Pentagon with a radius of 8 cm

Answer 1

To find the area of a regular pentagon, we can use the formula:

Area = (5/2) * (side length) * radius

Since the radius is given as 8 cm, we need to find the side length of the pentagon.

We know that a regular pentagon can be divided into 5 congruent triangles, with each triangle having an angle of 72 degrees at the center of the pentagon.

To find the side length of the pentagon, we can use trigonometry.

Let's consider one of the triangles within the pentagon and draw a perpendicular line from the center to one of the sides. This will divide the triangle in half, creating two right triangles with angles of 36 degrees, 54 degrees, and 90 degrees.

Let x be the length of the side of the pentagon, and let y be the length of the perpendicular line. Since we know the radius is 8 cm, we have a right triangle with hypotenuse of 8, one leg of x, and the other leg of y. Using trigonometry, we can set up the following equation:

sin(36) = x/8
x = 8 * sin(36)
x ≈ 4.9 cm

Now that we have found the length of the side of the pentagon to be approximately 4.9 cm, we can calculate the area using the formula:

Area = (5/2) * (4.9) * 8
Area ≈ 98 cm^2

Therefore, the area of the regular pentagon with a radius of 8 cm is approximately 98 cm^2.