Area and perimeter of 2D shapes (Grade 9) — Mathematics Grades 8–11

Calculating the perimeter and area of polygons and circles, including composite figures.

Gr 9 Term 2 — Area and perimeter of 2D shapesTerm 2

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What gets asked

Calculate the area of a triangle, parallelogram, rhombus, kite or trapezium.
Calculate the circumference or area of a circle.
Calculate the area or perimeter of a composite figure made of simple shapes.
Find an unknown side length given the area or perimeter.

You must be able to

Use the perpendicular height, not a slanted side, in an area formula.
Use pi as approximately 3,14 correctly in circumference and area formulas.
Break a composite figure into simple shapes before calculating.
Substitute into the correct formula and solve for the missing length.

Traps that cost marks

Using a triangle's slanted side as its height. The height must be perpendicular to the base, at a right angle, not a slanted side.
Substituting the diameter into the circle area formula without halving it first. Always halve the diameter to get the radius before using the area formula.
Including internal lines inside a composite figure as part of the perimeter. Perimeter is only the outside boundary. Leave out any lines drawn inside the figure.

Worked example

A rectangular school yard is 12 m by 8 m. A square vegetable bed with sides of 3 m is marked out inside it. Calculate the area of the yard that is left.

Area of the yard: $12 \times 8 = 96$ m².
Area of the vegetable bed: $3 \times 3 = 9$ m².
Area left: $96 - 9 = 87$ m².

Perimeter, area and units

How far around a shape, how much space inside it, and which units each one uses.

perimeter: The total distance 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 measured at a right angle to the base, not along a slanted side.
radius: The distance from the centre of a circle to its edge. It is half the diameter.
circumference: The perimeter of a circle: the distance all the way around it.

Perimeter is a length, so it is measured in mm, cm, m or km. Area is measured in squares, so its units are squared too. If your answer is an area and your unit has no square on it, something has gone wrong.

A rectangle's area is length times breadth. A triangle's is half of base times perpendicular height. The half is there because a triangle is half of the rectangle around it.

A parallelogram's area is base times perpendicular height, the same as a rectangle. Do not multiply the two side lengths: the slanted side is longer than the height.

For a circle, the circumference is $2 π r$ and the area is $π r^{2}$ . Questions usually give the diameter, so halve it first.

Converting area units is not the same as converting lengths. $1$ cm is $10$ mm, but a square centimetre is $10$ mm by $10$ mm, so $1$ cm $^{2}$ is $100$ mm $^{2}$ . Square the conversion factor.

Rules to remember

triangle: $A = \frac{1}{2} bh$
circle: $C = 2 π r$ and $A = π r^{2}$

Examples

A triangle has a base of

10

cm and a perpendicular height of

6

cm. Find its area.

Worked answer

Use $A = \frac{1}{2} bh$ .
Substitute: $A = \frac{1}{2} \times 10 \times 6$ .
$\frac{1}{2} \times 10 \times 6 = 30$ .
The area is 30 cm squared.

Answer: $30$ cm $^{2}$

Leaving out the $\frac{1}{2}$ would give $60$ , the area of the rectangle around it.

A circular tin lid has a diameter of

14

cm. Find its area, leaving your answer in terms of

π

Worked answer

The diameter is $14$ cm, so the radius is $7$ cm.
Use $A = π r^{2}$ .
Substitute the radius, not the diameter: $A = π \times 7^{2}$ .
$7^{2} = 49$ , so the area is $49 π$ cm squared.

Answer: $49 π$ cm $^{2}$

Putting $14$ into the formula gives $196 π$ , four times too big.

A rectangle measures

12

cm by

5

cm. Give its area in mm

^{2}

Worked answer

Area in cm squared: $12 \times 5 = 60$ .
Now convert. $1$ cm $^{2}$ is $100$ mm $^{2}$ , not $10$ .
So multiply by $100$ : $60 \times 100 = 6000$ .
The area is 6 000 mm squared.

Answer: $6000$ mm $^{2}$

Both sides of the square are converted, so the factor is used twice.

Traps

Using a slanted side of a triangle as its height. The height must meet the base at a right angle. A slanted side is longer, so it gives too big an area.
Multiplying the two side lengths of a parallelogram. Use base times perpendicular height, exactly as for a rectangle.
Putting the diameter into $A = π r^{2}$ without halving it. Halve the diameter first. A diameter of $14$ cm gives a radius of $7$ cm.
Treating $1$ cm $^{2}$ as $10$ mm $^{2}$ . Square the length factor. $1$ cm $^{2} = 100$ mm $^{2}$ and $1$ m $^{2} = 10000$ cm $^{2}$ .

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

Composite figures and scaling

Shapes built out of other shapes, working a missing side out backwards, and what doubling really does.

composite figure: A shape built from simpler shapes joined together, or with a piece taken out.
boundary: The outside edge of a figure. It is the line you would walk along.
dimension: One of the measurements of a shape, such as its length, its breadth or its height.

Break a composite figure into rectangles, triangles or parts of circles. Work out each piece on its own, then add the pieces up. If a piece has been taken OUT, subtract it instead of adding it.

Perimeter is the boundary and nothing else. A line you drew inside the figure to split it up is not part of the boundary, so it never goes into the perimeter.

Sometimes the area is given and a side is missing. Put what you know into the formula and solve for the side. For a rectangle that means dividing the area by the side you have.

For a rectangle, half the perimeter is the length PLUS the breadth. It is not the length. Take the side you know away from that half to get the other side.

Doubling does different things to perimeter and to area. Double every dimension and the perimeter doubles, but the area becomes four times as big. Double one dimension only and the area doubles instead.

Rules to remember

rectangle: $A = l \times b$
double every dimension: perimeter doubles, area becomes $4$ times bigger

Examples

A vegetable plot is

9

m by

5

m. A square water tank of side

2

m stands in one corner of it. Find the area left for planting.

Worked answer

Area of the whole plot: $9 \times 5 = 45$ square metres.
Area under the tank: $2 \times 2 = 4$ square metres.
The tank stands ON the plot, so take its area away.
$45 - 4 = 41$ square metres are left.

Answer: $41$ m $^{2}$

A piece that has been taken out is subtracted, never added on.

A triangle has an area of

24

^{2}

and a base of

8

cm. Find its perpendicular height.

Worked answer

Start from $A = \frac{1}{2} bh$ , with the area $24$ and the base $8$ .
$\frac{1}{2} \times 8 = 4$ , so $4 h = 24$ .
Divide both sides by $4$ : $h = 6$ .
The perpendicular height is 6 cm.

Answer: $6$ cm

Just halving the area would give $12$ cm, which is twice too big.

A rectangle is

8

cm by

3

cm. Only its length is doubled. What happens to the perimeter and the area?

Worked answer

Before: $2 \times (8 + 3) = 22$ cm round, and $8 \times 3 = 24$ square centimetres inside.
The length doubles to $16$ cm, while the breadth stays $3$ cm.
New perimeter: $2 \times (16 + 3) = 38$ cm, and double $22$ would have been $44$ .
New area: $16 \times 3 = 48$ , which IS double $24$ .

Answer: the area doubles; the perimeter grows from $22$ cm to $38$ cm

Only one of the two sides grew, so the area doubled and the perimeter did not.

Traps

Including internal lines inside a composite figure as part of the perimeter. Perimeter is only the outside boundary. Leave out any lines drawn inside the figure.
Adding on the area of a piece that was taken out. A piece that has been removed is subtracted. Work out the whole, then take the hole away.
Halving the perimeter of a rectangle and calling that the length. Half the perimeter is the length PLUS the breadth. Subtract the side you know from it.
Assuming that doubling every dimension doubles the area. Both sides are doubled, so the area comes out four times bigger, not twice.

Learn and practise “Composite figures and scaling” in the app →

Grade 9 Area and perimeter of 2D shapes

What gets asked

You must be able to

Traps that cost marks

Worked example

Perimeter, area and units

Rules to remember

Examples

Traps

Composite figures and scaling

Rules to remember

Examples

Traps

About this material