Year 11 General Shape And Measurement

Mensuration - Perimeter And Area

Q: What is the difference between perimeter and area?

Perimeter is the distance around the outside of a shape, measured in length units. Area is the surface the shape covers, measured in square units.

Q: What is the formula for the area of a rectangle?

Length times width. The perimeter is twice the length plus twice the width.

Q: What is the formula for the area of a triangle?

Half times base times perpendicular height. The height must be the vertical distance from the base to the opposite vertex, not the slanted side.

Q: What is the formula for the area of a trapezium?

Half times the sum of the two parallel sides, times the perpendicular height between them.

Q: How do you find the area of a compound shape?

Either split it into simple shapes like rectangles and triangles, add up each area; or compute the area of a bounding rectangle and subtract the unwanted parts.

Q: How do you convert between square metres and square centimetres?

One square metre equals 10,000 square centimetres because 1 metre equals 100 centimetres and the conversion is squared. To go from square metres to square centimetres, multiply by 10,000.

11 practice questions 2 video lessons Theory + worked examples

Theory

Perimeter is the total distance around the outside of a 2D shape, measured in length units. Area is the surface a shape covers, measured in square units. Each shape has its own formula; for compound shapes, split into pieces or subtract from a bounding rectangle. Watch unit conversions — $1$ m² = $10,000$ cm².

Perimeter is the total distance around the outside of a 2D shape — measured in length units (m, cm, mm). Area is the amount of surface a shape covers — measured in square units (m², cm², mm²).

The height in a triangle or parallelogram formula must be the perpendicular (vertical) height to the base — not the slanted side. The slanted side is generally longer and using it inflates the area.

For compound shapes (those made of rectangles, triangles, etc.), there are two common approaches: split the shape into simple pieces and add up their areas, or compute the area of a bounding rectangle and subtract the unwanted parts.

Unit conversions for area are squared. 1 m =100 cm, but 1 m² =100×100=10,000 cm². The conversion factor gets squared because area is two-dimensional.

Each shape has its own area formula

The perpendicular drops straight down, not along an edge

Square (side $s$ ):

A = s^{2}, P = 4 s

A = s^{2}

Rectangle (length $ℓ$ , width $w$ ):

A = ℓ \cdot w, P = 2 ℓ + 2 w

A = ℓ w

Triangle (base $b$ , perpendicular height $h$ ):

A = \frac{1}{2} \cdot b \cdot h

A = \frac{1}{2} b h

Parallelogram (base $b$ , perpendicular height $h$ ):

A = b \cdot h

A = b h

Trapezium (parallel sides $a$ and $b$ , perpendicular height $h$ ):

A = \frac{1}{2} (a + b) \cdot h

A = \frac{1}{2} (a + b) h

Common shapes at a glance:

Shape	Area	Perimeter
Square	$s^{2}$	$4 s$
Rectangle	$ℓ \cdot w$	$2 ℓ + 2 w$
Triangle	$\frac{1}{2} b h$	sum of 3 sides
Parallelogram	$b h$	sum of 4 sides
Trapezium	$\frac{1}{2} (a + b) h$	sum of 4 sides

Unit conversions are squared for area. 1 m² =10,000 cm². To go m² → cm², multiply by 10,000; cm² → m², divide by 10,000.

How to find the area or perimeter of any shape

Identify the shape — rectangle, triangle, parallelogram, trapezium, or a compound of these.
Read off the dimensions. For triangles and parallelograms, the height must be perpendicular to the base.
Apply the formula for area, or add up all the outer edges for perimeter.
For compound shapes, split into simple pieces and add their areas, or subtract from a bounding rectangle.
Check the units — area uses squared units (m², cm²).

Example 1 — Triangle area

A triangle has base

14

cm and perpendicular height

9

cm. Find its area.

Solution

Use $A = \frac{1}{2} b h$ .

$A$	$=$	$\frac{1}{2} \times 14 \times 9$
$A$	$=$	$63$ cm²

A = 63

The area is $63$ cm².

Example 2 — Trapezium area

A trapezium has parallel sides of

8

cm and

14

cm, and a perpendicular height of

6

cm. Find its area.

Solution

Use $A = \frac{1}{2} (a + b) h$ .

$A$	$=$	$\frac{1}{2} (8 + 14) \times 6$
$A$	$=$	$\frac{1}{2} \times 22 \times 6$
$A$	$=$	$66$ cm²

A = 66

The area is $66$ cm².

Example 3 — Compound (subtract)

A square plot of side

20

m has a square pond of side

6

m cut out of it. Find the area of the remaining lawn.

Solution

Lawn area = plot area − pond area.

$A_{plot}$	$=$	$20^{2} = 400$
$A_{pond}$	$=$	$6^{2} = 36$
$A_{lawn}$	$=$	$400 - 36 = 364$ m²

A_{lawn} = 364

The lawn area is $364$ m².

Example 4 — Fencing cost

A rectangular paddock is

80

m by

50

m. Fencing costs

$ 28

per metre. Find the total cost.

Solution

Find the perimeter, then multiply by the cost per metre.

$P$	$=$	$2 (80) + 2 (50)$
$P$	$=$	$160 + 100 = 260$ m
$cost$	$=$	$260 \times 28 = $ 7,280$

cost = 7280

The total fencing cost is $$7{,}280$ .

Common pitfalls

Using the slanted side as the height. The height in a triangle or parallelogram formula is the perpendicular drop from the base — not the length of a slanted edge. The slanted edge is longer and inflates the area.

Forgetting that area units are squared.

1

= 100

cm, but

1

m²

= 10,000

cm². Square the linear conversion factor.

Missing an edge on a compound shape's perimeter. Trace around the whole boundary — including any indented or cut-out edges. Use Pythagoras for any slanted sides if needed.

Confusing area and perimeter. Fencing costs scale with perimeter; turf, paint, and tiles scale with area. Read the problem carefully.

Frequently asked questions

What is the difference between perimeter and area?

Perimeter is the distance around the outside of a shape — a length. Area is the surface the shape covers — measured in square units.

What is the formula for the area of a rectangle?

$A = ℓ \cdot w$ (length times width). The perimeter is $2 ℓ + 2 w$ .

What is the formula for the area of a triangle?

$A = \frac{1}{2} \cdot b \cdot h$ . The height must be the perpendicular distance from the base to the opposite vertex, not the slanted side.

What is the formula for the area of a trapezium?

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

How do you find the area of a compound shape?

Split into simple shapes and add their areas, or compute the area of a bounding rectangle and subtract the unwanted parts.

How do you convert between square metres and square centimetres?

$1$ m² $= 10,000$ cm² because the linear conversion ( $\times 100$ ) is squared for area. m² to cm²: multiply by $10,000$ . cm² to m²: divide by $10,000$ .