Area, Perimeter, and Volume

Knowing how to calculate the perimeter, area, and volume of shapes is essential for the FSCE 11+ exam. These questions often involve compound shapes, unit conversions, and multi-step reasoning. This lesson teaches you every formula you need, shows you how to tackle compound shapes, and gives you six detailed worked examples.

Perimeter

Perimeter is the total distance around the outside of a shape. You find it by adding up all the side lengths.

Perimeter of Common Shapes

• Rectangle: P = 2 x (length + width) = 2(l + w)
• Square: P = 4 x side length = 4s
• Triangle: P = sum of all three sides
• Circle (circumference): C = π x diameter = πd, or C = 2 x π x radius = 2πr

Compound Shapes

For compound shapes (shapes made up of simpler shapes joined together), trace around the outside of the whole shape, adding up every edge length. Be careful: some edges may be shared (internal edges) and should NOT be included in the perimeter.

Area

Area is the amount of space a 2D shape covers, measured in square units (cm², m², etc.).

Area Formulae

Shape	Formula	Notes
Rectangle	A = length x width	l x w
Square	A = side x side	s²
Triangle	A = 1/2 x base x height	The height must be perpendicular (at right angles) to the base
Parallelogram	A = base x height	The height is the perpendicular distance between the parallel sides
Trapezium	A = 1/2 x (a + b) x h	a and b are the two parallel sides, h is the perpendicular height
Circle	A = π x radius²	A = πr²

Area of Compound Shapes

Strategy: Break the compound shape into simpler shapes (rectangles, triangles, etc.), calculate each area separately, then add them together. Sometimes you need to subtract — for example, an L-shape can be seen as a large rectangle minus a smaller rectangle.

Volume

Volume is the amount of 3D space an object occupies, measured in cubic units (cm³, m³, etc.).

Volume and Surface Area of a Cuboid

• Volume = length x width x height = l x w x h
• Surface area = 2(lw + lh + wh)

A cube is a special cuboid where all sides are equal:

• Volume = s³
• Surface area = 6s²

Units Table

Measurement	Units	Conversions
Perimeter (length)	mm, cm, m, km	10mm = 1cm, 100cm = 1m, 1000m = 1km
Area	mm², cm², m², km²	100mm² = 1cm², 10,000cm² = 1m²
Volume	mm³, cm³, m³	1,000mm³ = 1cm³, 1,000,000cm³ = 1m³
Capacity	ml, cl, l	1cm³ = 1ml, 1,000ml = 1l

Important: When converting area units, you must square the conversion factor. Since 1m = 100cm, then 1m² = 100 x 100 = 10,000cm². Similarly for volume: 1m³ = 100 x 100 x 100 = 1,000,000cm³.

Worked Examples

Worked Example 1: Perimeter of a Compound Shape

Question: An L-shaped room has the following dimensions: the overall shape is 8m long and 6m wide, with a rectangular piece removed from the top-right corner measuring 3m by 2m. Find the perimeter.

Step-by-step solution:

1. Sketch the L-shape. The outside edges are:
   • Bottom: 8m
   • Right side (lower part): 6m - 2m = 4m
   • The step going left: 3m
   • The step going up: 2m
   • Top: 8m - 3m = 5m
   • Left side: 6m
2. Perimeter = 8 + 4 + 3 + 2 + 5 + 6 = 28m

Answer: 28m

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Worked Example 2: Area of a Triangle

Question: A triangle has a base of 12cm and a perpendicular height of 7cm. Find its area.

Step-by-step solution:

1. Formula: A = 1/2 x base x height
2. A = 1/2 x 12 x 7
3. A = 1/2 x 84
4. A = 42cm²

Answer: 42cm²

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Worked Example 3: Area of a Compound Shape

Question: A shape consists of a rectangle (10cm by 6cm) with a right-angled triangle attached to one of its shorter sides. The triangle has a base of 6cm and a height of 4cm. Find the total area.

Step-by-step solution:

1. Area of rectangle = 10 x 6 = 60cm²
2. Area of triangle = 1/2 x 6 x 4 = 12cm²
3. Total area = 60 + 12 = 72cm²

Answer: 72cm²

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Worked Example 4: Area of a Trapezium

Question: A trapezium has parallel sides of 8cm and 14cm, and a perpendicular height of 5cm. Find its area.

Step-by-step solution:

1. Formula: A = 1/2 x (a + b) x h
2. A = 1/2 x (8 + 14) x 5
3. A = 1/2 x 22 x 5
4. A = 1/2 x 110
5. A = 55cm²

Answer: 55cm²

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Worked Example 5: Volume of a Cuboid