E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

How Changing Dimensions Affects Perimeter, Circumference and Area

E1.4 A1About 65 minutes

Learning goals

I can explain why multiplying every length by a scale factor k multiplies the perimeter (or circumference) by k and the area by k².
I can predict the new perimeter and area when only one dimension changes.
I can tell the difference between scaling (multiplying) and adding the same amount to each side.
I can work backwards from a change in area to the change in length, and use a table or spreadsheet to test patterns.

Words to know

dimension
A measurement of size such as length, width, height or radius.

scale factor (k)
The number every length is multiplied by. k > 1 enlarges; 0 < k < 1 reduces.

similar figures
Figures with the same shape, where every length is multiplied by the same scale factor.

circumference
The perimeter of a circle: C = 2πr = πd.

area
The amount of surface a 2D shape covers, in square units.

A hook: the pizza question

A 30 cm pizza costs $12. The 45 cm pizza is 1.5 times as wide. Should it cost 1.5 times as much ($18)? Or more?

You will see that the bigger pizza has 2.25 times as much pizza. The way area changes when lengths change surprises a lot of people, including people who price pizzas.

Remember from Grade 8

In Grade 8 you learned that when a scale drawing is enlarged, area changes by the scale factor squared. Now we explain why and apply it to every kind of shape, including circles, and to changes in only one dimension.

Doubling both dimensions of a 3 cm by 2 cm rectangle doubles the perimeter but makes 4 copies of the original area: 2² = 4.

Scaling all dimensions by k

If every length is multiplied by k:

perimeter or circumference is multiplied by k (it is a length),
area is multiplied by k² (area is length × length, and each length was multiplied by k).

For a circle, r becomes kr, so C = 2π(kr) = k × 2πr, and A = π(kr)² = k² × πr².

Doubling the radius doubles the circumference but multiplies the area by 4.

Changing only one dimension

If only the length of a rectangle is multiplied by k, the area is multiplied by k (only one factor changed), but the perimeter does not follow a simple rule; you need to recalculate it.

If you add the same amount to every side, that is not a scale: the shape changes and you must recalculate both perimeter and area.

Left to right: a 4 m by 3 m garden; the same garden with only the length doubled (area doubles); the garden with 1 m added to each dimension (a different shape, not a scale).

Radius (cm)	Circumference 2πr (cm)	Area πr² (cm²)	Area ÷ area when r = 1
1	6.28	3.14	1
2	12.57	12.57	4
3	18.85	28.27	9
4	25.13	50.27	16
5	31.42	78.54	25

A spreadsheet-style table. As r goes 1, 2, 3, 4, 5, the circumference grows in equal steps, but the area grows by 1, 4, 9, 16, 25 times: the square numbers.

Technology: build the table

In a spreadsheet, type radii 1 to 10 in column A. In B2 type **=2PI()A2 and in C2 type =PI()A2^2, then fill down. Add a column =C2/C$2* to see the area ratio. Try a column for a rectangle where only the length changes, and compare.

Worked example

Example 1: Scale a rectangle

A rectangle is 3 cm by 5 cm. It is enlarged by a scale factor of 3. Find the new perimeter and area, and compare.

Original: P = 2(3 + 5) = 16 cm, A = 3 × 5 = 15 cm².
New: 9 cm by 15 cm. P = 2(9 + 15) = 48 cm, A = 9 × 15 = 135 cm².
Compare: 48 ÷ 16 = 3 = k. 135 ÷ 15 = 9 = k².

P = 48 cm (× 3), A = 135 cm² (× 9)

Worked example

Example 2: The pizza question

Compare a 30 cm pizza and a 45 cm pizza. How many times as much pizza is the large one? Is $18 a fair price if the small one is $12?

Scale factor: 45 ÷ 30 = 1.5.
Area ratio: 1.5² = 2.25. (Check: π × 22.5² ≈ 1590 cm² and π × 15² ≈ 707 cm²; 1590 ÷ 707 ≈ 2.25.)
At the same price per cm², the large pizza would cost 2.25 × $12 = $27.

The large pizza has 2.25 times as much pizza, so $18 is a great deal.

Worked example

Example 3: Only one dimension changes

A garden is 4 m by 3 m. The length is doubled to 8 m. What happens to the area and perimeter?

Area: 12 m² becomes 8 × 3 = 24 m², so it doubles (× k with k = 2).
Perimeter: 2(4 + 3) = 14 m becomes 2(8 + 3) = 22 m. 22 ÷ 14 ≈ 1.57, so it does not double.

Area × 2; perimeter 14 m → 22 m

Worked example

Example 4: Working backwards

A circular rink has an area 9 times that of a smaller similar rink with radius 5 m. What is its radius?

Area ratio = k² = 9, so k = √9 = 3.
New radius = 3 × 5 = 15 m.

15 m

Worked example

Example 5: Two changes at once

A triangle has base 10 cm and height 6 cm. The base is tripled and the height is divided by 3. What happens to the area?

Original: A = ½ × 10 × 6 = 30 cm².
New: base 30 cm, height 2 cm. A = ½ × 30 × 2 = 30 cm².
The two changes cancel: × 3 and ÷ 3.

The area stays 30 cm².

Watch out!

A scale factor of 2 does not double the area; it multiplies it by 4.
Scaling only works when every length is multiplied. Adding 1 m to each side is not scaling.
Going backwards, take the square root of the area ratio to get the length ratio.

Hands-on activity

Grid-paper scaling lab

You need: grid paper (or draw a grid), a pencil, a ruler, and coins or round lids of two sizes.

Draw any shape on the grid using whole squares (an L shape works well). Count its perimeter and area.
Redraw it with every length × 2, then × 3. Count again. Make a table: k, perimeter, area.
Divide each new area by the original area. Do you see 4 and 9?
Measure the diameters of two round lids. Predict the ratio of their areas using (big ÷ small)², then check by tracing them on the grid and counting squares.

Model answer: an L made of 5 squares with perimeter 12 units becomes 20 squares and 24 units when k = 2, and 45 squares and 36 units when k = 3.

Just for fun

Pizza math: one 45 cm pizza has more pizza than two 30 cm pizzas (about 1590 cm² compared with about 1414 cm²), because area grows with the square of the size.

Your turn

A square has side 5 cm. It is enlarged by a scale factor of 4. Find the new perimeter and area.

A circle's radius changes from 3 cm to 6 cm. By what factor do the circumference and area change?

A rectangle is 2 cm by 7 cm. Only the width is tripled. What is the new area?

A shape has an area of 12 cm². It is enlarged by a scale factor of 2.5. Find the new area.

Two similar shapes have areas in the ratio 25 : 1. What is the ratio of their lengths?

A shape has perimeter 40 cm and area 80 cm². It is enlarged by 1.5. Find the new perimeter and area.

Is adding 2 cm to every side of a 4 cm by 6 cm rectangle a scale? Explain.

More practice: warm-up

All lengths × 2. Perimeter × ?

All lengths × 2. Area × ?

A circle's diameter is doubled. Area × ?

All lengths × 3. Area × ?

All lengths × ½. Area × ?

More practice: on your own

A 10 cm by 15 cm photo is enlarged to 20 cm by 30 cm. How many times as much paper does it need?

Compare a 25 cm and a 35 cm pizza. How many times as much pizza is the larger one?

A rink model is 60 cm by 26 cm at a scale of 1 : 100. Find the real area in m² and compare with the model's area.

A rectangle is 8 m by 5 m. Add 2 m to each dimension. Find the change in perimeter and area. Is it a scale?

A circle has an area of 50 cm². Its radius is tripled. What is the new area?

Challenge

A square's area increases by 44%. By what percent did its side length increase?

A rectangle is 6 m by 4 m. Its length increases by 50% and its width decreases by 50%. What happens to the area?

A circular garden has radius 5 m. You want to double its area. What should the new radius be?

Check your understanding

Quiz

Every length of a shape is tripled. The area is multiplied by:

A circle's radius is doubled. Its circumference is multiplied by:

A rectangle's length is doubled and its width stays the same. Its area is:

Two similar logos have areas 20 cm² and 180 cm². What is the scale factor from small to large?

A 20 cm pizza and a 40 cm pizza. The larger has how many times as much pizza?

A shape with area 10 cm² is reduced by a scale factor of ½. The new area is:

For parents and teachers

Big idea: E1.4 asks students to show how changing one or more dimensions affects perimeter, circumference, area, surface area and volume, using technology when appropriate. This lesson covers 2D shapes; the next covers 3D objects, and the one after uses these ideas to find the best (optimal) dimensions.

Common difficulties:

Expecting area to change by k. Draw the grid picture: doubling makes 4 copies.
Treating "add 2 cm to each side" as scaling.
Working backwards from an area ratio: remind your teen to take the square root.

How to help: The pizza question works at any restaurant. Compare prices per square centimetre of two sizes. Let your teen build the spreadsheet from the tip box.