What you'll learn
This topic covers the perimeter and area of 2D shapes, including the trapezium and compound shapes. In this guide you will learn the area formulae for common shapes, how to find perimeters, how to break compound shapes into simple parts, and how to handle parts of circles. These are core measurement skills used throughout maths.
Key terms and definitions
Perimeter — the total distance around the edge of a shape.
Area — the amount of surface a shape covers, measured in square units.
Compound shape — a shape made from two or more simple shapes.
Trapezium — a quadrilateral with one pair of parallel sides.
Perpendicular height — the height measured at right angles to the base.
Core concepts
Perimeter
The perimeter is the total distance around a shape, found by adding all the side lengths. For shapes with curved edges, the perimeter includes the arc length. Keep all lengths in the same unit before adding.
Area of common shapes
Key area formulae: rectangle = length × width, triangle = ½ × base × height, parallelogram = base × height, and circle = πr². The height must be the perpendicular height, not a slanted side.
Area of a trapezium
A trapezium's area is ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular distance between them. You average the two parallel sides and multiply by the height.
Compound shapes
For a compound shape, split it into simple shapes (rectangles, triangles, semicircles), find each area, and add them — or subtract a cut-out piece from a larger shape. Work methodically and label each part.
Circles and parts of circles
A circle has area πr² and circumference πd (or 2πr). For a semicircle halve these; for a quarter circle take a quarter. When finding the perimeter of part of a circle, remember to add the straight edges as well as the arc.
Worked examples
Example 1: Trapezium area
Find the area of a trapezium with parallel sides 6 and 10 and height 4.
½ × (6 + 10) × 4 = ½ × 16 × 4 = 32.
Example 2: Circle area
Find the area of a circle with radius 5 (to 1 d.p.).
π × 5² = 25π = 78.5.
Example 3: Compound shape
A shape is a 6 × 4 rectangle with a 2 × 2 square removed. Find its area.
24 − 4 = 20.
Common mistakes and how to avoid them
Using a slant height. Area needs the perpendicular height.
Mixing up area and perimeter. Area is square units; perimeter is a length.
Forgetting straight edges. Include them in the perimeter of part-circles.
Wrong trapezium formula. Average the parallel sides, then × height.
Mismatched units. Convert all lengths before calculating.
Exam technique for Perimeter and Area
Add all sides for perimeter, including arc lengths.
Use the correct area formula with the perpendicular height.
Use ½(a + b)h for a trapezium.
Split compound shapes into simple parts and add or subtract.
Halve or quarter circle formulae for semicircles and quarter circles.
Quick revision summary
The perimeter is the total distance around a shape (add all sides, including any arc length), while the area is the surface covered, in square units. Key formulae: rectangle = l × w, triangle = ½ × base × height, parallelogram = base × height, circle = πr² — always using the perpendicular height. A trapezium has area ½ × (a + b) × h, averaging the parallel sides and multiplying by the height. For a compound shape, split it into simple shapes and add, or subtract a cut-out from a larger shape. For parts of circles, halve (semicircle) or quarter the formulae, and remember to add the straight edges to the perimeter. The common errors are using a slant height, confusing area with perimeter, forgetting straight edges, and mismatched units. Add sides for perimeter, use the right formula with perpendicular height, apply ½(a + b)h for trapezia, and split compound shapes carefully.