Year 11 Maths: Area of sectors

What is the Area of Sectors?

In simple words, a sector is a part of a circle. Imagine you have a pizza and you take a slice out of it. That slice looks like a triangle with a curved base, right? That’s a sector!

So, how do we calculate the area of this pizza slice or sector?

Calculating the Area of Sectors

The formula to calculate the area of a sector is given by:

Where:

• A is the area of the sector
• r is the radius of the circle
• θ is the central angle of the sector in radians

Remember, we must convert the angle to radians when using this formula. We can convert degrees to radians using:

Key Rules

1. Always remember to convert the angle from degrees to radians before using the formula.
2. The radius of the sector is the same as the radius of the circle it comes from.

Tips and Tricks

1. You might not always be given the central angle of the sector. In these cases, you might need to use other properties of circles or triangles to find this angle first.
2. Drawing a diagram can often help you visualize the problem and see how to solve it.

Practice Questions

Now, let’s try some questions to apply what we’ve learned.

Easy Level

1. Calculate the area of a sector with a radius of 5 cm and a central angle of 60 degrees.
2. Calculate the area of a sector with a radius of 2 cm and a central angle of 45 degrees.
3. Calculate the area of a sector with a radius of 7 cm and a central angle of 90 degrees.
4. Calculate the area of a sector with a radius of 3 cm and a central angle of 30 degrees.
5. Calculate the area of a sector with a radius of 4 cm and a central angle of 120 degrees.

Medium Level

1. Calculate the area of a sector with a radius of 5 cm and a central angle of 72 degrees.
2. Calculate the area of a sector with a radius of 7 cm and a central angle of 144 degrees.
3. Calculate the area of a sector with a radius of 6 cm and a central angle of 36 degrees.
4. Calculate the area of a sector with a radius of 9 cm and a central angle of 108 degrees.
5. Calculate the area of a sector with a radius of 8 cm and a central angle of 180 degrees.

Hard Level

1. Calculate the area of a sector with a radius of 10 cm and a central angle of 30 degrees.
2. Calculate the area of a sector with a radius of 15 cm and a central angle of 60 degrees.
3. Calculate the area of a sector with a radius of 20 cm and a central angle of 90 degrees.
4. Calculate the area of a sector with a radius of 25 cm and a central angle of 120 degrees.
5. Calculate the area of a sector with a radius of 30 cm and a central angle of 150 degrees.

(The answers and explanations are given at the end)

Answers and Explanations

Easy Level

1. $$A = \frac{5^2 \times \left(\frac{60 \times \pi}{180}\right)}{2} = 13.09$$ cm²
2. $$A = \frac{2^2 \times \left(\frac{45 \times \pi}{180}\right)}{2} = 1.57$$ cm²
3. $$A = \frac{7^2 \times \left(\frac{90 \times \pi}{180}\right)}{2} = 27.45$$ cm²
4. $$A = \frac{3^2 \times \left(\frac{30 \times \pi}{180}\right)}{2} = 2.36$$ cm²
5. $$A = \frac{4^2 \times \left(\frac{120 \times \pi}{180}\right)}{2} = 16.76$$ cm²

Medium Level

1. $$A = \frac{5^2 \times \left(\frac{72 \times \pi}{180}\right)}{2} = 18.08$$ cm²
2. $$A = \frac{7^2 \times \left(\frac{144 \times \pi}{180}\right)}{2} = 50.27$$ cm²
3. $$A = \frac{6^2 \times \left(\frac{36 \times \pi}{180}\right)}{2} = 11.31$$ cm²
4. $$A = \frac{9^2 \times \right(\frac{108 \times \pi}{180}\right)}{2} = 86.39$$ cm²
5. $$A = \frac{8^2 \times \left(\frac{180 \times \pi}{180}\right)}{2} = 100.53$$ cm²

Hard Level

1. $$A = \frac{10^2 \times \left(\frac{30 \times \pi}{180}\right)}{2} = 26.18$$ cm²
2. $$A = \frac{15^2 \times \left(\frac{60 \times \pi}{180}\right)}{2} = 58.90$$ cm²
3. $$A = \frac{20^2 \times \left(\frac{90 \times \pi}{180}\right)}{2} = 157.08$$ cm²
4. $$A = \frac{25^2 \times \left(\frac{120 \times \pi}{180}\right)}{2} = 327.25$$ cm²
5. $$A = \frac{30^2 \times \left(\frac{150 \times \pi}{180}\right)}{2} = 590.74$$ cm²