Area of a Square: Formula, Perimeter and Diagonal

The area of a square measures the flat space inside its four sides. A square has four equal sides and four right angles. Because its base and height have the same length a, its area is a multiplied by itself.

Why is the area the side squared?

The rectangle formula is base times perpendicular height. In a square, both measurements equal the side length, so:

For a square with 10 cm sides, ten rows of ten unit squares cover the region. There are 100 unit squares altogether:

This is an area of 100 square centimetres, not a length of 100 cm. Multiplying two centimetre measurements produces square centimetres.

Area and perimeter measure different things

Perimeter is the distance around the boundary. Add the four side lengths, or multiply one side by four. A square with 5 cm sides has perimeter 20 cm:

Its area is 25 square centimetres. A frame around the square concerns perimeter; covering its interior concerns area. Do not choose a formula solely because it uses the one number supplied: decide which quantity the task asks for.

How do you find the diagonal?

A diagonal joins two opposite vertices. It divides the square into two congruent right triangles whose legs both equal a. Pythagoras gives the relation between the diagonal d and the side:

For a 10 cm side, the exact diagonal is 10 times the square root of two centimetres, approximately 14.14 cm. The diagonal must be longer than a side but shorter than two sides placed end to end, which helps catch an incorrect calculation.

Finding area when only the diagonal is known

Rearranging the diagonal relationship gives half the diagonal squared. If the area is known instead, its positive square root gives the side:

Finding a square's area from a 10 cm diagonal

A square has diagonal 10 cm. Find its area and then its side. This time 10 cm is not a side length.

1. Square the diagonal. This intermediate value is not yet the square's area.

   $$d^2={10}^2=100{\text{cm}}^2$$

2. Divide by two, using the relationship derived from Pythagoras. The resulting area is 50 square centimetres.

   $$A=\frac{100}{2}=50{\text{cm}}^2$$

3. Take the positive square root of the area to recover the side. Simplify the radical before rounding.

   $$a=\sqrt{50}=5\sqrt{2}\approx 7.07\text{cm}$$

The side is approximately 7.07 cm. It is shorter than the 10 cm diagonal, as required. Using 10 cm as the side would have produced the wrong area of 100 square centimetres.

For a diagonal of 10 cm, the area is 50 square centimetres. This is different from the 100-square-centimetre area of a square whose side is 10 cm. Label the supplied measurement before substituting it.

Converting units before squaring

A 50 cm side is 0.5 m. Its area is 2,500 square centimetres, equivalent to 0.25 square metres. Converting the side first avoids applying a length conversion factor directly to an area. One square metre contains 10,000 square centimetres.

Doubling the side doubles the perimeter but multiplies the area by four. The formulas assume a true square: four equal sides alone could describe a rhombus, whose area also depends on its shape. A face of a cube is a square, so the same area calculation is used six times for the cube's total surface.