E. Spatial Sense · Spatial Sense: Angles, Triangles, Transformations and Area

Same Area, Different Perimeters

E2.6 A1About 40 minutes

Learning goals

I can show that shapes with the same area can have different perimeters.
I can find all the rectangles with whole-number sides for a given area.
I can explain which shapes have the biggest and smallest perimeters for the same area.
I can solve problems about area and perimeter.

Words to know

perimeter
The distance all the way around a shape. Measured in units of length, like cm or m.

area
The amount of surface a shape covers. Measured in square units, like cm² or m².

dimensions
The measurements of a shape, like the length and width of a rectangle.

Remember from Grades 3 and 4

In Grade 3 you found the perimeter of shapes by adding up the lengths of all the sides. In Grade 4 you found the area of rectangles by multiplying length × width. They are different things:

Perimeter is the distance around (like a fence). Units: cm, m.
Area is the space inside (like the grass). Units: cm², m².

Today you'll explore a surprising fact: two shapes can have the same area but different perimeters.

Let's start with a class party

Your class has 12 square tables for a party. One person can sit at each open table edge. You push the tables together to make one big rectangle. Does it matter how you arrange them?

Yes! The area (12 tables) stays the same, but the number of seats around the outside, the perimeter, changes.

Three rectangles, each with an area of 12 squares: 12 by 1 (perimeter 26), 6 by 2 (perimeter 16) and 4 by 3 (perimeter 14).

Worked example

Example 1: Rectangles with an area of 12

Find every rectangle with whole-number sides and an area of 12 square units. Find each perimeter.

List pairs of numbers that multiply to 12: 1 × 12, 2 × 6, 3 × 4. (4 × 3 is the same as 3 × 4, just turned.)
1 by 12: P = 1 + 12 + 1 + 12 = 26.
2 by 6: P = 2 + 6 + 2 + 6 = 16.
3 by 4: P = 3 + 4 + 3 + 4 = 14.

All three have an area of 12, but the perimeters are 26, 16 and 14. For the party, the long line of tables seats 26 people!

The pattern

A long, skinny rectangle has the biggest perimeter.
A rectangle that is closest to a square has the smallest perimeter.

This works for other shapes too. Squares joined edge to edge make the smallest perimeter when they are bunched up together, and the biggest when they are spread out.

Four shapes each made of 4 squares. The 2 by 2 square has a perimeter of 8 units. The line, the L shape and the T shape each have a perimeter of 10 units.

Worked example

Example 2: Shapes made of 4 squares

Each shape in the picture is made of 4 squares, so each has an area of 4 square units. Find the perimeters.

Count the outside edges of each shape, going all the way around.
Square: 2 + 2 + 2 + 2 = 8 units.
Line: 4 + 1 + 4 + 1 = 10 units.
L shape and T shape: count the edges around: 10 units each.

Same area (4), but perimeters of 8 or 10. The bunched-up square has the smallest perimeter.

Worked example

Example 3: The least fencing

Grandpa wants a rectangular vegetable garden with an area of 36 m². Fencing is expensive. Which whole-number dimensions use the least fence?

List the rectangles with area 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.
Perimeters: 1 × 36 → 74 m. 2 × 18 → 40 m. 3 × 12 → 30 m. 4 × 9 → 26 m. 6 × 6 → 24 m.
The square, closest to the middle, uses the least.

A 6 m by 6 m garden needs only 24 m of fence.

Make an organized list

When a problem has many possible answers, make an organized list or a table. Start at 1 × something and work up. You'll know you're done when the pairs start repeating. Being organized helps you be sure you didn't miss any.

Length (m)	Width (m)	Area (m²)	Perimeter (m)
36	1	36	74
18	2	36	40
12	3	36	30
9	4	36	26
6	6	36	24

An organized list of all the whole-number rectangles with an area of 36 m².

Just for fun

Party planner tip: love company? Line up all 12 tables in one long row for 26 seats. Want a cozy chat? Bunch them into a 4 by 3 rectangle for only 14 seats. Same tables, different party!

Hands-on activity

Table arrangements

You need: 12 square things that are all the same size, like sticky notes, crackers, square coasters or paper squares, plus a pencil and paper.

Arrange all 12 squares into a rectangle. Count the edges around the outside (the perimeter). Write the length, width, area and perimeter.
Make a different rectangle with all 12. Record it.
Find all three rectangles (1 by 12, 2 by 6, 3 by 4).
Now make shapes that are not rectangles, joining squares edge to edge. What is the biggest perimeter you can make with 12 squares? The smallest?

(What to look for: a long line or a zigzag that never bunches up gives the biggest perimeter, 26. The 3 by 4 rectangle gives the smallest, 14.)

Math talk: "If you were planning a party, which arrangement would you choose? Why?"

Your turn

Find the rectangles with whole-number sides and an area of 8 square units, and their perimeters.

A 2 × 9 rectangle and a 3 × 6 rectangle both have an area of 18. Which has the bigger perimeter?

Which rectangle with an area of 16 has the smallest perimeter?

Five squares are joined in a straight line. What are the area and the perimeter?

Do shapes with the same area always have the same perimeter?

A dog run must have an area of 20 m². List the whole-number rectangles and their perimeters.

More practice: warm-up

What is perimeter?

What is the area of a 3 cm by 5 cm rectangle?

What is the perimeter of a 3 cm by 5 cm rectangle?

Is perimeter measured in cm or cm²?

What is the perimeter of a square with 4 cm sides?

More practice: on your own

List all the whole-number rectangles with an area of 24, and their perimeters.

You have 12 square tables. Which arrangement seats the most people? The fewest?

Shape A is a 2 × 3 rectangle. Shape B is 6 squares in a line. Compare their areas and perimeters.

A 4 cm × 6 cm rectangle and a 3 cm × 8 cm rectangle. Compare their areas and perimeters.

Mo says, "A bigger perimeter means a bigger area." Use an example to show he is not always right.

Challenge

You make shapes from 10 squares joined edge to edge. What is the biggest possible perimeter? What is the smallest?

A rectangle has an area of 30 cm² and a perimeter of 22 cm. What are its length and width?

Now flip the idea: you have 20 m of fence for a rectangular pen. Find two pens with the same perimeter but different areas.

Check your understanding

Quiz

Which two rectangles have the same area?

Which rectangle with area 12 has the biggest perimeter?

Which rectangle with area 36 uses the least fencing?

What is the perimeter of 6 squares in a straight line?

Shapes with the same area:

You have 12 square tables. How many people can sit around a 2 × 6 arrangement, one per open edge?

For parents and teachers

Big idea: The Ontario curriculum (E2.6) asks Grade 5 students to show that two-dimensional shapes with the same area can have different perimeters, and solve related problems. Children discover that long, thin shapes have larger perimeters and compact, square-like shapes have smaller ones.

Common difficulties:

Mixing up area and perimeter. Area is inside (square units); perimeter is around (length units).
Missing rectangles. Teach an organized list, starting at 1.
Counting squares instead of edges for perimeter. Trace around the outside with a finger.

How to help: Use crackers, sticky notes or tiles to build shapes. Plan a pretend garden or party seating together. Social-emotional learning (A1) is woven in: organized thinking and working through many cases patiently.