Find the ratio between the perimeter of the rhombus and the circumference of the circle in its simplest form, using 𝜋 equals 22 over seven.
So looking at the question, we have two shapes. We have a rhombus and a circle. We can recall that a rhombus is a quadrilateral that has all four sides of equal length. And we’re asked to find the perimeter of this rhombus. The perimeter of a shape is this sum of the outside edge lengths. So let’s start with our rhombus and see if we can work out its perimeter. We’re told that the edge length of 𝐴𝐷 is 2.5 metres. Therefore, all the other edges will be the same. To find the perimeter then, we would multiply this by four, which means that the perimeter of the rhombus is 10 metres.
Next, we need to look at the circumference of the circle. The circumference is the distance around the outside edge. And it could be calculated using the formula the circumference equals two 𝜋𝑟, where 𝑟 is the radius. We can see in our circle that we’re given the radius of 70 centimetres. We’re also told to use 22 over seven for 𝜋. So let’s substitute in our values for 𝜋 and the radius. This will give us two times 22 over seven times 70. We can multiply the two and the 70 to give 140 times 22 over seven. We can notice that the seven on the denominator is a factor of 140, meaning that we can simplify this as 20 times 22 to give us a circumference of 440 centimetres.
Now, we need to write the ratio of the perimeter of the rhombus and the circumference of the circle. But before we do this, we must notice one very important thing, and that is that the units of our values are different. We have a unit in metres and a unit in centimetres. We can compare these either in both metres or either in both centimetres. But they do need to be the same unit. We use the metric conversion that one metre is equal to 100 centimetres. Here, I’m going to change the units of perimeter in metres into centimetres. So 10 metres is 10 lots of 100 centimetres, which is 1000 centimetres.
Therefore, to write the ratio between the perimeter and the circumference, we would write this as perimeter to circumference. In a ratio then, the values are 1000 to 440. And to finish, we write it in its simplest form, so we simplify. Dividing both sides by 10 would give us the ratio 100 to 44. We could then notice that both of these are multiples of four. So we can divide both sides of our ratio by four, giving us 25 to 11. And so the ratio of the perimeter of the rhombus to the circumference of the circle in its simplest form is 25 to 11.