Answer has to be in exact form. Find the area of the region between a regular hexagon with sides of 6" and its inscribed circle.
You can split your hexagon up into 6 equal equilateral triangles with sides of 6
Drawing the height in one of them, let the height be h
h^2 + 3^2 = 6^2
h^2 = 36-9
h = √27 or 3√3
area of one is (1/2)(6)(3√3) = 9√3
so the area of the hexagon is 54√3
the inscribed circle has radius 3√3 or √27
area of that circle = π(√27)^2 = 27π
So area between circle and hexagon
= 54√3 - 27π
To find the area of the region between a regular hexagon and its inscribed circle, you can follow these steps:
1. Find the area of the hexagon:
- Using the formula for the area of a regular hexagon, A_hex = (3√3 / 2) * s^2, where s is the length of each side.
- Substitute the given side length of 6 inches into the formula:
A_hex = (3√3 / 2) * 6^2
A_hex = (3√3 / 2) * 36
A_hex = 54√3 square inches
2. Find the area of the circle:
- The radius of the inscribed circle can be found by dividing the side length of the hexagon by 2, so r = s / 2.
- Calculate the area of the circle using the formula A_circle = π * r^2:
A_circle = π * (6/2)^2
A_circle = π * 3^2
A_circle = 9π square inches
3. Find the area of the region between the hexagon and its inscribed circle:
- Subtract the area of the circle from the area of the hexagon:
A_region = A_hex - A_circle
A_region = 54√3 - 9π square inches
Thus, the area of the region between the regular hexagon with sides of 6 inches and its inscribed circle is 54√3 - 9π square inches.
To find the area of the region between the regular hexagon and its inscribed circle, we will need to calculate the areas of both shapes separately and then subtract.
First, let's find the area of the regular hexagon. A regular hexagon is composed of six equal equilateral triangles. We can calculate the area of one of these triangles and then multiply it by 6 to find the total area of the hexagon.
To find the area of an equilateral triangle, we use the formula A = (s^2√3) / 4, where A represents the area and s represents the length of each side.
In this case, the length of each side of the equilateral triangle (and the hexagon) is given as 6 inches. Plugging this into the formula, we get:
A_triangle = (6^2√3) / 4
= (36√3) / 4
= 9√3
Now we can calculate the area of the hexagon by multiplying the area of one triangle by 6:
A_hexagon = 6 * A_triangle
= 6 * (9√3)
= 54√3
Next, we need to find the area of the inscribed circle. The radius of the inscribed circle is equal to half the length of one side of the hexagon (since it touches the midpoints of all six sides). Therefore, the radius of the circle is 6 / 2 = 3 inches.
To find the area of the circle, we use the formula A = πr^2, where A represents the area and r represents the radius. Plugging in the values, we get:
A_circle = π * (3^2)
= 9π
Finally, we can find the area of the region between the hexagon and the circle by subtracting the area of the circle from the area of the hexagon:
Area = A_hexagon - A_circle
= 54√3 - 9π
So, the area of the region between the regular hexagon with sides of 6 inches and its inscribed circle is given by the expression 54√3 - 9π square inches.