Circles

A circle is the set of all points that are the same distance from one point in the middle. Wheels, plates, coins and clock faces are circles, because that shape rolls smoothly and looks the same from every side. Every circle, big or small, has the same special number inside it, called pi and written π. More than 2,200 years ago the Greek thinker Archimedes showed that π is a little more than 3.14 and a little less than 3.143. With π and one length, you can find everything about a circle.

Center, radius and diameter

The point in the middle is the center. A straight line from the center to the edge of the circle is the radius (we write r). A straight line from one side of the circle to the other, through the center, is the diameter (we write d). The diameter is made of two radii put end to end. ("Radii" is the word for more than one radius.)

The number pi

Take any circle and measure the distance around it. Then divide by its diameter. You always get the same number, about 3.14159. This number is π. Its digits go on forever and never repeat, so we usually keep the symbol π in the answer instead of writing the long decimal.

So an answer like 6π means "6 times π". It is exact. If you want a normal number, multiply by about 3.14: 6π is about 18.85. Most answers in this lesson are written with π, like 16π or 25π.

Circumference: the distance around

The circumference is the length of the edge of the circle, the distance all the way around. Because the circumference divided by the diameter is always π, the circumference is π times the diameter. And since the diameter is two radii, it is also 2 times π times the radius.

Area: the space inside

Why is this true? Cut a circle into many thin slices, like a cake. Lay the slices in a row, one pointing up, the next pointing down. Together they make a shape that is almost a rectangle. Its height is one radius. Its width is half of the circumference, which is π r. The more slices you cut, the closer it is to a real rectangle. So the area is π r × r = π r²π r × r = π r².

Working backwards from one measure

Sometimes you know the circumference and must find the area, or the other way round. The plan is always the same: first find the radius, then use it.

Half circles and quarter circles

A half circle is a circle cut through its center, along a diameter. Its area is half of the whole circle. A quarter circle is one of four equal pieces, like a slice that is a quarter of a pizza. Its area is a quarter of the whole circle. So find the area of the whole circle first, then divide.

A circle inside a square

When a circle fits exactly inside a square, it touches all four sides. So the circle is as wide as the square: its diameter is the side of the square.

Angles inside a circle

An arc is a piece of the edge of a circle. Draw two lines from the ends of an arc to the center: they make a central angle. Now draw two lines from the ends of the same arc to a point on the far side of the circle: they make an inscribed angle. We say both angles "stand on" the same arc.

A famous special case is named after Thales, a Greek thinker who lived about 2,600 years ago. If one side of a triangle is a diameter and the third corner is on the circle, the angle at that corner is always 90°. The reason: the diameter is a straight line through the center, so its central angle is 180°, and half of 180° is 90°.