Video Transcript
The perimeter of a square is twice the circumference of a certain circle. Given that the side of the square is 423.9 centimeters and using 3.14 to approximate 𝜋, determine the diameter of the circle.
First, let’s think about what we know about circles. To find the circumference of a circle, you multiply two times 𝜋 times the radius. And the radius is the distance from the center of the circle to any point on the outside of the circle. But we’re looking for the diameter. So it might be useful to try and find the circumference in relation to the diameter. The diameter is the distance from one side of the circle to the other side through the center. And it is two of the radius. Two 𝑟 is the same thing as the diameter. So we can say that the circumference of a circle is the diameter times 𝜋.
We also need to think about a square. The perimeter of a square is the distance all the way around the square. It’s the side plus side plus side plus side. But in a square, all the s’s, all the sides, are the same value. And so we usually just write that as four times the side length, four s. We’ve been told that the perimeter of a square is twice the circumference of a certain circle. If we take the circumference and multiply it by two, that will be equal to the perimeter of the square. The perimeter of the square is twice the circumference. And so two times the circumference equals the perimeter of the square. If we plug in what we know, the circumference is equal to — remember we want to use the formula that includes the diameter. Since that’s what we’re looking for. We can say that two times the diameter times 𝜋 is equal to four s.
From there, let’s plug in what we know. We’re going to use 3.14 to represent 𝜋. So we’ll have two times 3.14 times 𝑑. And we know that the side of the square is 423.9. So on the right side, we’ll have four times 423.9. When we multiply four times 423.9, we get 1695 and six-tenths. On the left, we need to multiply two times 3.14, which equals 6.28. And we bring down the 𝑑. To find out what the diameter is equal to, we need to divide both sides of the equation by 6.28. On the left, we’ll just be left with 𝑑. And 1695 and six-tenths divided by six and twenty-eight hundredths equals 270. The diameter is a measure of distance. And because the units of the square is centimeters, the units of the diameter will also be measured in centimeters. The diameter of this circle measures 270 centimeters.
If we wanted to do a quick check, we could find the circumference and the perimeter, respectively. If we multiply the diameter, 270 by 3.14, an approximation for 𝜋, we get 847 and eight-tenths centimeters. And then we wanna calculate the perimeter of the square. So we multiply four by 423 and nine-tenths, which, just like we saw earlier, is 1695 and six-tenths centimeters. The perimeter of the square should be two times the circumference of the circle. We wanna know is 847 and eight-tenths times two 1695 and six-tenths. When we multiply 847 and eight-tenths by two, we do get 1695 and six-tenths. And so we’ve correctly calculated a diameter that would make the perimeter of the square twice the circumference of the circle.
