A circle has a circumference of 40 centimeters. Calculate the radius of the circle accurate to three decimal places. Then, use this value to calculate the area of the circle, giving your answer accurate to one decimal place.
Let’s begin by reminding ourselves of some of the language in the question. The circumference of a circle is the distance all the way around the circle’s edge. And the radius of a circle is the distance from the center of that circle to any point on the circumference. In this question, we’re told that the circle has a circumference of 40 centimeters and we want to use this to calculate the radius. So, we need to recall the formula that we use for calculating the circumference of a circle. We can either use 𝜋𝑑, which means 𝜋 multiplied by the diameter of the circle. Or we can use the formula two 𝜋𝑟, two multiplied by 𝜋 multiplied by the radius of the circle. These two formulae are equivalent because the diameter of a circle is always twice the radius.
As it is the radius of the circle that we wish to calculate, we’ll use this second formula here. The circumference of the circle is 40 centimeters. So, we can substitute this value on the left-hand side, giving 40 is equal to two 𝜋𝑟. To solve this equation and leave 𝑟 on its own on the right-hand side, we need to divide by two 𝜋. But whatever we do to one side of an equation, we must also do to the other. So, we divide both sides of the equation by two 𝜋, which gives 40 over two 𝜋 is equal to 𝑟. We can simplify this fraction by dividing the numerator and denominator by two. 40 divided by two is 20 and two divided by two is one. So, we have 20 over one 𝜋 or just 𝜋.
Now, this is an exact value for the radius. But the question asks us to give our answer accurate to three decimal places. So, we need to use a calculator to evaluate 20 divided by 𝜋. We could also use our calculator to evaluate 40 divided by two 𝜋 if we haven’t simplified. We get that 𝑟 is equal to 6.36619. To round to three decimal places, we look at the number in the fourth decimal place. And we see that our deciding number is a one. As this is less than five, we’re rounding down. So, the six in the third decimal place will remain a six. So, the radius of the circle, correct to three decimal places, is 6.366. And the units for this radius will be the same as the units for the circumference, which are centimeters.
Next, we’re asked to calculate the area of the circle, which is the total amount of space inside the circle. We have a formula for this. Area is equal to 𝜋𝑟 squared, where 𝑟 represents the radius of the circle, which we’ve just calculated. So, we can substitute the value of 𝑟 we’ve just found, given that the area of this circle is equal to 𝜋 multiplied by 6.366 squared. Evaluating this on a calculator gives 127.3160. This time, we’re asked to give our answer to one decimal place. So, our deciding number is the digit in the second decimal place, which is a one, telling us that we’re rounding down. And so, the three in the first decimal place will remain a three. The area of the circle to one decimal place, then, is 127.3. And the units for this are square units. They’ll be centimeters squared because the lengths in the question were given in centimeters.
Now, it is just worth mentioning that we always want to be as accurate as possible. So, we could have used the exact value of 𝑟 from our calculator display in the next stage of the calculation. Rather than using the rounded value of 6.366, we could have used the value of 6.3661 continuing. If we were to use this unrounded value, we’d get a similar answer of 127.3239. The values do differ in the higher decimal places, but they do agree to one decimal place. They’re both 127.3 when rounded to one decimal place. The value that we used for the radius was to quite a high degree of accuracy, three decimal places. And as we’re only asked for the area to one decimal place, it was fine for us to use this rounded value. We have, then, that the radius of the circle, to three decimal places, is 6.366 centimeters. And the area of the circle, to one decimal place, is 127.3 centimeters squared.