Map Scales and Area

Why the area scale of a map is the square of its length scale, and how to find an actual area from a map or plan, a map area from an actual one, and the scale from a pair of areas.

Before you start

The scale $1 : 50\,000$ means that 1 cm on the map represents $50\,000$ cm on the ground, and $50\,000\text{ cm} = 500\text{ m} = 0.5\text{ km}$ (lesson 7.1).
The area of a square of side $a$ is $a \times a = a^{2}$ , and the side of a square of area $A$ is $\sqrt{A}$ : a square of area $36\text{ cm}^{2}$ has sides of $\sqrt{36} = 6$ cm.

Why the Area Scale Is the Square of the Length Scale

On a map with the scale $1 : 50\,000$ , 1 cm represents 0.5 km. Draw a square 1 cm by 1 cm on the map: its area is $1\text{ cm}^{2}$ . Each of its sides stands for 0.5 km, so it stands for a square 0.5 km by 0.5 km on the ground.

The square on the ground has an area of $0.5 \times 0.5 = 0.25\text{ km}^{2}\text{.}$ So $1\text{ cm}^{2}$ on the map represents $0.25\text{ km}^{2}$ , not $0.5\text{ km}^{2}$ . A length is changed by the scale once; an area is a length times a length, so it is changed twice.

The same is true in centimetres. The square on the ground is $50\,000$ cm by $50\,000$ cm, so its area is $50\,000 \times 50\,000 = 2\,500\,000\,000\text{ cm}^{2}\text{.}$ The two areas are in the ratio $1 : 2\,500\,000\,000\text{,}$ which is $1 : (50\,000)^{2}$ .

A shape that is not a square can be covered by small squares, each changed in the same way, so the rule holds for a shape of any kind.

Area scale

If 1 cm on a map represents $k$ km, then $1\text{ cm}^{2}$ on the map represents $k^{2}\text{ km}^{2}$ . The same holds in any unit: if 1 cm represents $k$ m, then $1\text{ cm}^{2}$ represents $k^{2}\text{ m}^{2}$ .

If the scale of a map is $1 : n$ , then map area : actual area $= 1 : n^{2}$ , with both areas in the same unit.

So, with 1 cm representing $k$ km, actual area (in $\text{km}^{2}$ ) = map area (in $\text{cm}^{2}$ ) × $k^{2}$ , and map area = actual area ÷ $k^{2}$ .

The table shows squares on the $1 : 50\,000$ map. Doubling the side multiplies the area by 4, and tripling it multiplies the area by 9, but the ground area divided by the map area is 0.25 every time.

Side on the map (cm)	$1$	$2$	$3$
Side on the ground (km)	$0.5$	$1$	$1.5$
Area on the map ( $\text{cm}^{2}$ )	$1$	$4$	$9$
Area on the ground ( $\text{km}^{2}$ )	$0.25$	$1$	$2.25$
Ground area ÷ map area	$0.25$	$0.25$	$0.25$

Example 1: The area scale

A map has the scale $1 : 20\,000$ . (a) Write the scale in the form 1 cm to $k$ km. (b) What area does $1\text{ cm}^{2}$ on the map represent? (c) Write the ratio map area : actual area.

Solution

(a) 1 cm represents $20\,000$ cm.

$20\,000\text{ cm} = 20\,000 \div 100\text{ m} = 200\text{ m}\text{,}$ and $200\text{ m} = 200 \div 1000\text{ km} = 0.2\text{ km}\text{.}$

(b) $1\text{ cm}^{2}$ represents $0.2 \times 0.2 = 0.04\text{ km}^{2}\text{.}$

(a) 1 cm to 0.2 km (b) $0.04\text{ km}^{2}$ (c) $1 : 400\,000\,000$

Units of Area and Hectares

Land is measured in square metres ( $\text{m}^{2}$ ), square kilometres ( $\text{km}^{2}$ ) or hectares (ha). A hectare is the area of a square 100 m by 100 m. Each change of unit for areas is the square of the change for lengths: 1 m = 100 cm, so $1\text{ m}^{2} = 100 \times 100 = 10\,000\text{ cm}^{2}\text{.}$

Units of area

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

$1\text{ ha} = 100\text{ m} \times 100\text{ m} = 10\,000\text{ m}^{2}$

$1\text{ km}^{2} = 1000\text{ m} \times 1000\text{ m} = 1\,000\,000\text{ m}^{2}$

$1\text{ km}^{2} = 1\,000\,000 \div 10\,000\text{ ha} = 100\text{ ha}$

Example 2: Changing units of area

Write (a) $0.04\text{ km}^{2}$ in $\text{m}^{2}$ and in hectares, (b) $35\,000\text{ m}^{2}$ in hectares, (c) 2.5 ha in $\text{km}^{2}$ .

Solution

(a) $0.04\text{ km}^{2} = 0.04 \times 1\,000\,000\text{ m}^{2} = 40\,000\text{ m}^{2}$

$40\,000\text{ m}^{2} = 40\,000 \div 10\,000\text{ ha} = 4\text{ ha}\text{.}$ Check: $0.04 \times 100 = 4$ ha.

(b) $35\,000\text{ m}^{2} = 35\,000 \div 10\,000\text{ ha} = 3.5\text{ ha}$

(a) $40\,000\text{ m}^{2}$ , or 4 ha (b) 3.5 ha (c) $0.025\text{ km}^{2}$

Actual Area from a Map Area

Find what 1 cm represents, square it to find what $1\text{ cm}^{2}$ represents, then multiply by the map area.

Example 3: A reservoir

On a map with the scale $1 : 50\,000$ , a reservoir covers $12\text{ cm}^{2}$ . Find its actual area in $\text{km}^{2}$ and in hectares.

Solution

1 cm represents $50\,000\text{ cm} = 500\text{ m} = 0.5\text{ km}\text{.}$

$1\text{ cm}^{2}$ represents $0.5 \times 0.5 = 0.25\text{ km}^{2}\text{.}$

$12\text{ cm}^{2}$ represents $12 \times 0.25 = 3\text{ km}^{2}$ .

$3\text{ km}^{2} = 3 \times 100\text{ ha} = 300\text{ ha}$

The reservoir covers $3\text{ km}^{2}$ , which is 300 ha.

Do not use the length scale for an area

In Example 3, $12 \times 0.5 = 6\text{ km}^{2}$ is wrong: 0.5 km is what 1 cm represents, and an area needs what $1\text{ cm}^{2}$ represents, $0.25\text{ km}^{2}$ .

Example 4: A park

A park covers $45\text{ cm}^{2}$ on a map with the scale $1 : 2000$ . Find its actual area in $\text{m}^{2}$ and in hectares.

Solution

1 cm represents 2000 cm, and $2000\text{ cm} = 2000 \div 100\text{ m} = 20\text{ m}\text{.}$

$1\text{ cm}^{2}$ represents $20 \times 20 = 400\text{ m}^{2}$ .

$45\text{ cm}^{2}$ represents $45 \times 400 = 18\,000\text{ m}^{2}\text{.}$

$18\,000\text{ m}^{2} = 18\,000 \div 10\,000\text{ ha} = 1.8\text{ ha}$

The park covers $18\,000\text{ m}^{2}$ , which is 1.8 ha.

Map Area from an Actual Area

Divide the actual area by what $1\text{ cm}^{2}$ represents, with both in the same unit.

Example 5: Sentosa

Sentosa has an area of about $5\text{ km}^{2}$ . About what area does it cover on a map with the scale $1 : 25\,000$ ?

Solution

1 cm represents $25\,000\text{ cm} = 250\text{ m} = 0.25\text{ km}\text{.}$

$1\text{ cm}^{2}$ represents $0.25 \times 0.25 = 0.0625\text{ km}^{2}\text{.}$

Map area $= 5 \div 0.0625 = 80\text{ cm}^{2}\text{.}$ Check: $80 \times 0.0625 = 5$ .

Sentosa covers about $80\text{ cm}^{2}$ on the map.

Finding the Scale from Two Areas

Work backwards: find what $1\text{ cm}^{2}$ represents, take the square root to find what 1 cm represents, and change it to centimetres.

Example 6: The scale from an area

On a map, an area of $72\text{ km}^{2}$ is shown as $18\text{ cm}^{2}$ . Find the scale of the map in the form $1 : n$ .

Solution

$1\text{ cm}^{2}$ represents $72 \div 18 = 4\text{ km}^{2}$ .

So 1 cm represents $\sqrt{4} = 2$ km: a square 2 km by 2 km has an area of $4\text{ km}^{2}$ .

$2\text{ km} = 2000\text{ m} = 2000 \times 100\text{ cm} = 200\,000\text{ cm}$

The scale is $1 : 200\,000$ .

Composite Shapes on a Plan

For a shape made of rectangles and triangles there are two ways. Find its area on the plan and change it with the area scale; or change each length with the length scale and find the area on the ground. Both give the same answer, so one checks the other.

Example 7: An L-shaped park

The plan of a neighbourhood park is drawn to the scale $1 : 500$ . All its corners are right angles, and its lengths on the plan are shown. Find the actual area of the park in $\text{m}^{2}$ and in hectares.

Solution

Method 1: the area on the plan first. A is 7 cm by 8 cm and B is $12 - 7 = 5$ cm by 5 cm.

Area on the plan $= 7 \times 8 + 5 \times 5 = 56 + 25 = 81\text{ cm}^{2}$

1 cm represents 500 cm, and $500\text{ cm} = 500 \div 100\text{ m} = 5\text{ m}\text{,}$ so $1\text{ cm}^{2}$ represents $5 \times 5 = 25\text{ m}^{2}$ .

Actual area $= 81 \times 25 = 2025\text{ m}^{2}$

Method 2: the lengths first. A is $7 \times 5 = 35$ m by $8 \times 5 = 40$ m; B is $5 \times 5 = 25$ m by 25 m.

Actual area $= 35 \times 40 + 25 \times 25 = 1400 + 625 = 2025\text{ m}^{2}\text{,}$ as before.

$2025\text{ m}^{2} = 2025 \div 10\,000\text{ ha} = 0.2025\text{ ha}$

The park covers $2025\text{ m}^{2}$ , which is 0.2025 ha.

Common Mistakes

Using the length scale for an area. With 1 cm to 0.5 km, $12\text{ cm}^{2}$ represents $12 \times 0.25 = 3\text{ km}^{2}$ , not $12 \times 0.5 = 6\text{ km}^{2}$ .
Doubling instead of squaring: the area ratio for $1 : 50\,000$ is $1 : 2\,500\,000\,000\text{,}$ not $1 : 100\,000$ .
Changing units of area by the length factor: $1\text{ m}^{2} = 10\,000\text{ cm}^{2}$ (not 100), $1\text{ km}^{2} = 1\,000\,000\text{ m}^{2}$ (not 1000), and $1\text{ ha} = 10\,000\text{ m}^{2}\text{.}$
Forgetting the square root when finding a scale: if $1\text{ cm}^{2}$ represents $4\text{ km}^{2}$ , then 1 cm represents 2 km, not 4 km.

Practice Questions

Try each question before you open its solution, and write every change of unit in full.

Question 1

A map has the scale $1 : 10\,000$ . (a) What does 1 cm on the map represent, in metres? (b) What does $1\text{ cm}^{2}$ represent, in $\text{m}^{2}$ and in hectares?

ShowHide solution

(a) $10\,000\text{ cm} = 10\,000 \div 100\text{ m} = 100\text{ m}$

(b) $1\text{ cm}^{2}$ represents $100 \times 100 = 10\,000\text{ m}^{2}\text{,}$ and $10\,000\text{ m}^{2} = 1\text{ ha}\text{.}$

(a) 100 m (b) $10\,000\text{ m}^{2}$ , which is 1 ha

Question 2

Write (a) $3.5\text{ km}^{2}$ in hectares, (b) $45\,000\text{ m}^{2}$ in hectares, (c) 2.4 ha in $\text{m}^{2}$ , (d) 80 ha in $\text{km}^{2}$ .

ShowHide solution

(a) $3.5 \times 100 = 350$ ha

(b) $45\,000 \div 10\,000 = 4.5$ ha

(d) $80 \div 100 = 0.8\text{ km}^{2}$

(a) 350 ha (b) 4.5 ha (c) $24\,000\text{ m}^{2}$ (d) $0.8\text{ km}^{2}$

Question 3

The scale of a map is 1 cm to 3 km. A forest covers $7\text{ cm}^{2}$ on the map. Find its actual area.

ShowHide solution

$1\text{ cm}^{2}$ represents $3 \times 3 = 9\text{ km}^{2}$ .

$7\text{ cm}^{2}$ represents $7 \times 9 = 63\text{ km}^{2}$ .

The forest covers $63\text{ km}^{2}$ .

Question 4

On a map with the scale $1 : 25\,000$ , a nature reserve covers $16\text{ cm}^{2}$ . Find its actual area in $\text{km}^{2}$ and in hectares.

ShowHide solution

1 cm represents $25\,000\text{ cm} = 250\text{ m} = 0.25\text{ km}\text{.}$

$1\text{ cm}^{2}$ represents $0.25 \times 0.25 = 0.0625\text{ km}^{2}\text{.}$

$16 \times 0.0625 = 1\text{ km}^{2}\text{,}$ and $1\text{ km}^{2} = 100$ ha.

$1\text{ km}^{2}$ , which is 100 ha

Question 5

A school field has an area of 1.5 ha. What area does it cover on a plan with the scale $1 : 1000$ ?

ShowHide solution

1 cm represents 1000 cm, and $1000\text{ cm} = 1000 \div 100\text{ m} = 10\text{ m}\text{.}$

$1\text{ cm}^{2}$ represents $10 \times 10 = 100\text{ m}^{2}$ .

$1.5\text{ ha} = 1.5 \times 10\,000\text{ m}^{2} = 15\,000\text{ m}^{2}$

Area on the plan $= 15\,000 \div 100 = 150\text{ cm}^{2}$

$150\text{ cm}^{2}$

Question 6

A 4-room HDB flat has a floor area of $92\text{ m}^{2}$ . What is its area on a floor plan with the scale $1 : 50$ ?

ShowHide solution

1 cm represents 50 cm, which is 0.5 m.

$1\text{ cm}^{2}$ represents $0.5 \times 0.5 = 0.25\text{ m}^{2}\text{.}$

Area on the plan $= 92 \div 0.25 = 368\text{ cm}^{2}\text{.}$ Check: $368 \times 0.25 = 92$ .

$368\text{ cm}^{2}$

Question 7

A reservoir with an area of $3.2\text{ km}^{2}$ covers $12.8\text{ cm}^{2}$ on a map. Find the scale of the map in the form $1 : n$ .

ShowHide solution

$1\text{ cm}^{2}$ represents $3.2 \div 12.8 = 0.25\text{ km}^{2}\text{.}$

1 cm represents $\sqrt{0.25} = 0.5$ km, because $0.5 \times 0.5 = 0.25$ .

$0.5\text{ km} = 500\text{ m} = 500 \times 100\text{ cm} = 50\,000\text{ cm}$

The scale is $1 : 50\,000$ .

Question 8

A bedroom with a floor area of $12\text{ m}^{2}$ is drawn as $48\text{ cm}^{2}$ on a floor plan. Find the scale of the plan in the form $1 : n$ .

ShowHide solution

$12\text{ m}^{2} = 12 \times 10\,000\text{ cm}^{2} = 120\,000\text{ cm}^{2}$

$1\text{ cm}^{2}$ represents $120\,000 \div 48 = 2500\text{ cm}^{2}\text{.}$

1 cm represents $\sqrt{2500} = 50$ cm, because $50 \times 50 = 2500$ .

The scale is $1 : 50$ .

Question 9

The plan of a school field is drawn to the scale $1 : 2000$ . The field has right angles at its two left-hand corners, and its lengths on the plan are shown. Find the actual area of the field in $\text{m}^{2}$ and in hectares.

ShowHide solution

Cut the plan into a rectangle 6 cm by 4 cm and a right-angled triangle with base $9 - 6 = 3$ cm and height 4 cm.

Area on the plan $= 6 \times 4 + \frac{1}{2} \times 3 \times 4 = 24 + 6 = 30\text{ cm}^{2}$

1 cm represents $2000\text{ cm} = 20\text{ m}$ , so $1\text{ cm}^{2}$ represents $20 \times 20 = 400\text{ m}^{2}$ .

Actual area $= 30 \times 400 = 12\,000\text{ m}^{2}\text{,}$ and $12\,000 \div 10\,000 = 1.2$ ha.

Check with the lengths: 180 m, 120 m and 80 m, so the area is $120 \times 80 + \frac{1}{2} \times 60 \times 80 = 9600 + 2400 = 12\,000\text{ m}^{2}\text{.}$

The field covers $12\,000\text{ m}^{2}$ , which is 1.2 ha.

Question 10

A housing estate covers $15\text{ cm}^{2}$ on a map with the scale $1 : 20\,000$ . What area does it cover on a map with the scale $1 : 50\,000$ ?

ShowHide solution

First map: 1 cm represents 0.2 km, so $1\text{ cm}^{2}$ represents $0.04\text{ km}^{2}$ , and the estate is $15 \times 0.04 = 0.6\text{ km}^{2}\text{.}$

Second map: 1 cm represents 0.5 km, so $1\text{ cm}^{2}$ represents $0.25\text{ km}^{2}$ .

Area on the second map $= 0.6 \div 0.25 = 2.4\text{ cm}^{2}$

$2.4\text{ cm}^{2}$

Question 11

The scale of a map is 1 cm to 0.4 km. Mei Ling says that a park covering $12\text{ cm}^{2}$ on the map has an area of $12 \times 0.4 = 4.8\text{ km}^{2}\text{.}$ Explain her mistake, and find the correct area in $\text{km}^{2}$ and in hectares.

ShowHide solution

She used the length scale. 0.4 km is what 1 cm represents; $1\text{ cm}^{2}$ represents $0.4 \times 0.4 = 0.16\text{ km}^{2}\text{.}$

Area $= 12 \times 0.16 = 1.92\text{ km}^{2}\text{,}$ and $1.92 \times 100 = 192$ ha.

She multiplied by the length scale instead of its square. The park covers $1.92\text{ km}^{2}$ , which is 192 ha.

Question 12

A rectangular plot of land is 8 cm by 5 cm on a map with the scale $1 : 25\,000$ . Find (a) its actual perimeter, in km, (b) its actual area, in $\text{km}^{2}$ and in hectares.

ShowHide solution

1 cm represents $25\,000\text{ cm} = 250\text{ m} = 0.25\text{ km}\text{.}$

The plot is $8 \times 0.25 = 2$ km by $5 \times 0.25 = 1.25$ km.

(a) Perimeter $= 2 \times (2 + 1.25) = 6.5$ km. A perimeter is a length, so it uses the length scale.

(b) Area $= 2 \times 1.25 = 2.5\text{ km}^{2}\text{,}$ and $2.5 \times 100 = 250$ ha.

Check with the area scale: $8 \times 5 = 40\text{ cm}^{2}$ , and $40 \times 0.0625 = 2.5\text{ km}^{2}\text{.}$

(a) 6.5 km (b) $2.5\text{ km}^{2}$ , which is 250 ha

Key Points

If 1 cm on a map represents $k$ km, then $1\text{ cm}^{2}$ represents $k^{2}\text{ km}^{2}$ .
The scale $1 : n$ gives map area : actual area $= 1 : n^{2}$ , in the same unit.
With 1 cm representing $k$ km: actual area (in $\text{km}^{2}$ ) = map area (in $\text{cm}^{2}$ ) × $k^{2}$ , and map area = actual area ÷ $k^{2}$ .
$1\text{ m}^{2} = 10\,000\text{ cm}^{2}\text{,}$ $1\text{ ha} = 10\,000\text{ m}^{2}\text{,}$ and $1\text{ km}^{2} = 1\,000\,000\text{ m}^{2} = 100\text{ ha}\text{.}$
To find a scale from two areas, find what $1\text{ cm}^{2}$ represents, then take its square root.
Lengths and perimeters use the length scale; areas use its square.