find the area of a regular pentagon with radius 5in
To find the area of a regular pentagon with a given radius, you can follow these steps:
Step 1: Determine the length of one side of the regular pentagon.
The radius of a regular pentagon is the distance from the center of the pentagon to any of its vertices. In this case, the radius is given as 5 inches.
Step 2: Use trigonometric functions to find the side length of the pentagon.
A regular pentagon can be divided into five congruent right triangles, with one of the triangle's legs as the radius and the hypotenuse as the side length of the pentagon. The angle formed by the radius and one side of the pentagon is 360 degrees divided by 5, as there are five sides in a pentagon.
Using trigonometry, we can find the side length by using the cosine function:
cos(angle) = adjacent/hypotenuse
cos(360°/5) = 5/hypotenuse
hypotenuse = 5 / cos(72°)
Step 3: Calculate the side length.
Using a scientific calculator, evaluate cos(72°) and substitute it into the equation:
hypotenuse = 5 / 0.309016994374947
hypotenuse ≈ 16.1803 inches
So, the side length of the regular pentagon is approximately 16.1803 inches.
Step 4: Find the area of the pentagon.
The formula to find the area of a regular pentagon is:
Area = (5/4) * side length^2 * tan(180°/5)
Substitute the given side length into the formula:
Area = (5/4) * (16.1803 inches)^2 * tan(180°/5)
Using a scientific calculator, evaluate tan(180°/5) and substitute it into the equation:
Area = (5/4) * (16.1803 inches)^2 * 1.376381920471173
Area ≈ 110.111 square inches
So, the approximate area of the regular pentagon with a radius of 5 inches is 110.111 square inches.
To find the area of a regular pentagon with a given radius, we need to know the length of one side of the pentagon. However, you have provided the radius and not the side length.
In a regular pentagon, the radius is the distance from the center of the pentagon to any of its vertices, while the side length is the distance between any two consecutive vertices.
If you can provide the side length of the regular pentagon, I would be happy to calculate the area for you.
from the centre I can draw 5 identical triangles
the central angle of each is 360/5 or 72°
the area of one of these = (1/2)(5)(5)sin72
= 11.8882...
so 5 of them or the pentagon has an area of
appr 59.44 in^2