The area of circle PQR with centre o is 72cm2.What is the area of sector POQ if angle POQ=40 degree?
Explain 72/pie
The notation "72/pie" means 72 divided by pi (π). This is often used in geometry to represent the area or circumference of a circle, as these measurements involve pi.
Pi is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. It is approximately equal to 3.14, but has an infinite number of decimal places.
When calculating the area of a circle, we use the formula A = πr^2, where A is the area, π is pi, and r is the radius of the circle. In this case, we are given that the area of circle PQR is 72 cm^2, so we can write:
72 = πr^2
To find the radius r, we can divide both sides of the equation by pi, giving:
72/π = r^2
Taking the square root of both sides, we get:
r = √(72/π)
This is the radius of the circle, which we will need to find the area of the sector POQ.
To find the area of a sector of a circle, we use the formula A = (θ/360)πr^2, where A is the area of the sector, θ is the angle (in degrees) of the sector, and r is the radius of the circle.
In this case, we are given that the angle POQ is 40 degrees, and we have just found the radius to be √(72/π). So we can plug these values into the formula:
A = (40/360)π(√(72/π))^2
Simplifying, we get:
A = (1/9)π(72/π)
The pi term cancels out, leaving:
A = 8
So the area of sector POQ is 8 cm^2.
To find the area of the sector POQ, you first need to understand the relationship between the area of a sector and the area of a full circle.
The formula for the area of a sector is A = (θ/360) x πr^2, where A is the area of the sector, θ is the central angle of the sector, and r is the radius of the circle.
In this case, you already know the area of the circle, which is 72 cm^2. The formula to find the area of a circle is A = πr^2.
Therefore, you can rearrange the formula for the area of the circle to solve for the radius (r):
72 = πr^2
To isolate r^2, divide both sides of the equation by π:
72/π = r^2
Now, to find the area of sector POQ, you need to substitute the values of the central angle (θ = 40 degrees) and the radius (r) into the formula for the area of a sector:
A = (θ/360) x πr^2 = (40/360) x π x (72/π)
Canceling out π, you're left with:
A = (40/360) x 72
Simplifying the calculation:
A = (1/9) x 72 = 8 x 8 = 64 cm^2
Therefore, the area of sector POQ is 64 cm^2.