Sector of a circle

A sector of a circle is a piece of a circle that looks like a slice of pizza or cake. You cut it out with two cuts that run from the centre to the edge. So a sector is bounded by two radii and a piece of the circumference between them.

A sector of a circle with radius r, angle at the centre α and arc l

The sector itself is shaded red, and its area is what we are going to find. The black line segments are the two radii r, that is, the two knife cuts. The angle α between the radii is marked in orange. It is called the angle at the centre, because its vertex is at the centre of the circle. The arc l, shown in blue, is the piece of the circumference that bounds the sector on the outside – on a pizza, it is the crust.

Formulae for a sector

If you only want the formula, then the area A of a sector with radius r and angle at the centre α in degrees is

$A = \frac{α}{360^{\circ}} \cdot π \cdot r^{2}$

and the length l of the arc of the same sector is

$l = \frac{α}{360^{\circ}} \cdot 2 \cdot π \cdot r$

For example, a sector with radius r = 6 and angle at the centre α = 60° has area

$A = \frac{60^{\circ}}{360^{\circ}} \cdot π \cdot 6^{2} = \frac{1}{6} \cdot π \cdot 36 = 6 π$

and its arc has length

$l = \frac{60^{\circ}}{360^{\circ}} \cdot 2 \cdot π \cdot 6 = \frac{1}{6} \cdot 12 π = 2 π$

If you do not like an answer with pi in it, you can substitute the approximate value 3.14:

$A \approx 6 \cdot 3.14 = 18.84, l \approx 2 \cdot 3.14 = 6.28$

How did we get this?

A full circle has an angle of 360° around the centre. A sector is the same fraction of the circle as its angle at the centre is of 360°. A sector with a 90° angle is a quarter of the circle, because 90° is a quarter of 360°. A sector with a 180° angle is half the circle, one with a 270° angle is three quarters, and one with a 360° angle is the whole circle:

As the angle at the centre grows, so do the sector and its arc: a quarter, a half, three quarters and the whole circle

So the fraction of the whole circle that the sector takes up is

$\frac{α}{360^{\circ}} .$

For 90° we get $\frac{90^{\circ}}{360^{\circ}} = \frac{1}{4}$ , and for 60° we get $\frac{1}{6}$ . The area of the sector is therefore the same fraction of the area of the whole circle. The area of a circle is

$A_{\circ} = π \cdot r^{2},$

so we get the area of the sector by multiplying it by this fraction:

$A = \frac{α}{360^{\circ}} \cdot A_{\circ} = \frac{α}{360^{\circ}} \cdot π \cdot r^{2}$

The arc works in exactly the same way. As the angle opens, the blue arc grows with it, and at 360° it becomes the whole circumference. So the arc is the same fraction of the circumference of the circle as the angle α is of 360°. The circumference is

$C = 2 \cdot π \cdot r,$

and the length of the arc is therefore

$l = \frac{α}{360^{\circ}} \cdot C = \frac{α}{360^{\circ}} \cdot 2 \cdot π \cdot r$

Example: pizza

A pizza has a diameter of 32 cm and is cut into 8 equal slices. What is the area of one slice?

A pizza cut into eight slices, with one slice pulled out

Each slice is a sector of a circle. First we need its angle at the centre. Eight equal slices share the full 360° between them, so one slice has angle

$α = 360^{\circ} \div 8 = 45^{\circ}$

Now watch out for a trap: 32 cm is the diameter of the pizza, not the radius. The formula needs the radius, which is half the diameter, so r = 16 cm. Substitute into the formula for the area of a sector:

$A = \frac{45^{\circ}}{360^{\circ}} \cdot π \cdot 16^{2}$

The fraction cancels to one eighth, which makes sense, because a slice is an eighth of the pizza. Square the 16:

$A = \frac{1}{8} \cdot π \cdot 256$

Divide 256 by 8:

$A = 32 π$

And finally substitute the approximate value of pi:

$A \approx 32 \cdot 3.14 = 100.48$

One slice has an area of approximately 100.48 square centimetres. If someone substituted the whole diameter 32 cm into the formula by mistake, they would get a slice four times as big, over 400 square centimetres. That would be a slice big enough to feed a whole family.

Example: clock

The minute hand on a wall clock is 15 cm long. How many centimetres does its tip travel in 20 minutes?

In 20 minutes the minute hand turns through α and its tip traces the arc l

The tip of the hand moves along a circle whose radius is the length of the hand, so r = 15 cm. The distance it travels is the blue arc l. We still need the angle. In 60 minutes the hand goes once round, that is, 360°. In 20 minutes it goes a third of the way round:

$α = \frac{20}{60} \cdot 360^{\circ} = 120^{\circ}$

Substitute into the formula for the length of an arc:

$l = \frac{120^{\circ}}{360^{\circ}} \cdot 2 \cdot π \cdot 15$

The fraction is one third and 2 · 15 = 30:

$l = \frac{1}{3} \cdot 30 π = 10 π$

Substitute the approximate value of pi:

$l \approx 10 \cdot 3.14 = 31.4$

In 20 minutes the tip of the minute hand travels approximately 31.4 cm.

Finding the angle when we know the arc

Sometimes it is the other way round: we know the radius and the length of the arc and want to find the angle at the centre. We start with the formula for the length of an arc and rearrange it to make the angle α the subject:

$\begin{array}{rcl} l & = & \frac{α}{360^{\circ}} \cdot 2 \cdot π \cdot r / \cdot 360^{\circ} \\ 360^{\circ} \cdot l & = & α \cdot 2 \cdot π \cdot r / \div (2 \cdot π \cdot r) \\ α & = & \frac{360^{\circ} \cdot l}{2 \cdot π \cdot r} \end{array}$

Example: A car enters a roundabout and drives along a circle of radius 10 metres. Before it leaves the roundabout again, it covers 15.7 metres along the circle. Through what angle has it turned about the centre of the roundabout?

Substitute l = 15.7, r = 10 and 3.14 for pi, because the arc length is given approximately:

$α = \frac{360^{\circ} \cdot 15.7}{2 \cdot 3.14 \cdot 10}$

Multiply out the denominator:

$α = \frac{360^{\circ} \cdot 15.7}{62.8}$

The number 15.7 is exactly a quarter of 62.8, so the fraction cancels:

$α = 360^{\circ} \cdot \frac{1}{4} = 90^{\circ}$

The car has turned through 90°, so it has gone a quarter of the way round the roundabout. We can check that we worked it out correctly: a quarter of the circumference is $\frac{1}{4} \cdot 2 \cdot 3.14 \cdot 10 = 15.7$ metres. That checks out.

Angle in radians

If your angle is given in radians, the formulae get simpler. A full circle is 360° = 2π radians, so the fraction that says how much of the circle the sector takes up is $\frac{α}{2 π}$ . After substituting, the pi cancels:

$l = \frac{α}{2 π} \cdot 2 \cdot π \cdot r = α \cdot r$

$A = \frac{α}{2 π} \cdot π \cdot r^{2} = \frac{1}{2} \cdot α \cdot r^{2}$

Watch out: these shortened formulae need the angle in radians, not degrees. If you substituted 45, say, you would get nonsense.

For example, the pizza slice from the example above has angle 45°, which is $\frac{π}{4}$ radians. The area of the slice is then $\frac{1}{2} \cdot \frac{π}{4} \cdot 16^{2} = 32 π$ , just as before.

Calculator

You can also work out the area of a sector and the length of its arc here. You can enter the angle in degrees or in radians, and you can switch the units for each field.