A circular birthday cake with a diameter of 22 centimeters is divided into 11 equal sectors. Using 3.14 as an approximation for 𝜋, find the area of one sector.
Recall, the formula for the area of a sector with an angle of 𝜃 radians and a radius 𝑟 is one-half 𝑟 squared 𝜃. To be able to calculate the area of one of the sectors, we’re going to need to calculate two dimensions.
The first is the radius. Remember, the radius is the length of the line that joins the centre of the circle to a point on its circumference. It’s half the length of the diameter. Since the birthday cake has a diameter of 22 centimeters, its radius can be found by halving this value. That’s 22 divided by two, which is 11 centimeters.
The other measure that we need to calculate is the size of the angle 𝜃. Angles around a point sum to two 𝜋 radians. And the cake’s been split into 11 equal pieces. We can therefore calculate the size of 𝜃 by dividing two 𝜋 by 11. That’s two elevenths of 𝜋.
All that’s left is to substitute what we know about our sector into the formula for area of a sector. The area of one sector is a half multiplied by 11 squared multiplied by two-elevenths 𝜋. We can cross cancel by a factor of two and 11. And that tells us that the area of one of the sectors is 11𝜋 centimeters cubed.
We were told however to use 3.14 as an approximation for 𝜋. So let’s substitute 3.14 into the expression for the area of a sector. It’s 11 multiplied by 3.14. To work out 11 multiplied by 3.14, we begin by calculating 11 multiplied by 314.
There are a number of formal written methods we could use. But remember, 10 multiplied by 314 is 3140. That means 11 multiplied by 314 is 3140 plus one more 314. That’s 3454. 3.14 is 100 times smaller than 314. So our answer is going to be 100 times smaller than 3454. 3454 divided by 100 is 34.54.
And the area of one sector is 34.54 centimeters squared.