Finding a missing side from the perimeter

When you reach for it

In Perimeter every side was known, and the sides were added to get the distance all the way around. Often the question comes the other way. Jurgis has 36 meters of fence for a rectangular horse corral and wants it 11 meters long: how wide can it be? Kotryna has a fixed length of border for a square herb garden: how long is each side? The perimeter is known, and so are some of the sides. One side, or one kind of side, is missing.

The idea that settles every such question is the same: the perimeter is the sum of the sides. If the total is known and some of the parts are known, the missing part is the total minus the known parts, exactly as with any sum where one addend is unknown. Subtraction undoes addition.

The lesson takes two days. Part 1 works with any shape with straight sides, a polygon, where one side is missing: subtract the known sides from the perimeter. It also writes the question as an equation with a letter for the missing side. Part 2 does rectangles and squares, where equal sides give shortcuts: half the perimeter of a rectangle is one length plus one width, and a square’s perimeter divided by 4 is its side. Part 2 ends with the cases that have no answer: no side in whole units, or no shape at all.

The procedure

Part 1: one side missing

Going all the way around a polygon walks each side once, so the perimeter is the sum of the side lengths. Suppose a shape has five sides, four of them known, and the perimeter is known too. Then

This is a sum with one addend missing, and the missing addend is found by subtraction:

There are two ways to carry out the subtraction, and both give the same answer. You can add the known sides first and take that one sum away from the perimeter. Or you can take the known sides away from the perimeter one at a time. The first way is usually shorter; the second shows the fence being used up, side by side, until only the last side’s share is left.

An equation with a letter. The question can be written down before it is solved, using a letter for the missing side, as in Using a letter for the unknown number. For a five-sided fence of 40 meters with sides 9, 6, 8 and 7 meters, call the missing side s:

The letter stands for one number, the one that makes the sentence true. Adding the known sides gives 30 + s = 40, and the related subtraction fact gives s = 40 - 30 = 10. A box in place of the letter, 30 + \square = 40, says the same thing.

Check by adding. Put the answer back with the other sides and add all the way around. If the sum is the perimeter, the side is right. The check costs one addition, and it catches a slip in the subtraction.

Two equal missing sides. Sometimes two sides are missing but are known to be equal, as in a triangle with two equal sides. Subtracting the known side leaves the length the two equal sides share. They share it in two equal parts, so divide by 2.

Part 2: rectangles and squares

A rectangle’s opposite sides are equal (see Rectangles, rhombuses and squares), so going around it walks length, width, length, width. The first length and width take you to the opposite corner, halfway around, and the second pair brings you back the same distance. So

That gives the quickest route to a missing side of a rectangle: halve the perimeter, then subtract the side you know. A second route says the same thing in a different order. Two lengths use 2 \times \text{length} of the fence; what is left covers the two widths, and half of that is one width. Both routes are worked in Example 4.

A square has four equal sides, so its perimeter is 4 \times \text{side}. Going backward, the side is the perimeter divided by 4, because division undoes multiplication (Multiplication and division undo each other): 4 \times s = 28 asks which number times 4 is 28, and 28 \div 4 is that number.

When there is no answer. A subtraction or a division can give a result that no real shape could have.

• If the perimeter does not divide evenly, there is no side in whole units. A square with 30 feet of border would need sides between 7 feet (which uses 28) and 8 feet (which uses 32).
• If the known sides already add up to the perimeter or more, nothing is left for the last side, and there is no shape. The same goes for a rectangle whose given length is half the perimeter or more.
• Even a missing side that comes out positive can be too long: the other sides together must stretch farther than it does, or they cannot reach around from one end of it to the other.

Checking by adding catches a wrong subtraction, but not these. They are caught by asking whether the shape could be built.

Worked examples

Examples 1 to 3 go with Part 1, and Examples 4 to 7 with Part 2.

Boundaries

Only one unknown length. Subtraction finds one missing side, or several sides known to be equal. If two sides are missing and nothing says how they compare, the perimeter alone cannot decide them: a four-sided fence of 30 meters with sides 8 and 9 leaves 13 meters for the other two, which could be 6 and 7, 5 and 8, or many other pairs.

A rectangle needs one side besides the perimeter. The perimeter gives only length plus width. A perimeter of 16 meters fits 7 by 1, 5 by 3, 4 by 4 and more.

The halving shortcut needs a rectangle. Length plus width is half the perimeter only because opposite sides are equal. For a four-sided shape whose sides are not two equal pairs, subtract all the known sides instead.

Dividing by 4 needs a square. The four sides must be equal. Likewise dividing by 3 needs a triangle with three equal sides, and in general by the number of sides when every side is equal.

All lengths in one unit. A perimeter in meters and a side in centimeters must be put into the same unit before subtracting (Metric length units, Customary length units).

The answer must make a shape. The missing side has to be more than zero, and it has to be shorter than all the other sides put together, or they cannot reach around from one of its ends to the other. If a whole number of units is required, the division for a square or for equal sides must come out even.

Where this goes next

Perimeter versus area keeps the perimeter fixed, as in Example 6, and asks how much ground rectangles with the same fence enclose. The answer is that it changes a lot. A different backward question starts from the area instead of the fence: Finding a missing side from the area knows how much ground a rectangle covers and one side, and finds the other by division rather than by halving and subtracting. Later, an equation with a letter for an unknown length becomes the ordinary way to write any such question, with the perimeter formula P = 2l + 2w solved for the length or the width.

Exercises

Part 1: one side missing

1. A triangle-shaped flag has a perimeter of 24 inches. Two of its sides are 7 inches and 9 inches. How long is the third side?
2. A four-sided herb bed has 30 feet of border. Three of its sides are 6 feet, 9 feet and 8 feet. Find the fourth side.
3. A six-sided fence at the duck pond is 52 meters long. Five of its sides are 8, 7, 9, 6 and 10 meters. How long is the sixth side?
4. A four-sided pen beside the corral has a perimeter of 64 meters, with three sides of 15, 18 and 12 meters. Write an equation with a letter for the fourth side, then solve it.
5. A four-sided field has a perimeter of 400 meters. Three of its sides are 125, 98 and 107 meters. Find the fourth side.
6. A five-sided pasture next to the corral has a perimeter of 612 meters. Four of its sides are 145, 118, 96 and 137 meters. Find the fifth side, and check by adding.
7. A triangle-shaped herb bed has 32 feet of border. One side is 10 feet and the other two are equal. How long is each equal side?
8. A triangle with three equal sides has a perimeter of 27 centimeters. How long is each side?
9. A four-sided fence has a perimeter of 45 meters, and three of its sides are 12, 10 and 14 meters. Is the fourth side 9 meters? Decide by adding all the way around.
10. A six-sided pen has all its sides equal and a perimeter of 42 meters. How long is each side?

Part 2: rectangles and squares

11. A rectangular corral has a perimeter of 26 meters and a length of 8 meters. Find its width.
12. A rectangular herb garden has 50 feet of border and is 9 feet wide. How long is it?
13. A square herb bed has 36 feet of border. How long is each side?
14. A square pen at the duck pond takes 32 meters of fence. How long is each side?
15. A rectangular paddock has a perimeter of 90 meters and a length of 30 meters. Find its width, both by halving first and by taking away the two lengths first.
16. A rectangular corral takes 140 meters of fence and is 25 meters wide. How long is it?
17. Can a square with whole-number sides, in meters, have a perimeter of 22 meters? Explain.
18. A four-sided fence is to have a perimeter of 25 meters, with three sides of 9, 8 and 10 meters. Find the fourth side, or explain why there is none.
19. A triangle is to have a perimeter of 30 meters, with two sides of 6 meters and 8 meters. Find the third side, or explain why there is none.
20. Jurgis wants a rectangular corral with 24 meters of fence, every side a whole number of meters. List every length and width he could choose. (Count 11 by 1 and 1 by 11 as the same corral.)
21. A rectangle has a perimeter of 40 meters, and its length is 4 meters more than its width. Find the length and the width.

Explain your thinking

22. Explain why half of a rectangle’s perimeter is always one length plus one width.
23. A rectangle is said to have a perimeter of 36 meters and a length of 20 meters. Explain why no such rectangle exists.
24. Can you find the length and width of a rectangle when you know only its perimeter? Explain, using an example.
25. Explain why the perimeter of a square with whole-number sides is always a number you say when counting by 4s, and use this to say why no such square has a perimeter of 30.