Circles Sectors and Arcs Worksheets, Questions and Revision

The formula for the area of a circle is \pi r^2. In this question we are given the diameter rather than the radius. Given that the diameter is twice the length of the radius:

\text{radius} = 5.24 \div 2 = 2.62 \text{cm}

Therefore,

\text{area } = \pi r^2 = \pi \times (2.62)^2 = 21.6 \text{cm}^2

The formula for the area of a circle is \pi r^2. The area of the sector is, 

\text{area} =\dfrac{50\degree}{360\degree} \times \pi (5)^2 = \dfrac{50\degree}{360\degree} \times 25 \pi = 10.9\text{m}^2

The formula for the area of a circle is \pi r^2. The area of this sector, 26.15 \text{m}^2, must be equal to \frac{x}{360} of the total area of the circle. So, as an equation, this looks like:

26.15 = \dfrac{x\degree}{360\degree} \times \pi \times 9^2 = \dfrac{x\degree}{360\degree} \times 81\pi

Dividing the left-hand side and right-hand side by 81 \pi gives

\dfrac{26.15}{81\pi} = \dfrac{x\degree}{360\degree}

Then, multiply both sides by 360 to get,

x=\dfrac{26.15}{81\pi}\times 360\degree

x = 37.0\degree (to 1d.p.)

The question asks for the total perimeter of the shape. We know that one side is 14 mm but the other two are missing. Immediately we can identify that the other straight side is also 14 mm. Then, all that remains is to calculate the arc length.

The angle in this sector is 165 degrees, meaning that the arc length will be equal to \frac{165}{360} of the total circumference. The formula for the circumference is \pi d, or alternatively (and more helpfully in this case), 2\pi r. So, we get:

Arc length = \dfrac{165}{360} \times 2 \pi \times 14 = 40.3 \text{mm}

Therefore,

\text{total perimeter} = 14 + 14 + 40.3 = 68.3 \text{mm}

The formula for the area of a circle is \pi r^2. The area of this sector, 160 \text{m}^2, must be equal to \frac{x}{360} of the total area of the circle. So, as an equation, this looks like:

160 = \dfrac{x\degree}{360\degree} \times \pi \times 9^2 = \dfrac{x\degree}{360\degree} \times 81\pi

Dividing the left-hand side and right-hand side by 81 \pi gives

\dfrac{160}{81\pi} = \dfrac{x\degree}{360\degree}

Then, multiply both sides by 360 to get,

x=\dfrac{160}{81\pi}\times 360\degree

x = 226\degree (to 3s.f.)