A circle has an area of 40 centimeters squared. Calculate the radius of the circle accurate to three decimal places. Use this value to calculate the circumference of the circle, giving your answer accurate to one decimal place.
Before jumping into this problem, let’s consider the information that we’ve been given and the information that we’re trying to find. We’ve been given the area of a circle, and we’re asked to find two different things. First, we’re asked to find the radius of that circle and then use that radius to calculate the circumference.
Let’s clear a little bit of space so that we can focus our attention on the first part of the question. Next, we need to recall the formula for the area of a circle, which is 𝐴 equal to 𝜋 times 𝑟 squared, where the variable 𝐴 is the area and the variable 𝑟 is the length of the radius. Since we are given that the area is 40 centimeters squared, we will substitute 40 for the variable 𝐴 in our equation. Then to solve for 𝑟, we divide both sides by the number 𝜋, resulting in the fraction 40 over 𝜋. Then we use the square root’s method to finish solving for 𝑟. We will only take the positive square root because the radius is a length or distance, which by definition must be positive.
It may be helpful at this point to notice that our area formula can be rearranged to isolate 𝑟 such that 𝑟 equals the positive square root of the area divided by 𝜋. Now that we have solved for 𝑟, we’re ready to calculate the approximate radius accurate to three decimal places by using our calculator. If our calculator does not have a button for the irrational number 𝜋, then let’s use at least the first six decimal places of 𝜋, which will be 3.141593.
Evaluating our expression on the calculator gives us approximately 3.568248, and those decimals continue on forever. But we are asked for only the first three decimal places. Those are five six eight. And we look at the fourth decimal place to see if that digit is five or greater. If this was the case, then we would round our eight to a nine. But since our fourth digit is a two, we do not round the third digit up. Therefore, given a circle with an area of 40 centimeters squared, we have calculated the radius to three decimal places, 3.568 centimeters.
Now we will clear some space and answer the second part of our question.
Use this value, meaning the radius, to calculate the circumference of the circle, giving your answer accurate to one decimal place.
To answer the second question, we’ll need to recall the circumference of a circle formula, which is 𝐶 equal to two times 𝜋 times radius. Now we are being asked to use our approximate radius value, which was rounded to three decimal places. Now this is quite unusual because normally we would want to keep all of the decimal places in our calculations for maximum accuracy and only round in our final answer. For maximum accuracy, we would substitute the exact radius value, which was square root of 40 divided by 𝜋.
Entering this expression into the calculator gives us an approximate value of 22.41996. But this is not what we are being asked to do; we are being asked to use the rounded value from the first part of the question, 3.568. So we’ll substitute that into our circumference formula. This gives us a slightly different and less accurate value of the circumference. By substituting our rounded value, we have preserved accuracy at least out to the second decimal place. But you can see that the digits are different after that point.
Now we are only asked for accuracy to one decimal place. So it turns out we would’ve gotten the answer of 22.4 centimeters whether we used the exact radius or the approximate radius in this case. But keep in mind we should use the exact value when we have the choice because that will preserve more accuracy in the later decimal places.
In conclusion, a circle with an area of 40 centimeters squared will have an approximate radius of 3.568 centimeters and the circumference of approximately 22.4 centimeters.