Answer:
case a) The approximate circumference of circle is
and the ratio of circumference to Diameter is ![\frac{C}{D}=2.95](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7BD%7D%3D2.95)
case b) The approximate circumference of circle is
and the ratio of circumference to Diameter is ![\frac{C}{D}=3](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7BD%7D%3D3)
case c) The approximate circumference of circle is
and the ratio of circumference to Diameter is ![\frac{C}{D}=3.12](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7BD%7D%3D3.12)
Step-by-step 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](https://tex.z-dn.net/?f=P%3D5s)
where
![s=1.18\ units](https://tex.z-dn.net/?f=s%3D1.18%5C%20units)
substitute
![P=5(1.18)=5.9\ units](https://tex.z-dn.net/?f=P%3D5%281.18%29%3D5.9%5C%20units)
therefore
The approximate circumference of circle is ![5.9\ units](https://tex.z-dn.net/?f=5.9%5C%20units)
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](https://tex.z-dn.net/?f=D%3D2r%3D2%281%29%3D2%5C%20units)
The ratio is equal to
![\frac{C}{D}=5.9/2=2.95](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7BD%7D%3D5.9%2F2%3D2.95)
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](https://tex.z-dn.net/?f=P%3D6s)
where
![s=1\ units](https://tex.z-dn.net/?f=s%3D1%5C%20units)
substitute
![P=6(1)=6\ units](https://tex.z-dn.net/?f=P%3D6%281%29%3D6%5C%20units)
therefore
The approximate circumference of circle is ![6\ units](https://tex.z-dn.net/?f=6%5C%20units)
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](https://tex.z-dn.net/?f=D%3D2r%3D2%281%29%3D2%5C%20units)
The ratio is equal to
![\frac{C}{D}=6/2=3](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7BD%7D%3D6%2F2%3D3)
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](https://tex.z-dn.net/?f=P%3D12s)
where
![s=0.52\ units](https://tex.z-dn.net/?f=s%3D0.52%5C%20units)
substitute
![P=12(0.52)=6.24\ units](https://tex.z-dn.net/?f=P%3D12%280.52%29%3D6.24%5C%20units)
therefore
The approximate circumference of circle is ![6.24\ units](https://tex.z-dn.net/?f=6.24%5C%20units)
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](https://tex.z-dn.net/?f=D%3D2r%3D2%281%29%3D2%5C%20units)
The ratio is equal to
![\frac{C}{D}=6.24/2=3.12](https://tex.z-dn.net/?f=%5Cfrac%7BC%7D%7BD%7D%3D6.24%2F2%3D3.12)
<em>Conclusion</em>
We know that
The exact value of the ratio
is equal to
![\frac{\pi D}{D}=\pi](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpi%20D%7D%7BD%7D%3D%5Cpi)
The approximate value of the circumference will be closer to the real one when the number of sides of the inscribed polygon is greater