Mensuration - Extended Maths

The three measures

Three questions come up again and again, and it pays to keep their units straight.

Perimeter: The total distance around the edge of a flat shape, measured in length units such as cm.

Area: The amount of flat surface a shape covers, measured in square units such as cm\(^2\).

Volume: The amount of space a solid fills, measured in cubic units such as cm\(^3\).

Compound 2D shapes

A compound shape is just simple shapes joined together, most often rectangles. The trick is to split it into pieces you know, work out each piece, then add or subtract. For an L-shape, you can find the area of the full rectangle and subtract the missing corner.

Worked example

An L-shape is a 10 cm by 8 cm rectangle with a 4 cm by 3 cm rectangle cut out of one corner. Find its area and its perimeter.

Area of the full rectangle: \(10 \times 8 = 80\) cm\(^2\).

Area of the cut-out corner: \(4 \times 3 = 12\) cm\(^2\).

Subtract to get the L-shape area: \(80 - 12 = 68\) cm\(^2\).

For the perimeter, notice a corner notch does not change the distance around, the two cut edges replace exactly the lengths they removed. So the perimeter equals that of the full rectangle: \(2(10 + 8) = 36\) cm.

Area \(= 68\) cm\(^2\), perimeter \(= 36\) cm

Find the hidden lengths first

Compound-shape questions often leave one or two edges unlabelled. Before calculating, work these out from the sides that are given: opposite runs of an L-shape must add up to the same total.

Volume of prisms and cylinders

A prism is a solid with the same cross-section all the way through. Its volume is simply that cross-sectional area multiplied by the length:

\[ V = \text{area of cross-section} \times \text{length} \]

A cylinder is a prism whose cross-section is a circle of radius \(r\). Since a circle has area \(\pi r^2\), the volume becomes:

\[ V = \pi r^2 h \]

Worked example

Find the volume of a cylinder with radius 3 cm and height 10 cm. Give your answer to 1 decimal place.

Write the formula: \(V = \pi r^2 h\).

Substitute \(r = 3\) and \(h = 10\): \(V = \pi \times 3^2 \times 10 = \pi \times 9 \times 10 = 90\pi\).

Evaluate: \(90\pi = 282.74\ldots\)

\(V \approx 282.7\) cm\(^3\)

Surface area

Surface area is the total area of every face, so it is measured in square units, not cubic ones. For a closed cylinder, the surface is two circles plus the curved wall, which unrolls into a rectangle of width \(2\pi r\) and height \(h\):

\[ \text{Surface area} = 2\pi r^2 + 2\pi r h \]

Worked example

Find the total surface area of a closed cylinder with radius 3 cm and height 10 cm. Give your answer to 1 decimal place.

Two circular ends: \(2\pi r^2 = 2\pi \times 3^2 = 18\pi\).

Curved surface: \(2\pi r h = 2\pi \times 3 \times 10 = 60\pi\).

Add them: \(18\pi + 60\pi = 78\pi = 245.04\ldots\)

Surface area \(\approx 245.0\) cm\(^2\)

Where this is assessed

Choosing the right formula and computing accurately is Criterion A (Knowing and understanding). Laying out each face or each piece with correct units earns Criterion C (Communicating), and real tasks such as sizing a water tank or a packaging design are classic Criterion D (Applying maths in context). Watch your units: mixing cm\(^2\) and cm\(^3\) is a common lost mark.

Check yourself

1. Find the volume of a cylinder with radius 5 cm and height 4 cm (to 1 d.p.). +

\(V = \pi r^2 h = \pi \times 5^2 \times 4 = \pi \times 25 \times 4 = 100\pi = 314.15\ldots\) So \(V \approx \mathbf{314.2}\) cm\(^3\).

2. An L-shape is a 12 cm by 9 cm rectangle with a 5 cm by 4 cm corner removed. Find its area. +

Full rectangle: \(12 \times 9 = 108\) cm\(^2\). Corner removed: \(5 \times 4 = 20\) cm\(^2\). Area \(= 108 - 20 = \mathbf{88}\) cm\(^2\).

3. A prism has a cross-sectional area of 15 cm\(^2\) and a length of 8 cm. Find its volume. +

\(V = \text{area of cross-section} \times \text{length} = 15 \times 8 = \mathbf{120}\) cm\(^3\).

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