15.03 Parts of circles

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.

Examples

Example 1

The angle \angle AXB is equal to 120\degree.

a

What fraction of the whole circle lies on the arc AB?

Worked Solution

Create a strategy

To find the fraction make the angle of the arc the numerator and the angle 360\degree the denominator.

Apply the idea

\displaystyle \text{Fraction}	\displaystyle =	\displaystyle \frac{120}{360}	Put the angle over 360
	\displaystyle =	\displaystyle \frac{1}{3}	Simplify the fraction

One third of the circle is covered by the arc.

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b

Find the exact length of the arc AB.

Worked Solution

Create a strategy

Use the formula C=2\pi r and multiply it by \dfrac{1}{3} to find the arc length.

Apply the idea

We multiply the circumference by \dfrac{1}{3} to find one third of it.

\displaystyle \text{Arc length }	\displaystyle =	\displaystyle 2\pi r \times \frac{1}{3}	Use the formula and multiply \dfrac{1}{3}
	\displaystyle =	\displaystyle 2\pi \times 9 \times \frac{1}{3}	Substitute the radius
	\displaystyle =	\displaystyle 6\pi\, \text{cm}	Simplify the fraction

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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.

Examples

Example 2

Find the perimeter of the sector shown, correct to two decimal places.

Worked Solution

Create a strategy

Add the two radii and the arc length.

Apply the idea

The radius of the circle is 7\text{ cm} which will be the lengths of our two straight sides.

The angle of the sector is 40\degree so we put this angle over 360\degree to find the fraction of the circumference for the arc length.

\displaystyle \text{Perimeter}	\displaystyle =	\displaystyle 2r + \frac{40}{360} \times 2\pi r	Add the radii to the arc length
	\displaystyle =	\displaystyle 2 \times 7+\frac{40}{360} \times 2\pi \times 7	Substitute the radius
	\displaystyle \approx	\displaystyle 18.89	Evaluate and round

The perimeter of this sector is 18.89\,cm.

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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.

Examples

Example 3

The sector in the diagram has an angle of 30\degree and a radius of 6\,cm.

a

What fraction of the circle's area is covered by this sector?

Worked Solution

Create a strategy

To find the fraction make the angle of the sector the numerator and 360\degree the denominator.

Apply the idea

\displaystyle \text{Fraction}	\displaystyle =	\displaystyle \frac{30\degree}{360\degree}	Divide the angle of arc by 360\degree
	\displaystyle =	\displaystyle \frac{1}{12}	Simplify the fraction

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b

Find the exact area of the sector.

Worked Solution

Create a strategy

Use the formula \text{A}=\pi r^2 and multiply the area by \dfrac{1}{12}.

Apply the idea

\displaystyle \text{Area of the sector }	\displaystyle =	\displaystyle \pi r^2 \times \frac{1}{12}	Multiply the area by \dfrac{1}{12}
	\displaystyle =	\displaystyle \pi \times 6^2 \times \frac{1}{12}	Substitute the radius
	\displaystyle =	\displaystyle \pi \times \frac{36}{12}	Evaluate the product
	\displaystyle =	\displaystyle 3\pi\, \text{cm}^2	Simplify

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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.

And these are not annuli.

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.

Examples

Example 4

The annnulus has an inner diameter of 10\, \text{cm} and an outer diameter of 18\,cm.

Find its exact area.

Worked Solution

Create a strategy

Subtract the smaller circle's area from the larger circle.

Apply the idea

The radius of the smaller circle is r=\dfrac{10}{2}=5\text{ cm,} and the radius of the larger circle is r=\dfrac{18}{2}=9\text{ cm}.

\displaystyle \text{Area of annulus}	\displaystyle =	\displaystyle \pi \times 9^2-\pi \times 5^2	Subtract the areas
	\displaystyle =	\displaystyle 81\pi -25\pi	Simplify
	\displaystyle =	\displaystyle 56\pi\, \text{cm}^2	Evaluate

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