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 11, VCE Units 1 & 2, General Mathematics, chapter 10: Measurement and similarity
In this chapter
Round to decimal places and significant figuresScientific notationMultiply and divide in scientific notationFind a side of a right-angled trianglePythagoras in three dimensions10C Perimeter and area: Convert measurement unitsPerimeter and area of standard shapesArea of a triangle from three sidesCircumference and area of a circlePerimeter and area of composite shapesArc length and sector area10E Volume and capacity: Convert measurement unitsVolume of prisms and cylinders10E Volume and capacity: Volume of cones, pyramids and spheresCapacity of a container10F Volume of pyramids: Volume of cones, pyramids and spheresSurface area of solidsSimilar figures and scale factorsTests for similar trianglesSimilar solids and scaling volume