Area and perimeter of 2D shapes (Grade 8) — Mathematics Grades 8–11

Calculating perimeter and area of polygons and circles, and converting units.

Gr 8 Term 3 — Area and perimeter of 2D shapesTerm 3

Practise Area and perimeter of 2D shapes in the app →

What gets asked

Calculate the perimeter or area of a square, rectangle or triangle.
Calculate the circumference or area of a circle.
Convert a measurement between square units.
Split a polygon into rectangles and triangles to find its area.

You must be able to

Choose the perimeter formula or the area formula, depending on what is asked.
Use the perpendicular height of a triangle, not a slanted side.
Use $π$ correctly in circumference and area formulae.
Convert between mm², cm², m² and km² using the correct squared factor.

Traps that cost marks

Leaving out the half when finding the area of a triangle. Area of a triangle is $\frac{1}{2} \times$ base $\times$ height. Do not forget the half.
Putting the diameter into $C = 2 π r$ , which doubles the circumference. $2 π r$ needs the radius. If you are given the diameter, halve it first, or use $C = π d$ .
Converting area units with the same factor as length units, e.g. treating $1 cm^{2}$ as $10 mm^{2}$ . Squaring the unit means squaring the factor too. $1 cm^{2} = 100 mm^{2}$ .
Calculating an area when the question is really asking for a perimeter, such as fencing. Fencing, edging and borders are perimeter questions. Covering or tiling questions are area.

Worked example

A rectangular school sports field is

30

m long and

20

m wide. Find its area.

Area of a rectangle $=$ length $\times$ width.
$30 \times 20 = 600$ .
Area $= 600$ m².

Perimeter and area

Fencing a plot and covering a plot are two different sums, with two different units.

perimeter: The distance all the way around the outside of a shape. It is a length.
area: The amount of flat space inside a shape. It is measured in squares.
perpendicular height: The height of a triangle measured at a right angle to its base, not along a slanted side.
decompose: To cut a shape into rectangles and triangles that do not overlap.

Perimeter is a walk round the edge, so add every side. A rectangle has two of each length, so double the sum of length and breadth. A square has four equal sides, so multiply one side by $4$ .

Area counts the squares that fit inside. A rectangle holds length times breadth of them. A triangle is half of the rectangle drawn around it, so its area is half of base times perpendicular height.

The unit tells you which one you found. Perimeter is a length: mm, cm, m or km. Area is squared, so its unit is squared too. An area answer with no square on its unit is a warning sign.

Converting area units is not the same as converting lengths. $1$ cm is $10$ mm, so one square centimetre is $10$ mm by $10$ mm. That makes $1 cm^{2} = 100 mm^{2}$ . Square the factor. In the same way $1 m^{2} = 10000 cm^{2}$ .

An odd shape has no formula of its own. Decompose it into rectangles and triangles, find each area, then add. Some inside lengths are not given. Work those out first, from the ones that are.

Rules to remember

rectangle: $P = 2 (l + b)$ and $A = l \times b$
triangle: $A = \frac{1}{2} bh$

Examples

A rectangular vegetable plot is

8

m long and

5

m wide. Find its perimeter and its area.

Worked answer

Perimeter: double the sum of the two measurements.
$2 \times (8 + 5) = 2 \times 13 = 26$ .
Area: multiply them instead.
$8 \times 5 = 40$ .
The perimeter is $26$ m and the area is $40$ m $^{2}$ .

Answer: $26$ m and $40$ m $^{2}$

One shape, two questions. The units keep the two answers apart.

A triangular flag has a base of

12

cm and a perpendicular height of

5

cm. Find its area in cm

^{2}

, and then in mm

^{2}

Worked answer

Half of base times perpendicular height.
$\frac{1}{2} \times 12 \times 5 = 30$ , so the area is $30$ cm $^{2}$ .
Now convert. One square centimetre holds $100$ square millimetres, not $10$ .
$30 \times 100 = 3000$ .

Answer: $30$ cm $^{2}$ , which is $3000$ mm $^{2}$

Both sides of every little square were converted, so the factor was used twice.

An L-shaped garden is

9

m along the bottom and

6

m up the left side. Its top right corner is missing:

4

m across and

3

m down. Find the area.

Worked answer

Decompose the L with one cut, straight down, into two rectangles.
The left one is $9 - 4 = 5$ m wide and $6$ m tall, so its area is $5 \times 6 = 30$ .
The right one is $4$ m wide and $6 - 3 = 3$ m tall.
Its area is $4 \times 3 = 12$ .
Add the two: $30 + 12 = 42$ .

Answer: $42$ m $^{2}$

The $5$ m and the $3$ m were never given. They had to be worked out first.

Traps

Adding only the two measurements given for a rectangle. A rectangle has four sides. Double the sum: $2 \times (8 + 5) = 26$ .
Leaving out the half when finding the area of a triangle. Area of a triangle is $\frac{1}{2} \times$ base $\times$ height. Do not forget the half.
Converting area units with the same factor as length units, e.g. treating $1 cm^{2}$ as $10 mm^{2}$ . Squaring the unit means squaring the factor too. $1 cm^{2} = 100 mm^{2}$ .
Adding up every length written on an L-shape as though one rectangle had those sides. Cut the shape into rectangles first. Find any missing inside length before you multiply.

Learn and practise “Perimeter and area” in the app →

Circles and pi

Every circle in the world hides the same number, and that number never finishes.

radius: The distance from the centre of a circle to its edge.
diameter: The distance right across a circle, through the centre. It is twice the radius.
circumference: The perimeter of a circle: the distance all the way round it.
pi: The circumference of any circle divided by its diameter. The answer is always about $3, 14$ .

Three words describe a circle. The centre is the middle point. The radius runs from the centre out to the edge. The diameter runs right across through the centre, so it is twice the radius. Most questions give you the diameter, so halve it first.

Take any circle. Divide its circumference by its diameter. The same number comes out for a bottle top and for a dam. That number is pi, written $π$ . It is irrational: its decimals run on for ever and never repeat. So $3, 14$ and $\frac{22}{7}$ are close, never exact.

The circumference is $C = 2 π r$ . Writing $C = π d$ says the same thing, because the diameter is two radii. The area is $A = π r^{2}$ . Square the radius, and do not swap the two formulas around.

Read the question before you choose one. A fence, an edging or a border asks for the circumference. Paint, tiles or grass ask for the area. Put every measurement into the same unit before you start.

Rules to remember

circle: $C = 2 π r$ , or $C = π d$
circle: $A = π r^{2}$

Examples

A round table top has a radius of

50

cm. Find its circumference. Use

π = 3, 14

Worked answer

The radius is given, so use $C = 2 π r$ .
$2 \times 3, 14 \times 50 = 314$ .
The circumference is $314$ cm.

Answer: $314$ cm

Using $3, 14$ makes the answer close, not exact. Pi has no last digit.

A round flower bed has a diameter of

6

m. Find its area. Use

π = 3, 14

Worked answer

Halve the diameter to get the radius: $6 \div 2 = 3$ .
The formula is $A = π r^{2}$ , so square the radius: $3^{2} = 9$ .
$3, 14 \times 9 = 28, 26$ .
The area is $28, 26$ m $^{2}$ .

Answer: $28, 26$ m $^{2}$

Putting the $6$ in without halving it gives four times too much.

A round farm dam has a radius of

10

m. A farmer fences right round it, and fencing costs R

45

a metre. What does the fence cost? Use

π = 3, 14

Worked answer

A fence runs round the edge, so this needs the circumference.
$2 \times 3, 14 \times 10 = 62, 8$ .
So $62, 8$ m of fencing is needed.
Cost: $62, 8 \times 45 = 2826$ .
The fence costs R $2826$ .

Answer: R $2826$

The word fence decided it. Finding the area here would buy nothing.

Traps

Putting the diameter into $C = 2 π r$ , which doubles the circumference. $2 π r$ needs the radius. If you are given the diameter, halve it first, or use $C = π d$ .
Writing that pi is exactly $3, 14$ , or exactly $\frac{22}{7}$ . Both are only close. Pi is irrational, so no decimal and no fraction lands on it.
Squaring the diameter instead of the radius when finding the area. Halve the diameter first. A diameter of $6$ m gives a radius of $3$ m.
Calculating an area when the question is really asking for a perimeter, such as fencing. Fencing, edging and borders are perimeter questions. Covering or tiling questions are area.

Learn and practise “Circles and pi” in the app →

Grade 8 Area and perimeter of 2D shapes

What gets asked

You must be able to

Traps that cost marks

Worked example

Perimeter and area

Rules to remember

Examples

Traps

Circles and pi

Rules to remember

Examples

Traps

About this material