Answer:
case a) The approximate circumference of circle is [tex]5.9\ units[/tex] and the ratio of circumference to Diameter is [tex]\frac{C}{D}=2.95[/tex]
case b) The approximate circumference of circle is [tex]6\ units[/tex] and the ratio of circumference to Diameter is [tex]\frac{C}{D}=3[/tex]
case c) The approximate circumference of circle is [tex]6.24\ units[/tex] and the ratio of circumference to Diameter is [tex]\frac{C}{D}=3.12[/tex]
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
[tex]P=5s[/tex]
where
[tex]s=1.18\ units[/tex]
substitute
[tex]P=5(1.18)=5.9\ units[/tex]
therefore
The approximate circumference of circle is [tex]5.9\ units[/tex]
step 2
Find the ratio of circumference to Diameter
we know that
The radius is half the diameter so
[tex]D=2r=2(1)=2\ units[/tex]
The ratio is equal to
[tex]\frac{C}{D}=5.9/2=2.95[/tex]
case B) the figure is an inscribed hexagon
step 1
Find the approximate circumference of circle
The perimeter of the hexagon is equal to
[tex]P=6s[/tex]
where
[tex]s=1\ units[/tex]
substitute
[tex]P=6(1)=6\ units[/tex]
therefore
The approximate circumference of circle is [tex]6\ units[/tex]
step 2
Find the ratio of circumference to Diameter
we know that
The radius is half the diameter so
[tex]D=2r=2(1)=2\ units[/tex]
The ratio is equal to
[tex]\frac{C}{D}=6/2=3[/tex]
case C) the figure is an inscribed dodecahedron
step 1
Find the approximate circumference of circle
The perimeter of the dodecahedron is equal to
[tex]P=12s[/tex]
where
[tex]s=0.52\ units[/tex]
substitute
[tex]P=12(0.52)=6.24\ units[/tex]
therefore
The approximate circumference of circle is [tex]6.24\ units[/tex]
step 2
Find the ratio of circumference to Diameter
we know that
The radius is half the diameter so
[tex]D=2r=2(1)=2\ units[/tex]
The ratio is equal to
[tex]\frac{C}{D}=6.24/2=3.12[/tex]
Conclusion
We know that
The exact value of the ratio [tex]\frac{C}{D}[/tex] is equal to
[tex]\frac{\pi D}{D}=\pi[/tex]
The approximate value of the circumference will be closer to the real one when the number of sides of the inscribed polygon is greater
