A triangle is placed in a semicircle with a radius of 3cm, as shown below.
Find the area of the shaded region. Use the value for 3.14, and do not round your answer. Be sure to include the correct unit in your answer.
25
81
To find the area of the shaded region, we need to calculate the area of the semicircle and subtract the area of the triangle.
1. Calculate the area of the semicircle:
The formula for the area of a semicircle is A = (π * r^2) / 2, where r is the radius.
In this case, the radius is given as 3 cm, so we can substitute the values into the formula:
A = (3.14 * 3^2) / 2
A = (3.14 * 9) / 2
A = 28.26 / 2
A = 14.13 cm^2
2. Calculate the area of the triangle:
The formula for the area of a triangle is A = (base * height) / 2.
To find the base and height of the triangle, we need to look at the diameter of the semicircle.
Since the radius of the semicircle is 3 cm, the diameter is twice the radius, which is 6 cm.
The base of the triangle is equal to the diameter of the semicircle, so it is 6 cm.
To find the height of the triangle, we need to use the Pythagorean theorem. The height is the length from the center of the semicircle to the edge of the semicircle, which is also the radius of the semicircle.
Using the Pythagorean theorem, we can calculate the height (h) as follows:
h^2 = r^2 - (base/2)^2
h^2 = 3^2 - (6/2)^2
h^2 = 9 - 3^2
h^2 = 9 - 9
h^2 = 0
h = 0 (since the square root of 0 is 0)
Since the height of the triangle is 0, the area of the triangle is 0.
3. Find the area of the shaded region by subtracting the area of the triangle from the area of the semicircle:
Shaded Area = Area of Semicircle - Area of Triangle
Shaded Area = 14.13 cm^2 - 0 cm^2
Shaded Area = 14.13 cm^2
Therefore, the area of the shaded region is 14.13 cm².