Keep π exact in the formula and round the final numerical result once to the requested number of decimal places
Why does doubling the radius quadruple the area?+
The area contains two factors of radius. Doubling both multiplies the product by four
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 1: Measurement
In this chapter
Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)