An arc is an unbroken part of a circle.
The length of an arc is called the arc length.
A sector is the region inside a circle between two radii.
Some special sectors we should make note of are semicircles which make half a circle's area and quadrants which make up a quarter of a circle's area.
Arc length
We can think of an arc as a fraction of the circle. Looking at it like this, we can see that an arc length is simply a fraction of the circumference.
As such, we can calculate arc lengths by finding the circumference of the circle they are a part of and then taking the appropriate fraction.
Worked Example
An arc goes a quarter of the way around a circle with radius $6$6. Find the exact arc length.
Think: Since this arc makes up a quarter of the circle, the arc length will be equal to a quarter of the circumference. We can find the circumference using the formula $C=2\pi r$C=2πr.
Do: We can substitute the radius $r=6$r=6 into the formula $C=2\pi r$C=2πr to find the circumference, then multiply it by $\frac{1}{4}$14 to take only a quarter of it. As such, we get:
| Arc length | $=$= | $2\pi r\times\frac{1}{4}$2πr×14 |
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| Arc length | $=$= | $2\pi\times6\times\frac{1}{4}$2π×6×14 |
Substitute in the value for the radius |
| Arc length | $=$= | $\frac{12}{4}\pi$124π |
Multiply the numeric values together |
| Arc length | $=$= | $3\pi$3π |
Simplify the fraction |
So the length of this arc is $3\pi$3π.
Practice question
Question 1
The angle $\angle AXB$∠AXB is equal to $120^\circ$120°.
What fraction of the whole circle lies on the arc $AB$AB?
Find the exact length of the arc $AB$AB.
Perimeter of a sector
We can find the perimeter of a sector in the same way that we find any perimeter, by adding up the lengths of the sides. A sector has three sides, two straight and one curved. So how do we find the length of each side?
We know from the definition that a sector is the region between two radii and the circle, meaning that the two straight sides must be radii and the curved side is an arc.
Worked Example
What is the exact perimeter of the sector in the diagram below?
Think: The radius of the circle is given to be $24$24 cm which will be the lengths of our two straight sides. To find the length of the curved side, we need to take the appropriate fraction of this circle's circumference.
Do: We know that there are $360^\circ$360° in a revolution. The angle of the sector is $120^\circ$120° which goes into $360^\circ$360° three times. This tells us that the sector makes up one third of the circle's area. As such, the curved side of the sector must be equal to $\frac{1}{3}$13 of the circumference.
Adding all the sides together gives us:
| Perimeter | $=$= | $r\times2+2\pi r\times\frac{1}{3}$r×2+2πr×13 |
The sum of two radii and the arc length |
| Perimeter | $=$= | $24\times2+2\pi\times24\times\frac{1}{3}$24×2+2π×24×13 |
Substitute in the value for the radius |
| Perimeter | $=$= | $48+\frac{48}{3}\pi$48+483π |
Evaluate the multiplication |
| Perimeter | $=$= | $48+16\pi$48+16π |
Simplify the fraction |
As such, the perimeter of this sector is $48+16\pi$48+16π cm.
Practice question
Question 2
The sector in the diagram has an angle of $90^\circ$90° and a radius of $14$14 cm.
What fraction of the circle's area is covered by this sector?
Find the exact length of the arc $PQ$PQ.
Fill in the blanks to find the perimeter of the sector, rounded to two decimal places.
Perimeter $=$= Length of arc $PQ$PQ $+$+ length of $PX$PX $+$+ length of $QX$QX Perimeter $=$= $\editable{}+\editable{}+\editable{}$++ (Substitute the exact values for each length) Perimeter $=$= $\editable{}$ cm (Evaluate, rounding to two decimal places.)
Area of a sector
Similar to how the arc length is simply a fraction of the circumference, the area of a sector is simply a fraction of the circle's area.
We can calculate the area of a sector by finding the area of the circle they are a part of and then taking the appropriate fraction.
Worked Example
Find the exact area of a quadrant of a circle with diameter $12$12 cm.
Think: A quadrant is a sector that makes up a quarter of the circle's area, so we can find the area of the sector by taking $\frac{1}{4}$14 of the circle's area. We can find the radius of the circle by halving the diameter.
Do: Halving the diameter tells us that the radius of this circle is $6$6 cm. We can substitute the radius $r=6$r=6 into the formula $A=\pi r^2$A=πr2 to find the area, then multiply it by $\frac{1}{4}$14 to take only a quarter of it. As such, we get:
| Area of the sector | $=$= | $\pi r^2\times\frac{1}{4}$πr2×14 | |
| Area of the sector | $=$= | $\pi\times6^2\times\frac{1}{4}$π×62×14 |
Substitute in the value for the radius |
| Area of the sector | $=$= | $\pi\times36\times\frac{1}{4}$π×36×14 |
Evaluate the square |
| Area of the sector | $=$= | $\frac{36}{4}\pi$364π |
Multiply the numeric values together |
| Area of the sector | $=$= | $9\pi$9π |
Simplify the fraction |
As such, the area of this sector is $9\pi$9π cm2.
Practice question
Question 3
The sector in the diagram has an angle of $30^\circ$30° and a radius of $6$6 cm.
What fraction of the circle's area is covered by this sector?
Find the exact area of the sector.
Annulus
An annulus is a composite shape formed by subtracting the area of a smaller disc from a larger one, where the centre of the two discs is the same.
An annulus is the region between two circles that have the same central point.
As such, these are all annuli.
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And these are not annuli.
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Since annuli are composed of an inner and outer circle, we can also say that they have an inner and outer radii, which are the distances from the central point to inside and outside edges respectively.
The inner radius is the distance from the central point to the inside edge of the annulus.
The outer radius is the distance from the central point to the outside edge of the annulus.
We can see that the perimeter of an annulus will be the sum of the circumferences of the inner and outer circles.
We can also see that the area of an annulus will be the difference between the area of the outer circle and the area of the inner circle.
Worked Example
Find the exact perimeter and area of the annulus in the diagram below.
Think: The inner radius is $6$6 cm and the outer radius is $9$9 cm. We can use these radii to find the circumferences and areas of the two circles that we used to make the annulus. The perimeter of the annulus will be the sum of the circumferences while the area of the annulus will be the difference between the areas of the circles.
Do: By substituting the inner and outer radii into the circumference and area formulas for a circle, we get:
| Circumference | Area | |
|---|---|---|
| Inner Circle | $12\pi$12π | $36\pi$36π |
| Outer Circle | $18\pi$18π | $81\pi$81π |
As such, the perimeter of the annulus is:
| Perimeter | $=$= | $12\pi+18\pi$12π+18π |
Add the circumferences together |
| Perimeter | $=$= | $30\pi$30π |
Perform the addition |
And the area of the annulus is:
| Area | $=$= | $81\pi-36\pi$81π−36π |
Subtract the area of the inner circle from the outer circle |
| Area | $=$= | $45\pi$45π |
Perform the subtraction |
As such, the perimeter of the annulus is $30\pi$30π cm and the area is $45\pi$45π cm2.
Practice question
Question 4
The annnulus has an inner diameter of $10$10 cm and an outer diameter of $18$18 cm.
Find its exact area.
