Mensuration

The One Thing

Mensuration requires selecting the correct formula for the geometric quantity being measured, using compatible units, and combining simpler parts accurately when a shape or solid is compound. Perimeter and area concern two-dimensional figures; surface area and volume concern three-dimensional solids.

Definitions and Results

• Perimeter: The total length of the boundary of a two-dimensional shape.
• Circumference: The perimeter of a circle, calculated using $C=2\pi r$ or $C=\pi d$.
• Area: The measure of the surface enclosed by a two-dimensional shape, expressed in square units.
• Surface area: The total area of all external faces or curved surfaces of a three-dimensional solid.
• Volume: The amount of three-dimensional space occupied by a solid, expressed in cubic units.
• Compound shape: A two-dimensional figure made from two or more familiar shapes joined together.
• Compound solid: A three-dimensional object formed from two or more familiar solids, such as cuboids, prisms, cylinders, cones or hemispheres.
• Prism: A solid with a constant cross-sectional shape along its length. Its volume is

$$V=\text{cross-sectional area}\times \text{length}.$$

• Net: A two-dimensional arrangement of the faces of a three-dimensional solid used to find its surface area.
• Radius: The distance from the centre of a circle or sphere to its boundary.
• Diameter: The distance across a circle or sphere through its centre;

$$d=2r.$$

• Slant height: The length measured along the sloping surface of a cone, used in its curved surface area.
• Scale factor: The multiplier relating corresponding lengths in similar figures or solids. If lengths have scale factor $k$, areas change by $k^2$ and volumes by $k^3$.
• Unit conversion: Changing measurements into compatible units before calculation. In particular,

$$1\text{ m}=100\text{ cm},1{\text{ m}}^2=10,000{\text{ cm}}^2,1{\text{ m}}^3=1,000,000{\text{ cm}}^3.$$

Two-dimensional formulae

• Rectangle:

$$P=2(l+w),A=lw.$$

• Square:

$$P=4s,A=s^2.$$

• Triangle:

$$A=\frac{1}{2}\times \text{base}\times \text{perpendicular height}.$$

• Parallelogram:

$$A=\text{base}\times \text{perpendicular height}.$$

• Trapezium:

$$A=\frac{1}{2}(a+b)h,$$ where $a$ and $b$ are the parallel sides and $h$ is the perpendicular height.

• Circle:

$$C=2\pi r=\pi d,A=\pi r^2.$$

• Arc length:

$$\text{arc length}=\frac{\theta }{360}\times 2\pi r,$$ where $\theta$ is the angle at the centre in degrees.

• Sector area:

$$\text{area}=\frac{\theta }{360}\times \pi r^2.$$

Three-dimensional formulae

• Cuboid:

$$V=lwh,$$ $$\text{total surface area}=2(lw+lh+wh).$$

• Prism:

$$V=\text{cross-sectional area}\times \text{length}.$$

• Cylinder:

$$V=\pi r^{2}h,$$ $$\text{curved surface area}=2\pi rh,$$ $$\text{total surface area}=2\pi r^2+2\pi rh.$$

• Pyramid:

$$V=\frac{1}{3}\times \text{base area}\times \text{perpendicular height}.$$

• Cone:

$$V=\frac{1}{3}\pi r^{2}h,$$ $$\text{curved surface area}=\pi rl,$$ $$\text{total surface area}=\pi r^2+\pi rl,$$ where $l$ is the slant height.

• Sphere:

$$V=\frac{4}{3}\pi r^3,\text{surface area}=4\pi r^2.$$

• Hemisphere:

$$V=\frac{2}{3}\pi r^3,$$ $$\text{curved surface area}=2\pi r^2,$$ $$\text{total surface area including the circular base}=3\pi r^2.$$

Units and supporting results

• Perimeter and length are measured in units such as $\text{cm}$ or $\text{m}$.
• Area and surface area are measured in square units such as ${\text{cm}}^2$ or ${\text{m}}^2$.
• Volume is measured in cubic units such as ${\text{cm}}^3$ or ${\text{m}}^3$.
• Use the value of $\pi$ given in the question or on the calculator, and round the final answer appropriately.
• Pythagoras’ theorem is

$$a^2+b^2=c^2.$$ It is used when a missing perpendicular length or slant height is required in a right-angled cross-section.

Worked Methods

1. Calculating the perimeter and area of familiar shapes

• Identify the two-dimensional shape.
• Select the relevant perimeter or area formula.
• Substitute the given measurements.
• Calculate and give the answer in the appropriate units.
• For area, use a perpendicular height where the formula requires one.

For a rectangle: $$P=2(l+w),A=lw.$$

For a square: $$P=4s,A=s^2.$$

For a triangle: $$A=\frac{1}{2}\times \text{base}\times \text{perpendicular height}.$$

For a parallelogram: $$A=\text{base}\times \text{perpendicular height}.$$

For a trapezium: $$A=\frac{1}{2}(a+b)h,$$ where $a$ and $b$ are the parallel sides and $h$ is the perpendicular height.

2. Calculating circumference, arc length and sector area

• Identify the radius $r$, diameter $d$, or central angle $\theta$.
• Convert the diameter to a radius if necessary using

$$r=\frac{d}{2}.$$

• Use the appropriate formula:

$$C=2\pi r=\pi d,$$ $$\text{arc length}=\frac{\theta }{360}\times 2\pi r,$$ $$\text{sector area}=\frac{\theta }{360}\times \pi r^2.$$

• Substitute the measurements and calculate.
• Round the final answer appropriately.

3. Calculating the volume and surface area of solids

• Identify the solid.
• Determine whether the question asks for volume, curved surface area, or total surface area.
• Select the corresponding formula.
• Check whether the dimensions use compatible units.
• Substitute and calculate.
• Give volume in cubic units and surface area in square units.

For example, for a cylinder: $$V=\pi r^{2}h,$$ $$\text{curved surface area}=2\pi rh,$$ $$\text{total surface area}=2\pi r^2+2\pi rh.$$

For a cone, distinguish between the perpendicular height $h$ used in $$V=\frac{1}{3}\pi r^{2}h$$ and the slant height $l$ used in $$\text{curved surface area}=\pi rl.$$

4. Finding a missing length using Pythagoras’ theorem

• Identify the right-angled triangle in the cross-section.
• Label the perpendicular sides $a$ and $b$, and the hypotenuse $c$.
• Use

$$a^2+b^2=c^2.$$

• Rearrange if the unknown is a shorter side:

$$a^2=c^2-b^2.$$

• Take the square root.
• Use the resulting perpendicular length or slant height in the relevant mensuration formula.

This method is particularly relevant when a cone’s slant height is needed for $$\text{curved surface area}=\pi rl.$$

5. Solving a compound shape

• Split the figure into simpler familiar shapes.
• Calculate the area or perimeter of each required part.
• Add areas when the parts form the whole.
• Subtract areas when a section has been removed.
• For perimeter, include only the boundary of the compound shape; do not include internal dividing lines.
• State the result in the correct units.

6. Solving a compound solid

• Divide the solid into familiar solids such as cuboids, prisms, cylinders, cones or hemispheres.
• Calculate the volume of each joined part.
• Add the volumes of joined parts, or subtract the volume of any removed part.
• For surface area, identify every exposed face or curved surface.
• Exclude faces hidden inside the joined solid.
• Combine the exposed areas and give the result in square units.

7. Applying scale factors

• Identify the scale factor $k$ between corresponding lengths.
• Multiply lengths by $k$.
• Multiply areas by $k^2$.
• Multiply volumes by $k^3$.
• Check that the resulting units match the quantity being measured.

8. Checking a mensuration answer

• Verify that all measurements were converted to compatible units.
• Check that the selected formula matches the shape and quantity.
• Confirm that perpendicular height, rather than a sloping side, was used where required.
• Check the units:

- length: units; - area or surface area: square units; - volume: cubic units.

• Estimate the expected size of the result.
• Check the value of $\pi$ and the final rounding.

Where It Goes Wrong

• Using a sloping side instead of the perpendicular height in triangle, parallelogram, trapezium, pyramid or cone volume calculations.
• Confusing radius and diameter; the required relationship is

$$d=2r.$$

• Including hidden internal faces when finding the surface area of a compound solid; only exposed surfaces are included.
• Combining parts of a compound shape or solid incorrectly instead of adding joined parts and subtracting removed parts where appropriate.
• Mixing incompatible units, particularly centimetres and metres, or linear, square and cubic units.
• Giving an answer with incorrect units or inappropriate rounding, or using a value of $\pi$ inconsistent with the question.

What Gets Asked

This material supports questions requiring students to:

• Find the perimeter or area of rectangles, squares, triangles, parallelograms, trapezia and circles.
• Calculate the circumference of a circle using $C=2\pi r$ or $C=\pi d$.
• Calculate arc length and sector area from a central angle.
• Find the volume or surface area of cuboids, prisms, cylinders, pyramids, cones, spheres and hemispheres.
• Distinguish between curved and total surface area.
• Use a net to determine the surface area of a three-dimensional solid.
• Use Pythagoras’ theorem to find a missing perpendicular length or cone slant height.
• Split compound shapes into familiar two-dimensional parts and combine their areas or perimeters.
• Split compound solids into familiar three-dimensional parts and add or subtract their volumes.
• Find the exposed surface area of a compound solid while excluding hidden joined faces.
• Convert between units such as centimetres and metres, square centimetres and square metres, or cubic centimetres and cubic metres.
• Apply length, area and volume scale factors $k$, $k^2$ and $k^3$.
• Show a complete solution through formula, substitution, calculation, units, suitable rounding and a reasonableness check.