Area & Perimeter

Spec reference: Geometry and measures - Mensuration and calculation
Key idea: Calculate the area and perimeter of 2D shapes including circles and compound shapes.

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Perimeter

The perimeter is the total distance around the outside of a shape.

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Area formulae

Shape	Area formula
Rectangle	$A=lw$
Triangle	$A=\frac{1}{2}bh$
Parallelogram	$A=bh$
Trapezium	$A=\frac{1}{2}(a+b)h$
Circle	$A=\pi r^2$

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Circles

$$\text{Circumference}=\pi d=2\pi r$$$$\text{Area}=\pi r^2$$

Example

Find the area and circumference of a circle with radius 5 cm.

$A=\pi \times 5^2=25\pi \approx \boxed{{\displaystyle 78.5{\text{ cm}}^2}}$

$C=2\pi \times 5=10\pi \approx \boxed{{\displaystyle 31.4\text{ cm}}}$

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Arc length and sector area

$$\text{Arc length}=\frac{\theta }{360}\times 2\pi r$$$$\text{Sector area}=\frac{\theta }{360}\times \pi r^2$$

where $\theta$ is the angle in degrees.

Example

Find the arc length of a sector with radius 8 cm and angle 135°.

$$\text{Arc length}=\frac{135}{360}\times 2\pi \times 8=\frac{3}{8}\times 16\pi =6\pi \approx \boxed{{\displaystyle 18.8\text{ cm}}}$$

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Compound shapes

Split into simpler shapes, calculate each area, then add (or subtract).

Example

Find the area of an L-shape made from two rectangles:

• Large rectangle: $8\times 5=40$
• Small rectangle removed: $3\times 2=6$

Area $=40-6=\boxed{{\displaystyle 34{\text{ cm}}^2}}$

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Exam tips

Watch out for

• Use the radius (not diameter) in area and circumference formulas
• Give answers in terms of $\pi$ when the question asks for an exact answer
• Always check the formula for trapezium: $\frac{1}{2}(a+b)h$ - the $h$ is perpendicular height

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Test Yourself

Question 1 of 5

Area of circle, radius 7 cm (in terms of $\pi$)?