Question

Find the shaded area. Explain the area of the whole circle first. then the area of the sector, then the area of the triangle. then the area of the shaded area.

Answer 1

uhhhhh 21?

Explanation:

Answer 2

The area is roughly 22.1106cm².

Explanation:

The area of the entire circle would be:

`\pi {(6)}^(2) = 36\pi`

However, we only have 120° of it, which out of 360° means we have 1/3 of the circle, and therefore the sector is:

`(36\pi)/(3) = 12\pi`

Using the Law of Cosines, it is possible to find the missing side of the triangle:

`{x}^(2) = {6}^(2) + {6}^(2) - 2(6)(6) \cos(120) \\ {x}^(2) = 36 + 36 - 2(36)( - (1)/(2) ) \\ {x}^(2) = 72 + 36 \\ {x}^(2) = 108 \\ x = √(3 * 9 * 4) \\ x = 3 * 2 * √(3) \\ x = 6 √(3)`

Then, to calculate the area of the triangle, we find its height using the Pythagorean Theorem.

`{ ((6 √(3) )/(2) )}^(2) + {x}^(2) = {6}^(2) \\ {(3 √(3) )}^(2) + {x}^(2) = {6}^(2) \\ 27 + {x}^(2) = 36 \\ {x}^(2) = 36 - 27 \\ {x}^(2) = 9 \\ x = 3`

Finally, we calculate the area of the triangle and subtract it from the sector.

`a = 12\pi - ( (3 * 6 √(3) )/(2) ) \\ = 12\pi - 9 √(3) = 22.1106`