Find the area of the segment of the circle. Leave your answer in terms of pi.
there is a circle with a 90 degree triangle and one side is 3 cm
possible answers are
a 9 pi / 4 - 9 units squared
b 6 pi / 4 - 4.5 units squared
c 9 pi / 4 units squared
d 9 pi /4 - 4.5 units squared
It’s a
To find the area of the segment of the circle, we need to calculate the area of the sector and subtract the area of the triangle.
First, let's find the area of the sector. The angle of the sector is 90 degrees, which is a quarter of the full circle.
The formula for the area of a sector is A = (θ/360) * πr², where A is the area, θ is the angle in degrees, and r is the radius.
Given that the radius of the circle is 3 cm and the angle is 90 degrees, we can substitute these values into the formula:
A_sector = (90/360) * π * (3 cm)²
Simplifying, we get:
A_sector = (1/4) * π * 9 cm²
A_sector = (9/4) * π cm²
Next, let's find the area of the triangle. The formula for the area of a triangle is A = (1/2) * base * height.
In this case, the base is the side of the triangle that is 3 cm, and the height is also 3 cm (since it is the radius of the circle).
A_triangle = (1/2) * 3 cm * 3 cm
A_triangle = (1/2) * 9 cm²
A_triangle = 4.5 cm²
Finally, let's subtract the area of the triangle from the area of the sector to find the area of the segment:
A_segment = A_sector - A_triangle
A_segment = (9/4) * π cm² - 4.5 cm²
A_segment = (9/4) * π cm² - (18/4) cm²
A_segment = (9π - 18)/4 cm²
Therefore, the area of the segment of the circle is (9π - 18)/4 cm².
To find the area of the segment of the circle, you first need to calculate the area of the circular sector formed by the 90-degree triangle. Then, subtract the area of the triangle from the area of the sector to get the desired area.
Here's how you can do it step by step:
1. Calculate the area of the circular sector:
- The formula for the area of a circular sector is (θ/360) × πr^2, where θ is the central angle and r is the radius of the circle.
- In this case, the central angle is 90 degrees (since it's a 90-degree triangle).
- The radius of the circle is not given directly, but we can find it using the Pythagorean theorem since we know one side of the triangle is 3 cm.
- Let's assume the side opposite the right angle (hypotenuse) is the radius. So, r = 3 cm.
- Plugging these values into the formula: (90/360) × π(3^2) = (1/4) × 9π = (9/4)π.
2. Calculate the area of the triangle:
- The formula for the area of a triangle is (base × height) / 2.
- In this case, the base of the triangle is the radius of the circle (r = 3 cm) and the height is the length of the side adjacent to the right angle.
- We are not given the length of the side adjacent to the right angle, so we can't calculate the exact area of the triangle. However, we can still write it as a general formula using "x" as the length of the side adjacent to the right angle.
- Plugging these values into the formula: (3x) / 2.
3. Subtract the area of the triangle from the area of the sector:
- Subtracting the two formulas we derived:
- (9/4)π - (3x) / 2.
So, the area of the segment of the circle, in terms of π, is (9/4)π - (3x) / 2, where "x" represents the length of the side adjacent to the right angle.