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Area of standard shapes

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Triangle with base 12.5 metres, left side 7.5 metres, right side 10 metres and a dashed perpendicular height of 6 metres from the apex to the base12.5 m7.5 m10 m6 m
Base
\(\displaystyle 12.5\text{ m}\)
Sloping sides
\(\displaystyle 7.5\text{ m and }10\text{ m}\)
Perpendicular height
\(\displaystyle 6\text{ m}\)

Find the area of the triangular bed from the labelled measurements

Year 10, Chapter 1, 4 difficulty levels, Calculator allowedBuild a worksheet

Area of standard shapes

Area measures the enclosed region in square units. Identify the shape and the dimensions its area formula needs

\[\begin{aligned}A_{\mathrm{triangle}}&=\frac12bh \\ A_{\mathrm{parallelogram}}&=bh\end{aligned}\]
  1. Use rectangle dimensions Multiply length by width to find the area of a rectangle \(\displaystyle A=8\times5=40\text{ cm}^{2}\)
  2. Halve a triangle product Multiply the base by its perpendicular height, then divide by 2 \(\displaystyle A=\frac{1}{2}\times10\times6=30\text{ cm}^{2}\)
  3. Use perpendicular height For a parallelogram, use the perpendicular height rather than a sloping side \(\displaystyle A=bh=10\times6=60\text{ cm}^{2}\)
  4. Use perpendicular diagonals For a rhombus or kite with perpendicular diagonals, add the areas of the two triangles. Use both complete diagonal lengths \(\displaystyle A=\frac12d_1d_2\)
  5. Trapezium area Add the two parallel side lengths a and b, multiply by the perpendicular height h, then halve \(\displaystyle A=\frac12(a+b)h\)

Worked example

A right-angled triangle has perpendicular sides of 9 cm and 12 cm and a hypotenuse of 15 cm. Find its area

\[\begin{gathered}b=9\text{ cm},\quad h=12\text{ cm} \\ s=15\text{ cm}\end{gathered}\]
  1. \(\displaystyle bh=9\times12=108\text{ cm}^{2}\)Multiply the base by the perpendicular height
  2. \(\displaystyle A=\frac{1}{2}\times108=54\text{ cm}^{2}\)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 height
Using 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 base
Confusing 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 the product of the perpendicular sides, which is 54 cm². The operations and units describe different measures

Difficulty levels

introFind the area of a square or rectangle from its side lengths
standardFind triangle, parallelogram and rhombus areas using perpendicular heights or diagonals
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 different parallelograms have the same area?+

Yes. Parallelograms with the same base and perpendicular height have equal areas even when their sloping sides differ

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)