A sector of a circle is an area of a circle where two of the sides are radii. An example of the sector (in red) is shown below:
A sector of a circle -StudySmarter Originals
An arc length is a part of the circle's circumference (perimeter). For the same sector, we could have arc as shown in green:
Arc length of a circle - StudySmarter Originals
Circle sector theorems where the angle is in degrees
You might already be familiar with this but let's look at calculating the area and arc length of a circle sector when the angle is given in degrees.
Calculating the area of a sector of a circle
The formula to calculate the area of a sector with an angle \(\theta\) is:
\(\text{Area of a sector} = \pi \cdot r^2 \cdot \frac{\theta}{360}\)
where r is the radius of the circle
Circle A has a diameter of 10cm. A sector of circle A an angle of 50. What is the area of this sector?
- First, we need to calculate the radius of the circle. This is because the formula for the area of a sector uses this value rather than the diameter.
\(\text{diameter = radius} \cdot 2\)
\(\text{radius} = \frac{\text{diameter}}{2} = \frac{10}{2} = 5 \space cm\)
- Then, substitute your values into the area of a sector formula.
\(\text{Area of a sector} = \pi \cdot r^2 \cdot \frac{50}{360} = 10.9 cm^2 (3 \space s.f.)\)
Calculating the arc length of a sector of a circle
The formula to calculate the arc length of a sector with an angle \(\theta\) is:
\(\text{Arc Length of a sector}: \pi \cdot d \cdot \frac{\theta}{360}\) where d is the diameter of the circle:
Circle B has a radius of 12cm. A sector within Circle B has an angle of 100. What is the length of the arc length of this sector?
- First, the formula for the arc length of a sector requires the diameter of the circle rather than the radius.
\(\text{Diameter} = r \cdot 2 = 2 \cdot 12 = 24 cm\)
- Then, you can substitute your values from the question into the formula
\(\text{Arc length of a sector} = \pi \cdot 24 \cdot \frac{100}{360} = 20.9 cm^2 \space (3 s.f.)\)
Circle sector theorems where the angle is in radians
You also need to be able to calculate the arc length and area of a sector of a circle where the angle is given in radians.
Radians are an alternative unit to degrees that we can use to measure an angle at the centre of the circle.
To recap, some common degree to radian conversions.
Calculating the area of a sector of a circle
To calculate the area of a sector of a circle with an angle \(\theta^r\), the formula you use is:
\(\text{Area of a sector} = \frac{1}{2} \cdot r^2 \cdot \theta\)
where r is the radius of the circle.
Circle C has a radius of 15cm. Within Circle C, there is a sector with an angle of 0.5 radians. What is the area of this sector?
- As all the variables are in the form required in the formula, you can substitute their values into the formula.
\(\text{Area of a sector} = \frac{ 1}{2} \cdot 15^2 \cdot 0.5 = 56.3 cm^2 \space (3 s.f.)\)
Calculating the arc length of a sector of a circle
To calculate the arc length of a sector of a circle with an angle \(\theta^r\), the formula you use is:
\(\text{Arc length of a sector} = r \cdot \theta\), where r is the radius of the circle.
A sector in Circle D has an angle of 1.2 radians. Circle D has a diameter of 19. What is the arc length of this sector?
- The formula requires the radius rather than the diameter.
\(\text{Diameter = Radius} \cdot 2\text{ Radius} = \frac{\text{Diameter}}{2} = \frac{19}{2} = 9.5\)
- You can then substitute these values into the formula \(\text{Arc length of a sector} = 9.5 \cdot 1.2 = 11.4 \space cm\)
Sector of a Circle - Key takeaways
- A sector of a circle is the proportion of a circle where two of the sides are radii. An arc length of the sector is the proportion of the circumference which runs the length of the sector of the circle.
- If the angle at the centre of the circle is in degrees, the formula for finding the area of the sector is: \(\text{Area of a sector} = \pi \cdot r^2 \cdot \frac{\theta}{360}\). To calculate the arc length, the formula is:
\(\text{Arc Length of a sector} = \pi \cdot d \cdot \frac{\theta}{360}\)
- If the angle of the circle is in radians, the formula for finding the area of the sector is: \(\text{Area of a sector} = \frac{1}{2} \cdot r^2 \cdot \theta\). For calculating the arc length of the sector, the formula is \(\text{Arc length} = r \cdot \theta\)
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