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:
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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
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.
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?
\(\text{diameter = radius} \cdot 2\)
\(\text{radius} = \frac{\text{diameter}}{2} = \frac{10}{2} = 5 \space cm\)
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?
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.
| Degrees | Radians |
| \(\frac{\pi}{6}\) | |
\(\frac{\pi}{4}\) | |
\(\frac{\pi}{3}\) | |
\(\frac{\pi}{2}\) | |
| \(\pi\) | |
\(\frac{3\pi}{2}\) | |
| \(2 \pi\) |
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?
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?
\(\text{Diameter = Radius} \cdot 2\text{ Radius} = \frac{\text{Diameter}}{2} = \frac{19}{2} = 9.5\)
\(\text{Arc Length of a sector} = \pi \cdot d \cdot \frac{\theta}{360}\)
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What is a sector of the circle?
A sector of a circle is a proportion of a circle where two sides are radii.
How do you find the sector of a circle?
To find the sector of a circle you need to use one of the formulas for the area of the sector. Which one you use is dependent on whether the angle at the centre is in radians or in degrees.
What are the formulas of the sector of the circle?
There are two formulas of a sector. One is to calculate the area of a sector of a circle. Area of a sector= pi × r^2 × (θ /360). The other one is to find the arc length of the sector of the circle. Arc length = pi × d × (θ /360)
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