Area of a Circle

The space inside a circle

The area of a circle is the space inside it, counted in unit squares, as for any other shape. A circle has no straight sides to multiply together, so we cut it into pieces and move the pieces into a shape we can measure.

Everything is worked out from the radius. A circle is every point that is the same distance from its center, and that distance is the radius, r.

Every point of the circle is the distance r from the center.

Cut it into wedges

Cut the circle into equal wedges from the center to the edge, like slices of a round pizza. Each wedge is called a sector. Its two straight sides are radii, and its curved edge is a piece of the circumference.

Nothing is added and nothing is thrown away. However the wedges are moved around, together they still cover exactly the area of the circle.

The circle cut into 12 equal wedges. Each one has two radii for its straight sides.

Lay them tip up, tip down

Lay the wedges in a row, one pointing up and the next pointing down, so that they fit together. The shape is nearly a rectangle, but it has bumps along the top and the bottom where the curved edges are.

Cut the circle into more wedges, each one thinner, and the bumps get smaller. The top and bottom edges straighten out, and the shape gets closer and closer to a rectangle.

Drag one handle to cut the circle into more sectors, and the other to lay them out. With 64 thin sectors, the shape they make is almost exactly a rectangle.

A rectangle $\pi r$πr wide and r tall

Now measure the rectangle. Its height is the straight side of one wedge, which is the radius, r.

Its top edge is made of the curved edges of the wedges that point down, and its bottom edge is made of the curved edges of the wedges that point up. Half of the circumference is along the top and half is along the bottom. The whole circumference is $2\pi r$2πr, so the top is half of that: $\pi r$πr.

The area of the rectangle is its width times its height: $\pi r\times r=\pi r^2$πr × r = πr². The wedges are the whole circle, so this is the area of the circle too: area $=\pi r^2$= πr². Read $\pi r^2$πr² as $\pi$π times r times r: $\pi$π lots of the square built on the radius.

With 40 thin wedges the shape is almost a rectangle: $\pi r$πr wide, which is half the circumference, and r tall.

Using the formula

A circle with a radius of 7 cm has an area of $\pi \times 7\times 7=49\pi$π × 7 × 7 = 49π cm². Using $\frac{22}{7}$22/7 for $\pi$π, that is $\frac{22}{7}\times 49=154$22/7 × 49 = 154 cm².

When a question gives the diameter, halve it first. A circle with a diameter of 10 cm has a radius of 5 cm, so its area is $\pi \times 5\times 5=25\pi$π × 5 × 5 = 25π, which is about 3.14 × 25 = 78.5 cm².

Double the radius

Double the radius of a circle and every length in it doubles, so its circumference doubles too. The area goes up 4 times, because the radius is used twice in r × r. A circle with a radius of 4 has an area of $16\pi$16π, which is 4 times the $4\pi$4π of a circle with a radius of 2.

Drag the radius from 2 to 4. The circle becomes twice as wide, so its circumference is twice as long, but it covers 4 times the area: $\pi \times 4\times 4=16\pi$π × 4 × 4 = 16π, against $\pi \times 2\times 2=4\pi$π × 2 × 2 = 4π.

The usual mistakes

Using the diameter in place of the radius. For a circle with a diameter of 10, $\pi \times 10\times 10=100\pi$π × 10 × 10 = 100π is 4 times the true area, which is $\pi \times 5\times 5=25\pi$π × 5 × 5 = 25π.

Mixing up the area and the circumference. $2\pi r$2πr is the length round the edge, measured in units such as cm. $\pi r^2$πr² is the space inside, measured in square units such as cm². For a radius of 7 cm, the circumference is $14\pi$14π cm and the area is $49\pi$49π cm².

Doubling instead of squaring. $r^2$r² means r × r, not 2 × r. For a radius of 7, $r^2$r² is 7 × 7 = 49, not 14.