Rectangle Area and Perimeter Calculator

This calculator finds the area and perimeter of a rectangle when the side lengths are known.

• A rectangle is a quadrilateral (4-sided shape) with 4 right angles.
• Every rectangle has opposite sides of equal length.
• If the sides are all the same length, then you have a special case of a rectangle called a square.

The area of a rectangle is the amount of space inside the rectangle.

The perimeter of a rectangle is the distance around the outside edge of the rectangle.

An example of this type of problem might be if we have a fence of a particular length and we want to enclosed the maximum rectangular space inside it.

It turns out that to get the largest area for a rectangle with a fixed perimeter is to make a square shape with equal sides.

If you look at the table below, you can see 3 rectangles, each with a perimeter of 12 cm.

• The rectangle with the largest area is the square with an area of 9 cm².

As the difference between the two adjacent sides of the rectangles become closer in value, the amount of space inside the rectangle increases.

As the difference between the two adjacent sides becomes greater, the amount of space inside the rectangle decreases.

This means that if we have a fixed perimeter, then to get the greatest area we need to divide the perimeter by 4 to get the length of each side of the square.

Sometimes, we know the area and perimeter of the rectangle but we need to find the length of the sides.

The formula for this is quite tricky. If you know that the lengths of the sides are integers, then Method 1) might be the easiest way to do this.

If you are confident using formulas or the side lengths have decimal or fractional values, then Method 2) will be better.

Method 1) Using factor pairs

Write down all the factor pairs that multiply together to make the area value.

Once you have written down the factor pairs, you can see if they make the correct perimeter by doubling the total of the pairs.

Example: A rectangle has an area of 36cm² and a perimeter of 26 cm.
What are the lengths of the two adjacent sides?

Factor pairs which multiply together to make 36 are:

• 1 and 36
• 2 and 18
• 3 and 12
• 4 and 9
• 6 and 6

The factor pairs represent the possibilities for the lengths of the two sides.

Now if we work out the perimeter of these pairs

• 1 and 36. Perimeter = 2 x (1 + 36) = 2 x 37 = 74 cm
• 2 and 18. Perimeter = 2 x (2 + 18) = 2 x 20 = 40 cm
• 3 and 12. Perimeter = 2 x (3 + 12) = 2 x 15 = 30 cm
• 4 and 9. Perimeter = 2 x (4 + 9) = 2 x 13 = 26 cm
• 6 and 6. Perimeter = 2 x (6 + 6) = 2 x 12 = 24 cm

The rectangle we need is the one with sides 4 cm and 9 cm long which have an area of 36 cm² and a perimeter of 26 cm.

Method 2) Using the formula

Alternatively, you can use the formula which is outlined below.

We know that \[ A = l \times w \] \[ P = 2(l + w) \]

If we rewrite the equations and substitute the new values into the equations, this gives us a formula for finding the length of the two sides.

The formula for finding the two sides is: \[ lengths = {P \over 4} \pm \sqrt{{P^2 \over 16} - A } \] this gives us both values of the two adjacent sides because of the plus/minus sign.

We can split this formula up into two separate formulas for each of the sides: \[ l = {P \over 4} + \sqrt{{P^2 \over 16} - A } \] \[w = {P \over 4} - \sqrt{{P^2 \over 16} - A } \] where l is the longer side of the rectangle and w is the shorter side

Example: A rectangle has an area of 21.7 cm and a perimeter of 19.4 cm.
What are the lengths of the sides?

Using the formula gives us: \[ lengths = {19.4 \over 4} \pm \sqrt{{19.4^2 \over 16} - 21.7} \] Simpliying this gives us: \[ lengths = 4.85 \pm \sqrt{23.5225 - 21.7} \] which means that \[ lengths = 4.85 \pm \sqrt {1.8225} = 4.85 \pm 1.35 \]

So the rectangle has a longer side of 4.85 + 1.35 = 6.2 cm
and a shorter side of 4.85 - 1.35 = 2.5 cm

We have a range of other area worksheets and support pages for a range of different 2d shapes.

We have a range of area and volume calculators to help you find the area and volumes of a range of different 2d and 3d shapes.

Each calculator page comes with worked examples, formulas and practice worksheets.

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