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Answer:
Area of Shaded Region: ( About ) 19.6
Step-by-step explanation:
~ To find the area of the shaded region, let us calculate the area of the circle, and the area of the hexagon, subtracting it's area from the area of the circle ~
1. Given a radius of 6 cm, let us calculate the area of the circle provided the area formula πr^2. Substitute the value of the radius, but keep π in terms of π until the end ⇒ π * ( 6 )^2 = 36π ⇒ ( About ) 113.1 units^2
2. To find the area of the hexagon, let us divide the hexagon into 6 triangles. All 3 of the sides of each triangle is 6 cm, provided these are equilateral triangles. We should calculate the area of each triangle, so let us draw an altitude for each of these Δs. Doing so, through Coincidence Theorem we split the segment drawn to the altitude into two ≅ parts, or 3 units each. That means that the altitude must be 3√3, by properties of 60-60-60 triangles. That being said, the area of one of the triangles: 1/2 * base * height ⇒ 1/2 * 6 * 3√3 ⇒ 9√3 units^2.
3. Knowing that all the 6 triangles are ≅, let us simply multiply the area of a of the triangles * 6 to recieve the area of the hexagon ⇒ 9√3 * 6 ⇒ 54√3 units^2
4. The area of the shaded region is now ⇒ 113.1 - 54√3 ⇒
Area of Shaded Region: ( About ) 19.6
By subtracting the area of the hexagon, from the area of the circle. The area of the shaded region is 19.6.
What is the area of the circle?
The area of the circle is defined as the product of the pie and the square of the radius.
The area of the circle = [tex]\pi r^{2}[/tex]
Given a radius of 6 cm,
The area of the circle = [tex]\pi r^{2}[/tex]
Substitute the value of the radius,
[tex]\pi ( 6 )^2 \\= 36 \times 3.14\\= 113.1 units^2[/tex]
The area of the circle is 113.1 units^2.
To find the area of the hexagon, let us divide the hexagon into 6 triangles. All 3 of the sides of each triangle are 6 cm, provided these are equilateral triangles.
The area of one of the triangles
[tex]=1/2 \times base \times height\\ \\= 1/2 \times 6 \times3\sqrt3 \\\\ =9\sqrt{3} units^2.[/tex]
The area of the triangles 6 times to receive the area of the hexagon
[tex]9\sqrt{3} \times 6 \\ \\54\sqrt{3} units^2[/tex]
The area of the shaded region = 113.1 - 54√3
= 19.6
Learn more about the area;
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