I am supposed to find the area of a segment of a circle. The circle is cut with two radi with an angle of 90 degrees. Then where the radi touch the circle another line is drawn to connect them. The triangle and the small section between the triangle and the edge of the circle are the shaded areas. It also tells me that the unshaded area of the circle is 3 m
If the area of the unshaded region in your diagram is 3 m^2, then the whole circle must be 4 m^2 (simple ratio)
and the area of the sector, which is your shaded region, must be 1 m^2
so πr^2 = 4
r^2 = 4/π
r = 2/√π
area of your triangle within your shaded region
= (1/2)r^2
= (1/2)4/π = 2/π
area of segment = 1 - 2/π or (π - 2)/π
To find the area of the shaded segment, we need to break it down into two parts: the area of the triangle and the area of the sector.
Step 1: Find the area of the triangle
To find the area of the triangle, we need to know the length of its base and height. In this case, the base of the triangle is the chord formed between the two radii, which is also the diameter of the circle. The height of the triangle can be found by drawing a perpendicular line from the center of the circle to the midpoint of the chord.
Step 2: Find the area of the sector
To find the area of the sector, we need to find the central angle formed by the two radii. In this case, the angle is given as 90 degrees, which is a quarter of the whole circle. Since the unshaded area of the circle is given as 3 m, we can use this information to find the radius of the circle.
Step 3: Calculate the area of the segment
Finally, we can subtract the area of the triangle and the area of the sector from the total area of the circle (which is πr^2) to find the area of the shaded segment.
Here's a step-by-step breakdown of the calculations:
1. Calculate the radius of the circle using the given unshaded area:
Area of the circle = πr^2
3 m = πr^2
r^2 = 3/π
r ≈ √(3/π) m
2. Calculate the area of the triangle:
Base of the triangle = Diameter of the circle = 2r
Height of the triangle = (r/2) (since it's the midpoint of the chord)
Area of the triangle = (1/2) * base * height
3. Calculate the area of the sector:
Central angle = 90 degrees (a quarter of the whole circle)
Area of the sector = (central angle/360 degrees) * πr^2
4. Calculate the area of the shaded segment:
Area of the shaded segment = Area of the sector - Area of the triangle
By following these steps, you should be able to find the area of the shaded segment of the circle.