Geometry & measures

Area and Perimeter of 2D Shapes

Year 7 · Year 8 · Year 9

✓By the end of this lesson students will be able to calculate the perimeter of 2D shapes.
✓By the end of this lesson students will be able to calculate the area of triangles.
✓By the end of this lesson students will be able to calculate the area of parallelograms.
✓By the end of this lesson students will be able to calculate the area of trapezia.
✓By the end of this lesson students will be able to calculate the circumference and area of circles.

Key concepts

Perimeter

The perimeter is the total distance around the outside edge of a 2D shape. It is found by adding the lengths of all its sides. Perimeter is measured in units of length (e.g., cm, m, km).

Area

The area is the amount of surface a 2D shape covers. It is measured in square units (e.g., cm², m², km²).

Area of a Triangle

The area of a triangle is half the product of its base and its perpendicular height. The perpendicular height is the shortest distance from the base to the opposite vertex, measured at a right angle to the base.

Area = (1/2) × base × perpendicular height or A = (1/2)bh

Area of a Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. The area of a parallelogram is the product of its base and its perpendicular height. The perpendicular height is the shortest distance between the base and the opposite parallel side.

Area = base × perpendicular height or A = bh

Area of a Trapezium

A trapezium is a quadrilateral with exactly one pair of parallel sides. The area is calculated by taking half the sum of the lengths of the parallel sides, and then multiplying by the perpendicular distance between them (the height).

Area = (1/2) × (a + b) × height or A = (1/2)(a + b)h (where 'a' and 'b' are the lengths of the parallel sides)

Circumference of a Circle

The circumference is the perimeter of a circle. It is calculated using the mathematical constant pi (π), which is approximately 3.14159. The radius (r) is the distance from the centre of the circle to any point on its circumference. The diameter (d) is the distance across the circle through its centre, so d = 2r.

Circumference = π × diameter or C = πd or C = 2πr

Area of a Circle

The area of a circle is calculated using pi (π) and the radius (r) squared.

Area = π × radius² or A = πr²

Key facts to remember

1Perimeter is the distance around a shape, measured in units of length (e.g., cm).
2Area is the space inside a shape, measured in square units (e.g., cm²).
3Always use the perpendicular height (not slant height) for calculating the area of triangles, parallelograms, and trapezia.
4For a circle, the diameter (d) is twice the radius (r): d = 2r.
5Pi (π) is a mathematical constant, approximately 3.142, used in circle calculations.
6Area of a Triangle = (1/2)bh
7Area of a Parallelogram = bh
8Area of a Trapezium = (1/2)(a + b)h
9Circumference of a Circle = πd or 2πr
10Area of a Circle = πr²

Worked examples

Example 1

A triangle has a base of 8 cm, a perpendicular height of 6 cm, and other sides of 7 cm and 10 cm. Calculate its area and perimeter.

ITo find the perimeter, add the lengths of all sides:

IIPerimeter = 8 cm + 7 cm + 10 cm

IIIPerimeter = 25 cm

IVTo find the area, use the formula: Area = (1/2) × base × perpendicular height

VArea = (1/2) × 8 cm × 6 cm

VIArea = 4 cm × 6 cm

VIIArea = 24 cm²

Answer

Perimeter = 25 cm, Area = 24 cm²

Example 2

Calculate the area of a trapezium with parallel sides of 5 m and 9 m, and a perpendicular height of 4 m.

IWrite down the formula for the Area of a Trapezium: A = (1/2)(a + b)h

IIIdentify the values: a = 5 m, b = 9 m, h = 4 m

IIISubstitute the values into the formula:

IVA = (1/2)(5 + 9) × 4

VA = (1/2)(14) × 4

VIA = 7 × 4

VIIA = 28 m²

Answer

28 m²

Example 3

A circle has a radius of 5 cm. Calculate its circumference and area. Give your answers to 1 decimal place.

ITo find the circumference, use the formula: C = 2πr

IISubstitute the radius r = 5 cm: C = 2 × π × 5

IIIC = 10π

IVUsing a calculator, C ≈ 10 × 3.14159... ≈ 31.4159...

VRounding to 1 decimal place: C ≈ 31.4 cm

VITo find the area, use the formula: A = πr²

VIISubstitute the radius r = 5 cm: A = π × 5²

VIIIA = π × 25

9A = 25π

10Using a calculator, A ≈ 25 × 3.14159... ≈ 78.539...

11Rounding to 1 decimal place: A ≈ 78.5 cm²

Answer

Circumference ≈ 31.4 cm, Area ≈ 78.5 cm²

Always use the π button on your calculator for the most accurate result, and only round your final answer.

Common mistakes

✗Using the slant height instead of the perpendicular height when calculating the area of triangles or parallelograms.
✗Confusing the radius and diameter when applying circle formulas (e.g., using diameter in A = πr²).
✗Forgetting to include the correct units (e.g., cm for perimeter, cm² for area) in the final answer.
✗Adding all four sides of a trapezium when the area formula only requires the two parallel sides.
✗Squaring the diameter instead of the radius when calculating the area of a circle.

Exam tips

★Always write down the correct formula first before substituting any values.
★If a diagram is not provided, sketch one and label all known dimensions clearly.
★Pay close attention to the units given in the question and ensure your final answer uses the correct units.
★Read the question carefully to determine whether you need to calculate perimeter, area, or both.
★Use the π button on your calculator for maximum accuracy, and only round your answer at the very end of the calculation to the specified degree of accuracy.

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