Radius of a circle with a sector area of 3 square ft determined by a central angle 1/4 radian
To find the radius of a circle given a sector area and a central angle, we can use the formula for the area of a sector:
A = (θ/2) * r^2
where A is the sector area, θ is the central angle in radians, and r is the radius of the circle.
In this case, we are given the sector area A = 3 square feet and the central angle θ = 1/4 radian. Let's substitute these values into the formula and solve for the radius r.
3 = (1/4) * r^2
To solve for r, we need to isolate it on one side of the equation. Let's start by multiplying both sides of the equation by 4:
4 * 3 = 4 * (1/4) * r^2
12 = r^2
Now, we take the square root of both sides of the equation to solve for r:
√12 = √r^2
√12 = r
Therefore, the radius of the circle is √12 (approximately 3.464 feet).
a = 1/2 r^2 θ
3 = 1/2 r^2 * 1/4
24 = r^2
r = √24