Grade by Grade
E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Scale Drawings: Real Lengths, Real Areas and New Scales

E1.3 A1About 50 minutes
Learning goals
  • I can use a scale drawing or map to calculate real lengths.
  • I can calculate real areas from a scale drawing, and explain why area changes by the scale factor squared.
  • I can reproduce a scale drawing at a different scale.

Words to know

scale drawing
A drawing in which every length is the real length multiplied (or divided) by the same number.
scale
The ratio of a drawing length to the real length, like 1 : 200. Both parts are in the same unit.
scale bar
A short labelled line on a map or plan that shows what real distance a length on the drawing stands for.
floor plan
A scale drawing of the rooms of a building, seen from above (a top view).
scale factor
The number you multiply drawing lengths by to get new lengths. Going from a 1 : 200 drawing to a 1 : 100 drawing, the scale factor is 2.

Remember from Grade 7

In Grade 7 you used a scale like 1 cm : 50 cm to go from real lengths to a drawing (divide) and back (multiply), and you made scale drawings of views. In the last lesson you chose a scale for a model. Now you'll calculate real areas from a drawing and redraw a plan at a different scale.

Let's start with a new apartment

A family is moving to a new apartment. The landlord sends a floor plan at a scale of 1 : 200. They want to know: Will our 3 m sofa fit along the living room wall? How much flooring do we need for the bedroom? Can we print a bigger copy so it's easier to plan the furniture? All three questions are about scale.

Bedroom4 m × 3 mKitchen4 m × 3 mLiving room6 m × 6 m10 m6 mscale 1 : 20002 m
A floor plan drawn to scale. The scale bar shows 2 m. Every room's drawn length and width match their labels: the bedroom and kitchen are each 4 m by 3 m, and the living room is 6 m by 6 m. On the printed plan at 1 : 200, the whole apartment is 5 cm by 3 cm.

Real lengths

At 1 : 200, every 1 cm on the drawing stands for 200 cm (2 m) in real life. Multiply a drawing length by 200 to get the real length, then change units.

Real areas

You could multiply the drawing area by 200, but that would be wrong. Each rectangle has a length and a width, and both are multiplied by 200. So the area is multiplied by 200 × 200 = 40 000.

The safest way: change the lengths to real lengths first, then find the area.

Redrawing at a new scale

To go from one scale to another, find the scale factor between them:

  • 1 : 200 → 1 : 100: the new drawing is 2 times as big (200 ÷ 100 = 2). Multiply every drawing length by 2.
  • 1 : 50 → 1 : 150: the new drawing is 1/3 as big (50 ÷ 150). Divide every drawing length by 3.

Redrawing at a new scale is a dilation, so the new drawing is similar to the old one: the angles stay the same. To copy curvy or irregular shapes, draw a grid over the old drawing and a grid with bigger (or smaller) squares on the new paper, then copy square by square.

Worked example

Example 1: Real length from a map

A hiking map has a scale of 1 : 50 000. A trail measures 7.2 cm on the map. How long is the real trail?

  1. Real length = 7.2 × 50 000 = 360 000 cm.
  2. 360 000 cm ÷ 100 = 3 600 m, and 3 600 m ÷ 1 000 = 3.6 km.

The trail is 3.6 km long.

Worked example

Example 2: Real area from a floor plan

On the 1 : 200 plan, the living room measures 3 cm by 3 cm. Find its real area.

  1. Real lengths: 3 cm × 200 = 600 cm = 6 m. So the room is 6 m by 6 m.
  2. Real area: 6 m × 6 m = 36 m².
  3. Check the other way: the drawing area is 9 cm², and 9 × 40 000 = 360 000 cm². Since 1 m² = 10 000 cm², that is 36 m² ✓

The living room is 36 m². (The family's 3 m sofa fits easily along a 6 m wall.)

Worked example

Example 3: Redraw at a larger scale

Redraw the 1 : 200 plan at 1 : 100. How big is the whole plan, and how much paper area does it cover compared with before?

  1. Scale factor: 200 ÷ 100 = 2. Every drawing length doubles.
  2. Old plan: 5 cm by 3 cm. New plan: 10 cm by 6 cm.
  3. Paper area: old 15 cm², new 60 cm². 60 ÷ 15 = 4 = 2².

The new plan is 10 cm by 6 cm, and it covers 4 times as much paper.

1 : 2005 cm3 cm15 cm² of paper× 21 : 10010 cm6 cm60 cm² of paper = 4 × 15 cm²
Example 3. Redrawing the 1 : 200 plan at 1 : 100 doubles every length, from 5 cm by 3 cm to 10 cm by 6 cm. Four copies of the old plan fit on the new one, so it uses 4 times the paper.
Worked example

Example 4: Redraw at a smaller scale

A garden plan is drawn at 1 : 50. A flower bed is 9 cm long on it. How long will the bed be on a new plan at 1 : 150?

  1. At 1 : 50, the real bed is 9 × 50 = 450 cm long.
  2. At 1 : 150, the drawing length is 450 ÷ 150 = 3 cm.
  3. Shortcut: 150 ÷ 50 = 3, so every length on the new plan is one third as long: 9 ÷ 3 = 3 cm ✓

3 cm.

Think like a mathematician

  • Lengths change by the scale factor. Areas change by the scale factor squared.
  • Convert lengths to real units before finding areas, and you won't need to remember the squared rule.
  • Reasonableness check: does the real room size make sense? A bedroom is usually a few metres across, not 30 m or 30 cm.
  • When a plan has a scale bar instead of a ratio, measure the scale bar first to find what 1 cm stands for.
Hands-on activity

Map your bedroom, twice

You need: a measuring tape, a ruler, a pencil, and two sheets of graph paper (or plain paper).

  1. Measure the length and width of your bedroom (or another room) to the nearest 10 cm. Measure the bed, the dresser and the door too.
  2. Choose a scale that fits the room on your paper, like 1 : 50 (2 cm on the paper = 1 m). Draw the floor plan with the furniture.
  3. Use your plan to find the room's real area in square metres. Check it by multiplying the real length by the real width.
  4. Now redraw the plan at half the size (1 : 100). Every length on the new plan is half as long as on the first plan.
  5. Compare the paper areas of the two plans. How many times bigger is the first one?

What to look for: the real area is the same from both plans. The 1 : 50 plan covers 4 times as much paper as the 1 : 100 plan, even though its lengths are only 2 times as long.

Math talk: "Could the bed fit along the other wall? How do you know from the plan?"

Funny but true

In a story by Lewis Carroll (the author of Alice in Wonderland), a character brags that his country made a map at a scale of "a mile to the mile". That's 1 : 1! They never unfolded it, because the farmers said it would cover the whole country and block out the sunlight.

Your turn

1

A plan has a scale of 1 : 200. A wall measures 4.5 cm on the plan. How long is the real wall?

2

On the 1 : 200 apartment plan, the bedroom is 2 cm by 1.5 cm. What are its real dimensions and area?

3

A map has a scale of 1 : 25 000. Two cabins are 8 cm apart on the map. How far apart are they?

4

A drawing at 1 : 100 is redrawn at 1 : 50. What happens to each length on the drawing?

5

A plan at 1 : 20 shows a table 6 cm long. How long will it be on a new plan at 1 : 40?

6

A drawing's lengths are tripled to make a new drawing. How many times as much paper area does the new drawing cover?

More practice: warm-up

1

At 1 : 100, what real length does 1 cm stand for?

2

At 1 : 100, a room is 5 cm long on the plan. How long is it really?

3

At 1 : 50, a real wall is 3 m long. How long is it on the drawing?

4

How many square centimetres are in 1 m²?

5

A drawing's lengths are doubled. Is the new drawing similar to the old one?

More practice: on your own

1

A backyard rink is drawn at 1 : 200 as a rectangle 6 cm by 4 cm. What is the real area of the ice?

2

A park plan at 1 : 500 shows a soccer field 18 cm by 12 cm. Find the real area in square metres.

3

The area of a room on a 1 : 50 plan is 24 cm². What is the real area?

4

A map at 1 : 50 000 is redrawn at 1 : 20 000. A lake is 4 cm long on the old map. How long is it on the new map?

5

A floor plan has a scale bar: 3 cm on the plan = 6 m. What ratio scale is this?

Challenge

1

Flooring costs $32 per m². On a 1 : 100 plan, a family room is an L-shape made of a 5 cm by 4 cm rectangle and a 2 cm by 3 cm rectangle (no overlap). How much will the flooring cost?

2

A poster is printed at 1 : 4 of a mural's size and covers 0.5 m². What is the mural's area?

3

Jamal says: "If I redraw a 1 : 300 plan at 1 : 100, I'll need 3 times as much paper." Is he right? Explain.

Check your understanding

Quiz

1.

A plan has a scale of 1 : 200. A wall is 3 cm long on the plan. How long is the real wall?

2.

A room is 2 cm by 3 cm on a 1 : 100 plan. What is its real area?

3.

When all the lengths of a drawing are multiplied by 4, the area is multiplied by:

4.

A plan at 1 : 50 is redrawn at 1 : 100. The new drawing's lengths are:

5.

A map has a scale of 1 : 100 000. Two towns are 5 cm apart on the map. How far apart are they?

6.

A drawing is redrawn at a new scale. Which stays the same?

For parents and teachers

Big idea: The Ontario curriculum (E1.3) asks Grade 8 students to use scale drawings to calculate actual lengths and areas, and reproduce scale drawings at different ratios. The key insight is that areas change by the scale factor squared.

Common difficulties:

  • Multiplying a drawing area by the scale number once. Encourage your child to find the real length and width first, then multiply.
  • Unit mix-ups (cm, m, km, cm², m²). Remember 1 m = 100 cm but 1 m² = 10 000 cm².
  • Getting the direction wrong when changing scales. 1 : 100 is a bigger drawing than 1 : 200 (each centimetre stands for less).

How to help: Look at real plans and maps together: a real estate floor plan, a trail map, a city map with a scale bar. Ask your child to find a real distance or the area of a room. The bedroom-plan activity is a practical way to plan a furniture rearrangement. This lesson also builds A1 skills: representing, and checking answers for reasonableness.