A figure is composed of a right triangle and a semicircle. What is the area of the shaded region?
the triangle is 8 by 10 with no hypontenuse number and a semi circle on the hypotanuse.
Please help!
r = hyp./2 = sqrt(8^2+10^2)/2 = 6.4 = radius.
A1 = hb/2 = 8*10/2 = 40 = Area of triangle.
A2 = Shaded area.
A3 = pi*r^2/2 = 3 = 3.14*6.4^2/2 = 64.3 = Area of semi-circle.
A1+A2 = A3
40 + A2 = 64.3
A2 =
@henry2 TYSM!
First find the hypotenuse
h^2 = 8^2 + 10^2
..
h = 2√41
which makes the radius of the semi-circle equal to √41
Now find the area of the triangle and add on half the area of the circle
To find the area of the shaded region, we need to calculate the combined areas of the right triangle and the semicircle, and then subtract the area of the overlap.
1. Calculate the area of the right triangle:
The formula for the area of a triangle is A = (base * height) / 2.
In this case, the base of the triangle is 8, and the height is 10.
Therefore, the area of the right triangle is (8 * 10) / 2 = 40 square units.
2. 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 of the semicircle.
To find the radius, we need to divide the hypotenuse of the triangle (the side opposite the right angle) by 2. The hypotenuse can be found using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the other two sides are 8 and 10.
Using the Pythagorean theorem, we can calculate the hypotenuse: c^2 = 8^2 + 10^2 = 64 + 100 = 164.
Taking the square root of 164, we get c ≈ 12.81 (rounded to two decimal places).
Therefore, the radius of the semicircle is half of c, so r ≈ 12.81 / 2 ≈ 6.41 (rounded to two decimal places).
Using this radius, we can calculate the area of the semicircle: A = (π * r^2) / 2 = (π * 6.41^2) / 2 ≈ 64.19 square units (rounded to two decimal places).
3. Calculate the area of the overlap:
The overlap occurs where the semicircle crosses over the hypotenuse of the triangle.
The hypotenuse has a length of 12.81, which is the diameter of the semicircle.
So, the overlap is a quarter of the area of the full circle with radius 12.81, since a semicircle is half of a circle.
The formula for the area of a circle is A = π * r^2, so the overlap area is (π * 12.81^2) / 4 ≈ 102.23 square units (rounded to two decimal places).
4. Find the area of the shaded region:
To calculate the area of the shaded region, subtract the overlap area from the combined area of the triangle and the semicircle:
Shaded area = (Triangle area + Semicircle area) - Overlap area =
(40 + 64.19) - 102.23 =
104.19 - 102.23 ≈ 1.96 square units (rounded to two decimal places).
Therefore, the area of the shaded region is approximately 1.96 square units.