Question

Part I: Measuring Parts of a Circle

a. Help Dakota with her homework. Use diagrams of inscribed polygons to approximate the circumference of each circle. Assume that all of the circles have a radius of 1. Then, calculate the ratio of the approximate circumference to the diameter.

Answer 1

case a) The approximate circumference of circle is
`5.9\ units` and the ratio of circumference to Diameter is
`(C)/(D)=2.95`

case b) The approximate circumference of circle is
`6\ units` and the ratio of circumference to Diameter is
`(C)/(D)=3`

case c) The approximate circumference of circle is
`6.24\ units` and the ratio of circumference to Diameter is
`(C)/(D)=3.12`

Explanation:

we know that

The approximate circumference of each circle, is equal to the perimeter of each inscribed polygon

case A) the figure is an inscribed pentagon

step 1

Find the approximate circumference of circle

The perimeter of the pentagon is equal to

`P=5s`

where

`s=1.18\ units`

substitute

`P=5(1.18)=5.9\ units`

therefore

The approximate circumference of circle is
`5.9\ units`

step 2

Find the ratio of circumference to Diameter

we know that

The radius is half the diameter so

`D=2r=2(1)=2\ units`

The ratio is equal to

`(C)/(D)=5.9/2=2.95`

case B) the figure is an inscribed hexagon

step 1

Find the approximate circumference of circle

The perimeter of the hexagon is equal to

`P=6s`

where

`s=1\ units`

substitute

`P=6(1)=6\ units`

therefore

The approximate circumference of circle is
`6\ units`

step 2

Find the ratio of circumference to Diameter

we know that

The radius is half the diameter so

`D=2r=2(1)=2\ units`

The ratio is equal to

`(C)/(D)=6/2=3`

case C) the figure is an inscribed dodecahedron

step 1

Find the approximate circumference of circle

The perimeter of the dodecahedron is equal to

`P=12s`

where

`s=0.52\ units`

substitute

`P=12(0.52)=6.24\ units`

therefore

The approximate circumference of circle is
`6.24\ units`

step 2

Find the ratio of circumference to Diameter

we know that

The radius is half the diameter so

`D=2r=2(1)=2\ units`

The ratio is equal to

`(C)/(D)=6.24/2=3.12`

Conclusion

We know that

The exact value of the ratio
`(C)/(D)` is equal to

`(\pi D)/(D)=\pi`

The approximate value of the circumference will be closer to the real one when the number of sides of the inscribed polygon is greater