Video Transcript
A circle has a circumference of 90
centimeters. Work out the square of the
circumference. Divide the square of the
circumference by four 𝜋, giving your answer accurate to two decimal places. Work out the radius of the circle
to three decimal places. Work out the area of the
circle. Round your answer to two decimal
places.
The first part of this question
asks us to work out the square of the circumference. We’re told that the circumference
is 90 centimeters. So the square of this value is 90
squared, which is 8,100.
The second part of the question
asks us to divide the square of the circumference by four 𝜋. So we take the value we’ve just
calculated and divide it by four 𝜋. As a decimal, this is equal to
644.577 continuing. We’re asked to give the answer to
two decimal places. So, as the digit in the third
decimal place is a seven, we round up. The square of the circumference
divided by four 𝜋 is equal to 644.58 to two decimal places.
Next, we’re required to calculate
the radius of the circle. To do so, we can recall that the
circumference of a circle is equal to two 𝜋 times its radius. Rearranging this equation by
dividing by two 𝜋, we have that the radius of the circle is equal to its
circumference over two 𝜋. Substituting 90 for the
circumference of this circle gives 𝑟 equals 90 over two 𝜋. As a decimal, this is equal to
14.3239 continuing. We’re asked to give this value to
three decimal places. And as the digit in the fourth
decimal place is a nine, we round up. The radius of the circle to three
decimal places is 14.324 centimeters.
The final part of the question asks
us to calculate the area of the circle. We recall that the area of a circle
is equal to 𝜋 multiplied by its radius squared. Substituting the value we’ve just
found for the radius and using the exact value before rounding if possible gives
that the area is equal to 𝜋 multiplied by 14.323 continuing squared. That’s 644.5775 continuing, which
to two decimal places is 644.58, and the units for this area are square
centimeters.
Now, there is an alternative
approach we could take, in which we use some of the work we’ve already done to
answer this last part of the question. In the third part of the question,
we found that the radius of a circle is equal to the circumference divided by two
𝜋. So we could substitute this
expression in place of 𝑟 in the area formula. The square of 𝐶 over two 𝜋 is 𝐶
squared over four 𝜋 squared. Canceling a factor of 𝜋 in the
numerator and denominator gives the simplified expression 𝐶 squared over four
𝜋.
Looking back at the second part of
the question, this is the expression that we evaluated when dividing the square of
the circumference by four 𝜋. Notice that the answers to parts
two and four of the question are the same, because the expressions are
equivalent. So, if we had made this connection
immediately, we wouldn’t actually have needed to carry out any further work in order
to answer the final part of the question.
Our answers to the four parts of
the question are 8,100, 644.58, 14.324 centimeters, and 644.58 square
centimeters.
