Perimeter and area of standard shapes
Perimeter is the total length around a shape. Area is the amount of surface inside it, so use the formula for the shape and the correct height
\[\begin{gathered}P=\text{sum of side lengths} \\ A_{\mathrm{triangle}}=\frac12bh\end{gathered}\]
- Count each boundary side once Add all side lengths around the outside to find perimeter \(\displaystyle P=8+5+8+5=26\text{ cm}\)
- Use rectangle dimensions Multiply length by width to find the area of a rectangle \(\displaystyle A=8\times5=40\text{ cm}^{2}\)
- Halve a triangle product Multiply the base by its perpendicular height, then divide by 2. \(\displaystyle A=\frac{1}{2}\times10\times6=30\text{ cm}^{2}\)
- Use perpendicular height For a parallelogram, use the perpendicular height rather than a sloping side \(\displaystyle A=bh=10\times6=60\text{ cm}^{2}\)
- Use perpendicular diagonals For a rhombus or kite with perpendicular diagonals, add the areas of the two triangles. Use both complete diagonal lengths \(\displaystyle A=\frac12d_1d_2\)
Worked example
A right-angled triangle has perpendicular sides of 9 cm and 12 cm and a hypotenuse of 15 cm. Find its perimeter and area
\[\begin{gathered}b=9\text{ cm},\quad h=12\text{ cm} \\ s=15\text{ cm}\end{gathered}\]
- \(\displaystyle P=9+12+15=36\text{ cm}\)Add all three boundary sides to find the perimeter
- \(\displaystyle bh=9\times12=108\text{ cm}^{2}\)Multiply the base by the perpendicular height
- \(\displaystyle A=\frac{1}{2}\times108=54\text{ cm}^{2}\)Halve the base-height product to find the triangle area
Common mistakes
Dropping the triangle half
✕\(\displaystyle A=8\times6=48\text{ cm}^{2}\)✓\(\displaystyle A=\frac{1}{2}\times8\times6=24\text{ cm}^{2}\)
A triangle has half the area of a parallelogram with the same base and perpendicular heightUsing a sloping side as height
✕\(\displaystyle A=\frac{1}{2}\times8\times10=40\text{ cm}^{2}\)✓\(\displaystyle A=\frac{1}{2}\times8\times6=24\text{ cm}^{2}\)
The height in an area formula must be perpendicular to the chosen baseConfusing perimeter and area
✕\(\displaystyle A=12+9+15=36\text{ cm}^{2}\)✓\(\displaystyle A=\frac{1}{2}\times12\times9=54\text{ cm}^{2}\)
For this right triangle, adding the 12 cm, 9 cm and 15 cm sides gives a perimeter of 36 cm. The area is half of 12 × 9, which is 54 cm². The operations and units describe different measuresDifficulty levels
introFind perimeter or area for a square or rectangle from its side lengths
standardFind perimeter and area for triangles, parallelograms and rhombuses using the appropriate dimensions
challengeWork backwards with a trapezium and convert between mixed area units when required
examRecognise the correct area of a triangle or rhombus from distinct same-unit results
Questions about this topic
What is the difference between perimeter and area?+
Perimeter measures the length around the boundary, while area measures the surface inside the boundary
Why must the height be perpendicular?+
The area formulas use the shortest distance between parallel base lines, which is the perpendicular height
How is the area of a trapezium found?+
Add the lengths of the parallel sides, multiply by the perpendicular height, and divide by 2.
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 5: Measurement
In this chapter
5A Length conversion and perimeter: Convert measurement units5A Length conversion and perimeter: Perimeter and area of standard shapesFind an unknown side from a perimeter5A Length conversion and perimeter: Solve a linear equation5A Length conversion and perimeter: Substitute values into a formula5A Length conversion and perimeter: Round measurements to decimal places5B Circumference, sectors and exact π: Circumference and area of a circle5B Circumference, sectors and exact π: Arc length and sector areaPerimeter of a sector5B Circumference, sectors and exact π: Simplify exact expressions involving pi5B Circumference, sectors and exact π: Substitute values into a formula5B Circumference, sectors and exact π: Round measurements to decimal places5C Area formulas and units: Convert measurement units5C Area formulas and units: Perimeter and area of standard shapes5C Area formulas and units: Circumference and area of a circle5C Area formulas and units: Arc length and sector area5C Area formulas and units: Simplify exact expressions involving pi5C Area formulas and units: Substitute values into a formula5C Area formulas and units: Round measurements to decimal placesPerimeter and area of composite shapes5D Composite perimeter and area: Find a side of a right-angled triangle5D Composite perimeter and area: Substitute values into a formula5D Composite perimeter and area: Round measurements to decimal places5E Right-prism surface area: Surface area of cuboidsSurface area of a triangular prism5E Right-prism surface area: Find a side of a right-angled triangle5E Right-prism surface area: Substitute values into a formula5E Right-prism surface area: Round measurements to decimal placesPyramid surface area: Surface area of square pyramids (optional extension)5F Cylinder surface area: Surface area of cylindersSurface area of a cylinder portion5F Cylinder surface area: Substitute values into a formula5F Cylinder surface area: Round measurements to decimal places5G Volume and capacity: Convert measurement units5G Volume and capacity: Volume of right prisms5G Volume and capacity: Capacity of a container5G Volume and capacity: Solve a linear equation5G Volume and capacity: Substitute values into a formula5G Volume and capacity: Round measurements to decimal places5H Cylinder volume and capacity: Volume of cylinders5H Cylinder volume and capacity: Capacity of cylindersVolume of a cylinder portion5H Cylinder volume and capacity: Substitute values into a formula5H Cylinder volume and capacity: Round measurements to decimal placesSmallest unit and limits of accuracy