Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 7, lesson 1 (perimeter and area of rectangles): a rectangle is 9 m long and 4 m wide. What are its perimeter and area?
Answer
Perimeter 26 m (9 + 4 + 9 + 4) and area 36 m² (9 × 4). - From Unit 1, lesson 3 (factors, multiples and primes): list all the factors of 18.
Answer
1, 2, 3, 6, 9 and 18. - From Unit 1, lesson 2 (powers, squares and roots): what is 7²?
Answer
49. 7 × 7 = 49.
Words used in this lesson
- fixed
- not allowed to change. For example: The fence is fixed at 24 m while the width changes.
- enclose
- to be all the way round something. For example: A fence encloses a pen.
- whole number
- a number with no fraction or decimal part. For example: 5 is a whole number; 5.5 is not.
- square
- a rectangle with four equal sides. For example: 6 m by 6 m.
- factor pair
- two whole numbers that multiply to make a number. For example: 4 and 9 are a factor pair of 36.
- compare
- to look at two things and say how they are alike and different. For example: Compare the areas of two pens.
One roll of fence
Lucía has a roll of fence wire for a llama pen outside Cusco. Whatever shape she makes, the roll fixes the perimeter: the pen's edge is exactly as long as the roll. So does every pen she can make hold the same number of llamas? Another group has the opposite problem: they need a pen of a given area, and fence costs effort to carry up the hill. For them the question is which shape needs the least fence.
Predict first: with a fence of 24 m, which is bigger, a pen 2 m wide or a pen 6 m wide? Then choose Fence 24 m, slide the width and watch the bars.
The fence is 32 m, so a pen 5 m wide is 11 m long and holds 55 m². The square, 8 m by 8 m, holds 64 m². This pen is 9 m² short: its width is 3 m from 8 m, and 3 × 3 = 9.
Show me the steps
- The fence is 24 m, so length plus width is 12 m. At width 4 m the length is 8 m and the area is 32 m². The square would be 6 m by 6 m and hold 36 m², so this pen is 4 m² short.
- Make the width 1 m. The length is 11 m and the area is only 11 m², 25 m² short of the square: the same fence, a thin strip.
- Make the width 6 m. The length is 6 m too, a square, with the most area, 36 m². Nothing is short of the square.
- Choose a fence of 16 m and a width of 2 m. The length is 6 m, the area 12 m², and the square (4 m by 4 m) holds 16 m², so it is 4 m² short.
With the perimeter fixed, the area is biggest when the rectangle is a square. The nearer a rectangle is to a square, the more it encloses.
Why the square wins. Start with a square, a metres on each side. To keep the same fence, make one side d metres shorter and the next side d metres longer. Cutting a strip d wide off the square loses d × a. Laying it on the end gains only d × (a − d), because the pen is now d shorter. The net loss is d × a − d × (a − d) = d × d. With a = 6 and d = 2 that is 12 − 8 = 4, the same 4 m² as the first figure shows. Only d = 0, the square, loses nothing.
Turn it round. With the area fixed, the square needs the least perimeter. Here is why. Take area 36: the 4 by 9 rectangle needs 26 m of fence. A square with a 26 m fence is 6.5 m by 6.5 m and holds 42.25 m², which is more than 36. So a square that holds exactly 36 m² must need less than 26 m of fence: it needs 24 m. The same argument works for any rectangle: the square made from its fence holds at least as much, so the square that holds exactly its area needs no more fence.
Back to Lucía: her best pen is a square, and the group carrying fence up the hill should make their patch as close to a square as the ground allows.
The other way round: fixed area
Now keep the area, 36 m² or 48 m², and list every rectangle with whole-number sides. They all cover the same ground. How much edge does each one need?
Every rectangle here has an area of 36 m². The one chosen, 1 m by 36 m, has a perimeter of 74 m. The least is 24 m.
Show me the steps
- Area 36 m², rectangle 1: 1 m by 36 m. The perimeter is 2 × (1 + 36) = 74 m, a lot of edge for the same ground.
- Choose rectangle 4: 4 m by 9 m. The perimeter is 2 × (4 + 9) = 26 m.
- Choose rectangle 5: 6 m by 6 m, the square. The perimeter is 2 × (6 + 6) = 24 m, the least.
- Switch to area 48 m². The last rectangle is 6 m by 8 m, with perimeter 2 × (6 + 8) = 28 m. It is the closest to a square a whole number allows, and it is the least.
A common mistake: “This rectangle has the longer perimeter, so it must be the bigger one.” A long thin rectangle can have a long perimeter and a small area. Compare 12 m by 1 m (perimeter 26 m, area 12 m²) with 4 m by 4 m (perimeter 16 m, area 16 m²): the second has the shorter perimeter and the bigger area. Always work out both before you compare.
Worked example: the best pen from 30 m
A farmer has 30 m of fence and wants a rectangular pen with whole-number sides and the largest area. What should the sides be?
- The sides add to half the perimeter: 30 ÷ 2 = 15, so length + width = 15.
- List the pairs and their areas: 1 × 14 = 14, 2 × 13 = 26, 3 × 12 = 36, 4 × 11 = 44, 5 × 10 = 50, 6 × 9 = 54, 7 × 8 = 56.
- The biggest is 7 × 8, the pair closest together. 15 is odd, so no pair is exactly equal, and a square with whole sides is not possible.
- 7 m by 8 m, area 56 m².
Your go. The first step is given. A label must have an area of 30 cm² with whole-number sides. Which size has the shortest perimeter, and what is it?
- List the pairs of factors of 30: 1 × 30, 2 × 15, 3 × 10, 5 × 6.
- Now find the perimeter of each pair yourself, pick the least, then check.
Check the full answer
The perimeters are 62, 34, 26 and 22 cm. The least is 22 cm, from 5 cm by 6 cm, the pair closest together.
What if the sides need not be whole numbers? The 30 m fence in the worked example can also make a pen 7.5 m by 7.5 m, a square. It holds 7.5 × 7.5 = 56.25 m², a little more than the best whole-number pen, 56 m². The square is still the best.
Check before you go on
1. A rectangular pen is made from 22 m of fence, with whole-number sides. Which pen holds the most?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A rectangular pen is made from 28 m of fence. One side is 9 m long. What is the area of the pen in square metres?
3. A square pen uses 40 m of fence. What is its area in square metres?
Level 2 · Building
4. A rectangle has a perimeter of 26 m and whole-number sides. What is the greatest area it can have, in square metres?
5. A rectangular patch must have an area of 60 m² with whole-number sides. What is the least perimeter it can have, in metres?
Level 3 · Stretch
6. A rectangle has a perimeter of 40 m and an area of 84 m². How long is its longer side, in metres?
7. How many different rectangles have a perimeter of 34 m and whole-number sides? (A rectangle 3 m by 14 m is the same as one 14 m by 3 m.)
Something to think about
A rectangle is 10 m by 4 m, and a second rectangle is 7 m by 7 m. Which has the bigger perimeter, and which has the bigger area? Can you find a rectangle that has both a bigger perimeter and a bigger area than the 7 m by 7 m square? Find one.
One way to think about it
The first has perimeter 28 m and area 40 m². The second has perimeter 28 m and area 49 m²: the same fence, more ground. For both bigger, make a rectangle 12 m by 5 m: perimeter 34 m and area 60 m², which beats the 7 by 7 square on both. Perimeter and area do not have to move in opposite directions: they just do not decide each other.
Exit ticket
8. Rectangle P is 10 m by 2 m. Rectangle Q is 6 m by 5 m. Which statement is true?