A map scale of 1 cm to 5 km means every centimeter on the map is 5 km in the world.
Step 1: Picture It
Start with something you already know.
A bigger brownie pan
- A small square pan holds one batch of brownies.
- A big pan is longer and wider, both by the same scale factor.
- Set small pans inside it: the longer side fits more pans in each row.
- The wider side fits more rows, so the big pan gains both ways.
The batches it holds are pans per row times rows, and the factor sets both.
So area changes by the scale factor times itself, not by the factor alone.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A small rug is 2 ft by 3 ft. A big rug is a scale copy of it with scale factor 2. What is the area of the big rug, in square feet?
square feet
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Lengths are easy
7 cm on that map is 7 × 5 = 35 km.
Area is not
Doubling every side of a rectangle makes the area four times bigger, not twice.
Why four
Both the length and the width doubled, and area multiplies them, so 2 × 2 = 4.
Step 3: Try It Yourself
Tap and try it out.
Area · squares inside
12
Perimeter · walk around
14
4 × 3 = 12 · 4 + 3 + 4 + 3 = 14
Area counts the cm inside. Perimeter measures the walk all the way round.
Step 4: In Real Life
A floor plan
A plan drawn at half size has floors a quarter of the area. Doubling the sides of a room quadruples the carpet. Builders learn this the expensive way.
Step 5: Watch an Example
One step at a time.
Watch Ana Scale an Area
A rectangle 2 cm by 5 cm is scaled by 4. Ana wants the area of the copy.
- Step 1
Ana writes the original area down first: 2 × 5 = 10 square cm. That is the number the copy will be compared against.
Watch Dev Read a Patio Plan
A plan of a patio uses 1 in to 5 ft. On the plan the patio is 3 in by 4 in. Dev starts, and you finish.
- Step 1
Dev reads the scale first. Every 1 in on the plan stands for 5 ft on the ground, so the plan numbers are small and the ground numbers are big.
Step 6: Your Turn
Practice makes it stick.
The Deck
Problem 1 of 17
Sam's deck is 4 ft by 6 ft. He builds a new deck that is a scale copy with scale factor 2. Decking boards cost $3 for every square foot. What do the boards for the new deck cost?
dollars
The Garden Fence
Problem 2 of 17
A plan of a garden uses 1 in to 3 ft. On the plan the garden is 4 in by 7 in. Fencing costs $8 for every foot. What does a fence right around the garden cost?
dollars
Lengths and Areas
Map to ground multiplies. Ground to map divides. An area uses the factor twice.
Problem 3 of 17
A map's scale is 1 cm to 4 km. A road is 9 cm long on the map. How long is the real road, in km?
km
Problem 4 of 17
On a map with scale 1 cm to 25 km, two towns are 200 km apart. How far apart are they on the map, in cm?
cm
Problem 5 of 17
A drawing uses 1 in to 10 ft. A wall is 6.5 in long on the drawing. How long is the real wall, in ft?
ft
Problem 6 of 17
A shape is scaled by 5. Its area comes out how many times as big?
times
Problem 7 of 17
A photo covers 20 square inches. A copy is made with scale factor 3. What area does the copy cover, in square inches?
square inches
Whose Step?
One line in each piece of working is a slip. Find it, then fix it.
Problem 8 of 17
Maya found the area of a copy of a 5 cm by 6 cm rectangle, scaled by 3. One line is a slip. Which?
Maya's copy of a 5 cm by 6 cm rectangle, scale factor 3
- 1Rectangle 5 cm by 6 cm, scale factor 3
- 2Area of the original = 5 × 6 = 30 square cm
- 3Area factor = 3
- AnswerArea of the copy = 30 × 3 = 90 square cm
Problem 9 of 17
Maya's rectangle is 5 cm by 6 cm and the scale factor is 3. What area does the copy cover, in square cm?
square cm
Problem 10 of 17
Priya worked out how long 56 km is on a map with scale 1 cm to 8 km. One line is a slip. Which?
Priya's map length for 56 km at 1 cm to 8 km
- 1Scale 1 cm to 8 km, real distance 56 km
- 2Map length = 56 × 8
- 3= 448
- AnswerThe map length is 448 cm
Problem 11 of 17
On a map with scale 1 cm to 8 km, how many cm stand for 56 km?
cm
Problem 12 of 17
Leo found the area of a copy of a 3 ft by 7 ft rug, scaled by 4. One line is a slip. Which?
Leo's copy of a 3 ft by 7 ft rug, scale factor 4
- 1Rug 3 ft by 7 ft, scale factor 4
- 2New long side = 7 ft
- 3New short side = 3 × 4 = 12 ft
- AnswerNew area = 12 × 7 = 84 square ft
Problem 13 of 17
Leo's rug is 3 ft by 7 ft and the scale factor is 4. How long is the copy's long side, in ft?
ft
Pick the Move
Choose the move before you make it.
Problem 14 of 17
A plan of a stage uses 1 in to 6 ft. On the plan the stage is 2 in by 3 in. Which move gives the real area, in square feet?
Problem 15 of 17
A room is scaled up by a factor of 3. For which of these do you multiply by 9 rather than by 3?
Same Thing?
Decide without any drawing. Ask each time whether the thing named is a length or a space to cover, and which way the scale runs.
Problem 16 of 17
Select every statement that is true when a shape is scaled by 2.
Problem 17 of 17
A map uses 1 cm to 8 km. Select every statement that is true.
Step 7: Quick Check
Show what you know.
Question 1 of 2
A shape is scaled by 3. How many times bigger is its area?
Question 2 of 2
Why does doubling the sides quadruple the area?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
A photo is enlarged. The copy's area comes out 49 times the original's area. What is the scale factor?
Stretch 2 of 2
A plan of a hall uses 1 in to 4 ft. On the plan the hall is 4 in long. The real floor covers 192 square feet. How wide is the hall on the plan, in inches?
in
What You Learned
- A scale converts map lengths to real ones.
- Lengths scale by the factor.
- Area scales by the factor squared.
For the grown-up
The misconception this lesson targets is that an area scales the way a length does: double the sides and the area doubles. It is stubborn because the scale factor is sitting right there in the question, and using it once feels like using it. An area is made from two lengths, and scaling hits both of them, so the area factor is the scale factor multiplied by itself. Two smaller slips ride alongside it: multiplying a real distance by the scale number when it should be divided, which draws a map several meters wide, and scaling one side of a rectangle while leaving the other alone. The cure for all three is one habit — say out loud whether the thing being asked for is a length or a space to cover, before any arithmetic starts.
Area scales by the square of the factor — Double every length and the area becomes four times as big, not twice. Triple the lengths and the area is nine times. The area factor is the length factor squared, because area involves two dimensions and each is being scaled.
Seeing why — A 3 by 4 rectangle has area 12. Doubled, it becomes 6 by 8 with area 48 — four copies of the original would fit inside. Drawing the four copies inside the enlarged rectangle makes the squaring obvious rather than something to accept.
Volume scales by the cube — Double every edge of a box and its volume becomes eight times as large. Three dimensions, so the factor is cubed. This is why a scale model that looks twice as big weighs eight times as much, and why large animals are not simply scaled-up small ones.
The mistake this prevents — People routinely assume that doubling a map's scale doubles the land area shown, or that a pizza twice the diameter feeds twice as many. Both are wrong by a factor of two. Knowing that area squares is genuinely useful outside the classroom.
Area scales by the square — Original 3 by 4: area 12. Scale by 2: 6 by 8, area 48. The sides doubled but the area went up four times. Because both dimensions grew: 2 × 2 = 4. Scale factor k means area factor k².