M30 · Perimeter, area and circles | Edexcel GCSE Maths

Revise the key ideas

Perimeter is the total boundary length; area measures the enclosed surface. Perimeter uses units such as cm; area uses cm². Add only actual outside edges for a composite shape.
A rectangle has area lw. A triangle has area 12bh, where h is perpendicular to the base, even if its foot lies outside the triangle.
A triangle with perpendicular height
Enlarge diagram: A triangle with perpendicular height
A parallelogram has area bh, using perpendicular height rather than sloping side. A trapezium has area 12(a + b)h, where a and b are its parallel sides.
Worked example: A trapezium with parallel sides 5 cm and 9 cm and height 4 cm has area 12 × (5 + 9) × 4 = 28 cm².
For a composite area, split into familiar shapes or subtract a missing region from a larger one. Label dimensions and check that pieces neither overlap nor leave gaps.

A radius joins centre to circumference; a diameter passes through the centre and is twice the radius. A chord joins two circumference points; a tangent touches at one point. An arc is part of the circumference.
Radius, diameter, chord and tangent
Enlarge diagram: Radius, diameter, chord and tangent
A sector is enclosed by two radii and an arc. A segment is enclosed by a chord and an arc. They are different regions.
Circumference C = 2πr = πd; area A = πr². Use the radius in the area formula, not the diameter.
Worked example: For radius 4 cm, circumference is 8π cm ≈ 25.13 cm and area is 16π cm² ≈ 50.27 cm². Keep π when an exact answer is required.
A semicircle's perimeter includes its diameter: πr + 2r. Its curved edge alone has length πr. For r = 3 cm, perimeter is 3π + 6 cm.

For angle θ in degrees, arc length = θ360 × 2πr and sector area = θ360 × πr². These use the fraction of a full turn.
A sixty-degree sector of radius six
Enlarge diagram: A sixty-degree sector of radius six
Worked example: A 60° sector of radius 6 cm has arc length 2π cm and area 6π cm². Its perimeter is 12 + 2π cm, including both radii.
To find θ from a sector area, rearrange θ = 360 × sector areaπr². Check whether the question describes a minor or major sector.
Convert lengths before calculating area. Doubling a length unit conversion without squaring it gives an incorrect area conversion.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.