Question

Find the missing dimensions of other circular objects by completing the ratio table. Diameter, d 42 mm 259 301 mm Circumference, C 132 mm 198 mm

Answer 1

Diameter, d 42 mm d 301 mm

Circumference, C 132 mm 198 mm C

We need to find the diameter of a circle that has a circumference measuring 198 mm.

Also, we need to find the circumference of another circle that has a diameter measuring 301 mm.

First, let's remember that the diameter is the measure of any segment that goes from one point in the circumference to another point, passing through its center:

And the circumference is the length of the curve that forms the circle itself. If we walked on the circle, when we were back to the point we begin, then we would have walked a measure equal to the circumference of the circle.

Now, the diameter d and the circumference C of the same circle are related by the formula:

C = d * π

In this formula, π is the constant that has an approximate value of 3.14159.

Notice that the number 132 in the table is obtained by multiplying 42 by 3.14159:

42 * 3.14159 = 131.94678 ≅ 132

Another way of solving this problem, without having to use the value of π is the following:

The ratio d/C is always the same for any circle. So, we can equate the ratio of those values for two different circles:

`(42)/(132)=(d)/(198)`

Now, we can cross multiply the numbers to solve this equation:

`132* d=42*198`

Now, if we divide both sides by 132, we find the value of d (132 divided by 132 is 1, so we will get only d on the left side of the equation):

`\begin{gathered} (132* d)/(132)=(42*198)/(132) \\ \\ 1* d=(42*198)/(132)\text{ (132 : 132 = 1)} \\ \\ d=(21*198)/(66)\text{ (42 : 2 = 21 and 132 : 2 = 66)} \\ \\ d=(21*99)/(33)\text{ (198 : 2 = 99 and 66 : 2 = 33)} \\ \\ d=(21*33)/(11)\text{ (99 : 3 = 33 and 33 : 3 = 11)} \\ \\ d=(21*3)/(1)\text{ (33 : 11 = 3)} \\ \\ d=63 \end{gathered}`

Notice that we solved this problem by simplifying the fraction from one step to another.

Now, let's find the missing circumference C using this same method:

`(42)/(132)=(301)/(C)`

When we cross multiply the numbers, we find:

`42* C=132*301`

Now, we can divide both sides by 42 to find the value of C (42 divided by 42 is 1, so we will get only C on the left side of the equation):

`\begin{gathered} (42* C)/(42)=(132*301)/(42) \\ \\ C=(132*301)/(42) \\ \\ C=(66*301)/(21)\text{ (1}32\colon2=66\text{ and 42 : 2=21)} \\ \\ C=(66*43)/(3)\text{ (301 : 7 = 43 and 21 : 7 = 3)} \\ \\ C=(22*43)/(1)\text{ (66 : 3 = 22)} \\ \\ C=946 \end{gathered}`

Now, we need to remember that the measures are expressed in mm. So the answers are:

Diameter, d 42 mm 63 mm 301 mm

Circumference, C 132 mm 198 mm 946 mm