How can I find the missing side length of a rectangle from its perimeter?
Quick answer
A rectangle has two sides of each length, so its perimeter is twice the known side plus twice the missing side: P = 2a + 2x. Substitute the perimeter and known side, solve for x, then add all four sides to check. For example, if P = 34 cm and one side is 7 cm, then 34 = 2(7) + 2x, so x = 10 cm; 7 + 10 + 7 + 10 = 34 cm. [S1]
The question, the details, the next step.
I know a rectangle’s perimeter and one side, but not the other side. How do I work out the missing length?
Use the fact that a rectangle’s opposite sides are equal. If the known side is a and the missing side is x, the four sides are a, x, a, and x. That gives the perimeter equation P = 2a + 2x. [S1]
Put the values you know into the equation, then solve for x. You can also subtract the two known sides from the perimeter, then divide the remainder by 2, because the missing side occurs twice. Finally, add all four sides to confirm that they reproduce the stated perimeter. [S1]
Could you show the steps with actual numbers? I want to see how the check works too.
Suppose the perimeter is 34 cm and one side is 7 cm. We’ll call the missing side x. These are example values; use your own measurements if they differ.
- Start with P = 2a + 2x.
- Substitute: 34 = 2(7) + 2x.
- Simplify: 34 = 14 + 2x.
- Subtract 14 from both sides: 20 = 2x.
- Divide by 2: x = 10 cm.
Now rebuild the perimeter: 7 + 10 + 7 + 10 = 34 cm. The result checks out. The equation works because each of the two side lengths appears twice in a rectangle’s perimeter. [S1]
What if the side I know is called the width, or the problem gives a different set of units?
The labels “length” and “width” don’t change the calculation. Call the known side a and the missing side x; opposite sides match, so the perimeter is still 2a + 2x. [S1]
Keep units consistent. For example, if the perimeter is 2 metres and the known side is 40 centimetres, convert one measurement so both are in the same unit before calculating. Convert 2 m to 200 cm: 200 = 2(40) + 2x, so x = 60 cm. Check: 40 + 60 + 40 + 60 = 200 cm. This is an illustrative calculation using the rectangle perimeter relationship. [S1]
Can you give me a short template to use on a worksheet, and how can I tell if I made a mistake?
Copy and fill in this pattern:
Known side a = ___ [unit] Perimeter P = ___ [same unit] Equation: P = 2a + 2x Substitute: ___ = 2(___) + 2x Solve: x = ___ [unit] Check: a + x + a + x = ___ [unit]
For the check, the final sum should equal the given perimeter. Also use the same unit throughout and check that the answer is a positive side length. If the check fails, review the substitution and arithmetic. This method assumes the shape is a rectangle and uses the stated perimeter and side measurement. Adding all four sides checks the calculation against the perimeter. [S1]
A practical checklist
Write down the known side and perimeter with their units.
Convert measurements so they use one consistent unit.
Set up P = 2a + 2x, where x is the missing side.
Solve for x and include the unit.
Add the four side lengths and confirm the total equals P.
A few more useful answers
Why do I divide the leftover perimeter by 2?
After subtracting the two known sides, what remains is the combined length of the two equal missing sides. Divide that remainder by 2 to get one missing side. [S1]
What if my calculation gives a negative side length?
A negative length is not a valid result for this rectangle problem. Recheck that the perimeter and known side use the same units and that you substituted the values correctly.
Does this method work for any four-sided shape?
This equation relies on opposite sides being equal, as they are in a rectangle. For a different shape, use the side relationships and perimeter information given for that shape. [S1]
Sources
References used for this guide.
- [S1] OpenStax Prealgebra 2e: Use Properties of Rectangles, Triangles, and Trapezoids openstax.org · Checked