Find in the simplest form the ratio between the circumference of a circle whose radius is 63 centimetres and the perimeter of a square whose side length is seven centimetres, where 𝜋 is equal to 22 over seven.
So we have two things that we need to work out: firstly the circumference of a circle with radius 63 centimetres and secondly the perimeter of a square with side length seven centimetres. Let’s begin with the circle.
The formula for finding the circumference of a circle is 𝜋𝑑 or two 𝜋𝑟, where 𝑟 represents the radius of the circle. In this question, we’re told to use 22 over seven as an approximation for 𝜋. So the circumference is two multiplied by 22 over seven multiplied by 63. Both seven and 63 can be divided by seven to give a simplified calculation of two multiplied by 22 multiplied by nine. This gives 44 multiplied by nine.
Now recall that 44 multiplied by 10 is 440. So if we subtract 44 from this, we’ll have the result of multiplying 44 by nine. It’s 396. So we have the circumference of the circle. And next we need to work out the perimeter of a square.
The perimeter of a square is just found by multiplying its side length 𝑠 by four, as all four sides are the same length. So in this question, the perimeter of the square is four multiplied by seven, which is 28. So now we know the circumference of the circle and the perimeter of the square.
The question asked us to find in the simplest form the ratio between these two values. So the initial ratio between the circumference of the circle and the perimeter of the square is 396 to 28. Now both of these can be divided by four. 28 divided by four is seven. And if we recall that four multiplied by 100 is 400, then we can see easily that 396 divided by four is 99.
This ratio can’t be simplified any further as seven is a prime number and it isn’t a factor of 99. Therefore, the ratio between the circumference of the circle and the perimeter of the square in its simplest form is 99 to seven.