Problem : Howm do you find the area of a circle circumscribed about a regular pentagon with a perimeter of 50 inches.
So far I each side of pentagon is 10. How do I find the radius of the circle? Is this the correct direction to go?
Ahh the golden ratio, and now you will know why followers of Pythagoras used the pentagon as a secret symbol: it harbors secret numbers.
http://www.treenshop.com/Treenshop/ArticlesPages/FiguresOfInterest_Article/The%20Regular%20Pentagon.htm
To find the radius of the circle circumscribed about a regular pentagon, you are on the correct track. Here's the step-by-step process you can follow:
1. Recall that in a regular pentagon, all sides are equal. Since the perimeter of the pentagon is given as 50 inches and each side is 10 inches, it means that there are 5 sides in total.
2. The perimeter of a regular pentagon can be calculated by multiplying the length of one side by the total number of sides. In this case, the perimeter is 10 inches × 5 = 50 inches (as given).
3. The distance from the center of the circle to any vertex of the pentagon is the radius. Let's denote it as r (unknown).
4. Draw two radii intersecting each other at the center of the circle and forming an angle of 72 degrees (360 degrees divided by 5 sides) at the center. Connect the center to any two consecutive vertices of the pentagon.
5. This will create an isosceles triangle with two radial segments (radius) of length r and a base of length 10 inches (one side of the pentagon).
6. Using basic trigonometry, we can determine the length of the base of the triangle (10 inches) and the central angle (72 degrees) to calculate r.
- The base is opposite the vertex angle, so we can use the formula: 2r × sin(72°) = 10 inches.
- Rearranging the equation, we get: r = 10 inches / (2 × sin(72°)).
7. Now, substitute the values into the equation and compute:
r = 10 inches / (2 × sin(72°)) ≈ 10 inches / (2 × 0.9511) ≈ 10 inches / 1.9022 ≈ 5.26 inches.
Therefore, the radius of the circle circumscribed about the regular pentagon is approximately 5.26 inches.
To find the radius of the circle circumscribed about a regular pentagon, you can use a formula called the apothem-radius ratio. The apothem is the line segment drawn from the center of the pentagon to the midpoint of one of its sides.
In a regular pentagon, the apothem and the radius of the circumscribed circle are related by the following formula:
r = a/cos(θ/2)
Where:
- r is the radius of the circle
- a is the apothem of the pentagon
- θ is the central angle of the pentagon, which can be found using the formula θ = 360° / n, where n is the number of sides (in this case, 5)
Since you mentioned that each side of the pentagon is 10 inches, we can calculate the apothem and the radius using the formula above.
First, find the central angle of the pentagon:
θ = 360° / 5 = 72°
Then, calculate the apothem:
a = s / 2tan(θ/2) = 10 / 2tan(36°) ≈ 5.5377 inches
Finally, find the radius:
r = a / cos(θ/2) = 5.5377 / cos(36°/2) ≈ 5.8817 inches
Therefore, the radius of the circle circumscribed about the regular pentagon is approximately 5.8817 inches.