In a diagram O is the centre of the circle and PQ is the diameter.triangle ^RSO is equilateral of side 4cm.calculate the area of the shaded portion
What diagram? Cannot copy and paste here.
To calculate the area of the shaded portion, we need to determine the area of the equilateral triangle and then subtract the area of the segment of the circle.
Step 1: Calculate the area of the equilateral triangle.
Since we know that the side length of the equilateral triangle is 4 cm, we can use the formula for the area of an equilateral triangle:
Area = (sqrt(3) / 4) * side^2
Area = (sqrt(3) / 4) * 4^2
Area = (sqrt(3) / 4) * 16
Area = (1.732 / 4) * 16
Area = 0.433 * 16
Area = 6.928 cm^2
Step 2: Calculate the area of the segment of the circle.
The segment of a circle, in this case, is the region enclosed by the circle and the triangle.
To find the area of this segment, we need to find the area of the sector formed by the central angle (angle ROS) and then subtract the area of the triangle (∆RSO).
The central angle ROS can be found by using the fact that the equilateral triangle has angles of 60 degrees.
So, the central angle = 360 degrees - 60 degrees
Central angle = 300 degrees
The area of the sector can be calculated using the formula:
Area of sector = (θ/360) * π * r^2
In this case, r is the radius, which is half the diameter, so r = PQ/2
Since PQ is a diameter and O is the center, PO = OQ = PQ/2
Now, let's substitute the values into the formula:
Area of sector = (300/360) * π * (PQ/2)^2
Step 3: Calculate the area of the shaded portion.
Finally, we can calculate the area of the shaded portion by subtracting the area of the segment from the area of the equilateral triangle:
Shaded area = Area of equilateral triangle - Area of segment
Now, you can substitute the values calculated in the previous steps to find the area of the shaded portion.