Perimeter, Area and Circles

This lesson covers the essential formulae for perimeter and area of 2D shapes, as well as circles, sectors and arcs, for AQA GCSE Mathematics. Area and perimeter questions appear on every exam paper, from simple rectangles to compound shapes and circle problems. You must know all the formulae and be confident applying them in context.

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Perimeter

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

• For most shapes, add up all the side lengths.
• For a rectangle: Perimeter = 2(l + w), where l = length and w = width.
• For a circle: the perimeter is called the circumference.

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Area of Standard Shapes

Shape	Formula	Notes
Rectangle	Area = length x width	Also applies to squares
Triangle	Area = (1/2) x base x height	The height must be perpendicular to the base
Parallelogram	Area = base x perpendicular height	NOT the slant height
Trapezium	Area = (1/2)(a + b) x h	a and b are the parallel sides; h is the perpendicular height
Kite	Area = (1/2) x d1 x d2	d1 and d2 are the diagonals

graph TD
    A[Area Formulae] --> B[Rectangle: l x w]
    A --> C[Triangle: half x b x h]
    A --> D[Parallelogram: b x h]
    A --> E[Trapezium: half x a+b x h]
    A --> F[Circle: pi x r squared]

Worked Example 1: Triangle

Find the area of a triangle with base 12 cm and perpendicular height 8 cm.

Solution: Area = (1/2) x 12 x 8 = (1/2) x 96 = 48 cm squared

Worked Example 2: Trapezium

A trapezium has parallel sides of 5 cm and 9 cm, and a perpendicular height of 6 cm. Find its area.

Solution: Area = (1/2)(5 + 9) x 6 = (1/2)(14) x 6 = 7 x 6 = 42 cm squared

Exam Tip: The most common mistake with trapeziums is forgetting to halve the sum of the parallel sides. Remember the formula as "half the sum of the parallel sides times the height."

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Compound Shapes

A compound shape (or composite shape) is made up of two or more simple shapes. To find its area:

1. Split the shape into rectangles, triangles, or other standard shapes.
2. Calculate the area of each part.
3. Add (or subtract) the areas.

Worked Example 3

An L-shaped room has the following dimensions: the overall shape is 10 m by 8 m, with a 4 m by 3 m rectangle cut from one corner.

Solution: Total area of full rectangle = 10 x 8 = 80 m squared Area of cut-out = 4 x 3 = 12 m squared Area of L-shape = 80 - 12 = 68 m squared

Exam Tip: When dealing with compound shapes, always check whether you should add or subtract areas. If a piece is "cut out" or "removed," subtract. If the shape is "made up of" simpler shapes, add.

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Circles: Circumference

The circumference is the perimeter of a circle.

Formulae:

• Circumference = pi x d (where d = diameter)
• Circumference = 2 x pi x r (where r = radius)

Remember: the diameter is twice the radius, so d = 2r.

Worked Example 4

Find the circumference of a circle with radius 7 cm. Give your answer to 1 decimal place.

Solution: C = 2 x pi x 7 = 14 x pi = 44.0 cm (1 d.p.)

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Circles: Area

Formula: Area = pi x r squared

Worked Example 5

Find the area of a circle with diameter 10 cm. Give your answer to 1 decimal place.

Solution: Radius = 10 / 2 = 5 cm Area = pi x 5 squared = pi x 25 = 78.5 cm squared (1 d.p.)

Worked Example 6 (Reverse)

The area of a circle is 154 cm squared. Find the radius to 1 decimal place.

Solution: pi x r squared = 154 r squared = 154 / pi = 49.0197... r = square root of 49.0197... = 7.0 cm (1 d.p.)

Exam Tip: Always check whether the question gives you the radius or the diameter. If you are given the diameter, halve it before substituting into the area formula. This is one of the most common errors in the exam.

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Sectors and Arcs

A sector is a "slice" of a circle (like a slice of pizza). An arc is the curved edge of a sector.

Arc Length

Arc length = (angle / 360) x 2 x pi x r

Sector Area

Sector area = (angle / 360) x pi x r squared

Worked Example 7

A sector has radius 8 cm and angle 45 degrees. Find the arc length and sector area.

Solution:

Arc length = (45 / 360) x 2 x pi x 8 = (1/8) x 16 x pi = 2 x pi = 6.28 cm (2 d.p.)

Sector area = (45 / 360) x pi x 8 squared = (1/8) x 64 x pi = 8 x pi = 25.13 cm squared (2 d.p.)

Worked Example 8

Find the perimeter of a sector with radius 10 cm and angle 120 degrees.

Solution: Arc length = (120 / 360) x 2 x pi x 10 = (1/3) x 20 x pi = 20pi / 3 = 20.944... cm

Perimeter of sector = arc length + 2 x radius = 20.944... + 10 + 10 = 40.9 cm (1 d.p.)

Exam Tip: The perimeter of a sector is NOT just the arc length. You must add the two straight edges (both radii) as well. This is a very common mistake.

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Semicircles and Quarter-Circles

These are special cases of sectors.