Perimeter, Area and Volume

Topic Overview

The 'perimeter' of a shape is the distance around it. In order to calculate the perimeter of a shape, you must add up the lengths of all its sides. For example, if a rectangle has a width of 5cm and a length of 3cm, its perimeter would be:

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The 'area' of a shape is the number of square units which cover it, i.e. the size of the surface of a shape.

Due to the fact that the area of a shape is calculated by multiplying a shape's length by its width, it is measured in 'square units' .For example, the area of a square which is 1 metre on each side is 1 metre x 1 metre = 1 square metre or m2.

Other examples of square units include: millimetres squared (mm2) and centimetres squared (cm2).

For example, if a rectangle has a width of 5cm and a length of 3cm, its area would be:

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There are several shapes which follow simple area formulae:

The area of a triangle = 1/2 x base x height

The area of a rectangle = base x height

The area of a parallelogram = base × height

The 'volume' of a shape is the number of cubic units which occupy it, i.e. the amount of 3D space which the shape occupies.

Due to the fact that the volume of a shape is calculated by multiplying a shape's length by its width by its depth, it is measured in 'cubic units'. For example, the volume of a square which is 1 metre in length, 1 metre in width and 1 metre in depth is 1 metre x 1 metre x 1metre = 1 cubic metre or m3.

Other examples of cubic units include: millimetres cubed (mm3) and centimetres cubed (cm3).

For example, if a cuboid has a width of 5cm, a length of 3cm and a depth of 2cm, its volume would be:

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Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems involving perimeter, area and volume. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

  • Calculate the perimeter of various shapes
  • Calculate the area of various shapes
  • Calculate the volume of various shapes

Listed below are a series of summaries and worked examples to help you solidify your knowledge about perimeter, area and volume.

Worked Examples

1 - Calculating the area of a trapezium
A trapezium is a 4-sided shape with straight sides and a pair of parallel sides. In order to calculate the area of a trapezium, you must follow the rule:

Area of a trapezium =

\[\frac{{a + b}}{2}*h\]

Where 'a' and 'b' are the two side lengths of the trapezium and 'h' is its height.

Example
(a) - Calculate the area of the following trapezium:

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Solution
(a) - Using the formula for the area of a trapezium, you can calculate that:

Area =

\[\frac{{7 + 11}}{2}*5 = 9*5 = 45\]

Therefore the area of the trapezium =

\[45c{m^2}\]

2 - Calculating the area of a circle
During your GCSE maths exam, you will be required to calculate the area of a circle. In order to calculate a circle's area, you need to know the values of some of its parts. For example,

If you know the radius of a circle, you can calculate its area using the formula:

\[Area = \pi *radiu{s^2}\]

If you know the diameter of a circle, you can calculate its area using the formula:

\[Area = \frac{\pi }{4}*diamete{r^2}\]

If you know the circumference of a circle, you can calculate its area using the formula:

\[Area = circumferenc{e^2}/4\pi \]

The 'radius' of a circle is the distance from its centre to its edge.

The 'diameter' of a circle is the distance from one edge of a circle through its centre to the edge on the other side of the circle.

The 'circumference' of a circle is the distance around its edge.

(Note: When you divide the circumference of a circle by its diameter you get 3.141592654... which is the value of Pi (π)).

Example
(a) - Calculate the area of a circle which has a radius of 4cm

Solution
(a) - From the question, you know that the circle has a radius of 4cm. As a result you can use the formula

\[Area = \pi *{4^2} = 16\pi = 50.265482....\]

to calculate its area:

Therefore, the area of a circle to 2dp, with radius 4cm =

\[50.27c{m^2}\]

(Note: Always remember to present your answer using the correct units of measurement and approximated to a suitable degree of accuracy)

3-Calculating the lengths of arcs and the areas of sectors
In order to calculate the length of an arc or the area of a sector, you must calculate the value of the angle which is made by the arc or sector at the centre of a circle.

For example, if the angle is a right angle (90°), then the arc in question is a quarter of the circumference of a circle and that sector area is a quarter of the area of the circle.

Therefore, you can use the formulae for the circumference of circle and the area of a circle in order to calculate the length of the arc and the area of the sector respectively:

\[Circumference = 2*\pi *radius\]
\[Area = \pi *radiu{s^2}\]

Example
(a) - Calculate the length of the arc and the area of the sector of a circle which has a radius of 4cm.

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Solution
(a) - The angle which is made by the arc or sector at the centre of the circle is a right angle (90°).

Therefore, you can calculate the length of the arc and area of the sector using the formulae mentioned below:

Arc length = 1/4 x circumference of the circle

Sector area = 1/4 x area of the circle

Therefore:

Arc length = 1/4 x (2 x π x 4) = 8 π / 4 = 2π = 6.2831...

Sector area = 1/4 x (π x 42) = 16 π / 4 = 4 π = 12.5663....

Therefore, the length of the arc and the area of the sector to 2dp are,

\[6.28cm\]
\[12.57c{m^2}\]

As a rule, you can calculate the length of an arc and the area of a sector by finding the angle which is made by the arc or sector at the centre of the circle and calculating what proportion it is of a whole turn (360°).

Once you know this proportion, you can multiply the circumference of the circle and the area of the circle by this proportion in order to calculate the arc length and sector area respectively:

Example
(a) - Calculate the length of the arc and the area of the sector

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Solution
(a) - First you need to calculate the proportion of this sector in relation to a whole rotation. You can do so by dividing the angle which is made by the arc or sector at the centre of the circle by 360°:

360/ 40 = 9

Therefore, the sector is 1/9 of the entire circle. Using this information, you can calculate the arc length and sector area:

Arc length = 1/9 x (2 x π x 10) = 20 π / 9 = 6.981...

Sector area = 1/9 x (π x 102) = 100 π / 9 = 34.906...

Therefore the length of the arc and the area of the sector to 2dp are

\[6.98cm\]
\[34.91c{m^2}\]

4 - Calculating the volume of a cone
A cone is a shape which has a flat base, one curved side, and a curved surface. During your GCSE maths exam, you may be required to calculate the volume of a cone. You can do so by using the following formula:

Volume of a cone =

\[\pi *radiu{s^2}*(\frac{{height}}{3})\]

Example
(a) - Calculate the volume of the following cone:

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Solution
(a) - Using the formula mentioned above, you can calculate the volume of the cone:

\[Volume = \pi *{(8)^2}*(5/3) = 64\pi *(5/3)\]

Volume = 335. 10321....

Therefore the volume of the cone to 1 dp =

\[335.1c{m^3}\]

5 - Calculating the volume of a cylinder
A cylinder has a flat base, a flat top and one curved side. The base of a cylinder is the same as the top of the cylinder.

During your GCSE maths exam, you may be required to calculate the volume of a cylinder. You can do so by using the following formula:

Volume of a cylinder=

%\[\pi *radiu{s^2}*height\]

The volume of a cylinder is very similar to the volume of a cone. The exception is that, if a cone and cylinder share the same radius, then the volume of the cylinder will be three times larger than the volume of the cone.

Example
(a) - Calculate the volume of the following cylinder:

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Solution
(a) - The diagram tells us the radius and height of the cylinder. As a result, you can use the formula mentioned earlier to calculate the cylinder's volume:

Volume of cylinder=

\[\pi *{(8)^2}*15 = \pi *960 = 3015.92894745\]

Therefore the volume of the cylinder =

\[3015.93c{m^3}\]

6 - Calculating the volume of a sphere
A sphere is a perfectly symmetrical shape with no edges or corners. All of the points on the surface of a sphere are the same distance from the centre of the sphere.

During your GCSE maths exam, you may be required to calculate the volume of a sphere. You can do so by using the following formula:

Volume of a sphere =

\[(4/3)*\pi *radiu{s^2}\]

Example
(a) - Calculate the volume of the following sphere:

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Solution
(a) - From the diagram, you know that the diameter of the sphere is 30cm. Given that:

Diameter of a circle = 2 x radius of a circle

You know that the radius of the sphere = 30/2 = 15cm.

Now that you know the radius of the sphere, you can use the formula mentioned earlier to calculate its volume:

Volume of sphere =

\[(4/3)*\pi *{(15)^2} = 14137.1669412\]

Therefore the volume of the sphere to 2dp =

\[14137.17c{m^3}\]

7 - Calculating the volume of a pyramid
A pyramid is a shape with triangular outer surfaces which converge to a single point at the top. The base of a pyramid can be any shape; i.e. a triangle, square or pentagon.

During your GCSE maths exam, you may be required to calculate the volume of a pyramid. You can do so by using the following formula:

Volume of a pyramid = 1/3 × [Base Area] × Height

Example
(a) - Calculate the volume of the following pyramid:

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Solution
(a) - The base of the pyramid is a square. Therefore, its area =

\[6*6 = 36{m^2}\]

Now that you know the value of the base area, you can calculate the volume of the pyramid:

Volume of the pyramid = 1/3 × 36 × 15 = 540/3 = 180

Therefore the volume of the pyramid =

\[180{m^3}\]

8 - Calculating the volume of a prism
A prism is a shape which has the same cross section all along its length. The volume of a prism is the area of one of the prism's sides times the length of the prism:

Volume = [Base Area] x Length

Example
(a) - Calculate the volume of the following prism:

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Solution
(a) - First you need to calculate the area of the base of the prism.

The base is a triangle so its area

\[\frac{1}{2}*base*height = \frac{1}{2}*3*6 = \frac{{18}}{2} = 9c{m^2}\]

Now that you know the area of the base, you can calculate the volume of the prism:

Volume of prism =

\[9*10 = 90c{m^3}\]

Exam Tips

  1. Make sure you memorise the individual formulae for the shapes mentioned in the worked examples.
  2. Remember that perimeter means the distance around a shape.
  3. Remember that area is the size of the surface of a shape, and that area is measured in square units.
  4. Remember that volume is the amount of 3D space which a shape occupies, and that volume is measured in cubic units.
  5. When calculating the perimeters, areas and volumes of various shapes, make sure you write down ALL of your working out.

Topic Summary

Calculating the perimeter, area and volume of various shapes requires you to recognise and use the properties of numerous different shapes. Therefore, it is advisable that you revise the 'Polygons' and 'Circle Theorem Problems' topics in order to refresh your knowledge of the properties of these various shapes.

Furthermore, when calculating the perimeters, areas and volumes of these shapes, it is strongly advised that you write down ALL of your working out. This enables you to double check that you have used the correct formulae, that you have approximated to a suitable degree of accuracy, and that you have used the correct units of measurements. By doing so, you can earn the maximum amount of marks for this topic during your actual GCSE exam.

Related Topics

  • Polygons
  • Congruent Triangles
  • Trigonometry and Pythagoras
  • Construction and Loci Problems
  • Transformations
  • Circle Theorem Problems
  • Proof