Lesson 45: Areas and Perimeters

Lesson 45 · GCSE Mathematics · Higher focus

Area measures the space inside a two-dimensional shape; perimeter measures the distance around its boundary. This lesson develops both ideas through compound shapes, circles, sectors, segments and algebra, with worked examples and a complete downloadable lesson pack.

Area or perimeter?

Use length units such as cm or m for perimeter, and square units such as cm² or m² for area. A rectangle measuring 12 cm by 6 cm has perimeter 2(12 + 6) = 36 cm and area 12 × 6 = 72 cm².

Make units match before calculating. For a 1.4 m by 60 cm rectangle, convert 60 cm to 0.60 m: area = 1.4 × 0.60 = 0.84 m², and perimeter = 2(1.4 + 0.60) = 4 m. Remember: 1 cm² = 100 mm² and 1 m² = 10,000 cm².

Essential area formulas

• Rectangle: A = length × width.
• Square: A = s²; perimeter = 4s.
• Triangle: A = ½bh.
• Parallelogram: A = bh.
• Trapezium: A = ½(a + b)h, where a and b are the parallel sides.
• Kite or rhombus with perpendicular diagonals: A = ½d₁d₂.

In these formulas, height means perpendicular height, not a sloping side. A triangle with base 10 cm and perpendicular height 7 cm has area ½ × 10 × 7 = 35 cm². A trapezium with parallel sides 8 cm and 14 cm and height 5 cm has area ½ × (8 + 14) × 5 = 55 cm².

Compound shapes: split or subtract

For an L-shaped region, suppose the full width is 12 cm, full height 9 cm, top width 7 cm and lower right vertical side 4 cm. The missing corner measures (12 − 7) cm by (9 − 4) cm, or 5 cm by 5 cm.

Area = 12 × 9 − 5 × 5 = 83 cm².
Trace the six outside edges for perimeter: 12 + 4 + 5 + 5 + 7 + 9 = 42 cm. Internal dividing lines are not part of the perimeter. If the question asks for the total boundary of a region with a hole, include the hole’s boundary too.

Circles and semicircles

For radius r and diameter d = 2r, circle area is πr² and circumference is 2πr = πd. Keep π in exact answers and round only at the end.

A semicircle has area ½πr², but its perimeter is πr + 2r: half the circumference plus the straight diameter. For diameter 10 cm, r = 5 cm, so area = 12.5π cm² and perimeter = 5π + 10 cm.

If a semicircle is attached to a 6 cm side of a 10 cm by 6 cm rectangle, its radius is 3 cm. Combined area = 60 + 4.5π cm²; outside perimeter = 26 + 3π cm. The shared diameter is internal.

Sectors and arc lengths

A sector with central angle θ degrees is θ/360 of a full circle:

• Sector area = (θ/360) × πr².
• Arc length = (θ/360) × 2πr.
• Sector perimeter = arc length + 2r.

For radius 9 cm and angle 120°, the fraction is ⅓. Area = 27π cm², arc length = 6π cm and perimeter = 6π + 18 cm.

Reverse problems use the same formula. A sector of radius 12 cm and area 30π cm² satisfies (θ/360) × 144π = 30π, giving θ = 75°. For a major sector, use the larger angle: if the minor angle is 80°, the major angle is 280°.

Higher extension: segments

A segment lies between a chord and its arc. For a minor central angle θ, subtract the triangle from the sector:

Minor segment area = (θ/360)πr² − ½r² sin θ.

Use degree mode. With r = 8 cm and θ = 90°, sector area = 16π cm² and triangle area = 32 cm². The minor segment is 16π − 32 cm², approximately 18.27 cm². The major segment is the rest of the circle: 48π + 32 cm².

Segment perimeter is arc length plus chord length, not two radii. In this example, the chord is 8√2 cm, giving minor segment perimeter 4π + 8√2 cm.

Algebra and reverse problems

A trapezium with parallel sides 9 cm and 15 cm and area 84 cm² satisfies 84 = ½(9 + 15)h = 12h, so h = 7 cm.

For a rectangle of width x cm, length (x + 5) cm and area 84 cm²: x(x + 5) = 84, so x² + 5x − 84 = 0 and (x + 12)(x − 7) = 0. Reject the negative length. The dimensions are 7 cm by 12 cm, and perimeter is 38 cm.

A ring between concentric circles of radii 10 cm and 6 cm has area π(10² − 6²) = 64π cm². Its total boundary length includes both circles: 20π + 12π = 32π cm.

Check your understanding

1. A parallelogram has base 13 cm, perpendicular height 6 cm and sloping side 8 cm. Find its area and perimeter.
2. A trapezium has parallel sides 7 cm and 17 cm and area 96 cm². Find its height.
3. A semicircle has diameter 14 cm. Find its exact area and perimeter.
4. A sector has radius 12 cm and angle 150°. Find its exact area, arc length and perimeter.

Download the Lesson 45 pack

Use the illustrated notes first, attempt the questions independently, then check your method against the worked answers.

• Illustrated lesson notes and practice solutions (PDF)
• 27 Higher Tier past-paper questions — 103 marks (PDF)
• Separate fully worked answers (PDF)
• 16-slide teaching presentation (PowerPoint)

The past-paper set includes source references and original examination diagrams. Its coverage page identifies the selected papers; it is not an exhaustive archive of every board and session. The teaching examples and practice above are original QLS material.

Before you finish: check units, use perpendicular heights, distinguish radius from diameter, include every required boundary, reject negative lengths and round only at the end.