Find the area of the region between a regular hexagon with sides of 6" and its inscribed circle.
find area of hexagon
subtract area of circle
https://rechneronline.de/pi/hexagon.php
note that will give you incircle radius of 3 sqrt 3 = 5.2 for side length 6
To find the area of the region between a regular hexagon and its inscribed circle, we can break it down into two parts: the area of the hexagon and the area of the circle.
First, let's find the area of the hexagon. A regular hexagon consists of six equilateral triangles. Since each side of the hexagon is given as 6 inches, the height of each equilateral triangle is the distance from the center of the hexagon to any of its vertices. In this case, the height is also the radius of the inscribed circle.
To find the height (radius), we can use the formula:
radius = (side length)/2 * tan(30 degrees)
In this case, radius = (6 inches)/2 * tan(30 degrees).
Now, we can calculate the area of one of the equilateral triangles using the formula:
area of triangle = (base * height)/2
Since the base of each triangle is the side length of the hexagon (6 inches) and the height is the radius of the inscribed circle, the area of one triangle is:
area of triangle = (6 inches * radius)/2
Since there are six equilateral triangles in a hexagon, the total area of the hexagon is:
area of hexagon = 6 * (6 inches * radius)/2
Next, let's find the area of the inscribed circle. The area of a circle is given by the formula:
area of circle = π * (radius)^2
In this case, radius is the same as the height of the equilateral triangles we calculated earlier.
Now, we can calculate the area of the region between the hexagon and its inscribed circle by subtracting the area of the circle from the area of the hexagon:
area of region = area of hexagon - area of circle
Now, let's substitute the values we calculated earlier and find the final answer.