Question

Find the central angle theta which forms a sector of area 6 square meters of a circle of radius 6 meters.

Answer 1

The central angle θ that forms a sector with an area of 6 m² in a circle of radius 6 meters is 60 degrees. This angle is found using the formula for the sector area, solving for θ, and converting radians to degrees.

Step-by-step explanation:

To find the central angle θ that forms a sector of area 6 square meters in a circle of radius 6 meters, we use the formula for the area of a sector, A = (θ2π) * r², where A is the area of the sector, r is the radius of the circle, and θ is the central angle in radians.

First, we solve for θ using the given values:

• A = 6 m²
• r = 6 m

Plugging these into the formula, we get:

6 = (θ/2π) * 6²

To isolate θ, we perform the following steps:

1. Multiply both sides by 2π to get rid of the fraction: 12π = θ * 36
2. Divide both sides by 36 to solve for θ: θ = (12π) / 36
3. Simplify θ: θ = π/3 radians

To convert this angle to degrees, we multiply by (180/π):

θ in degrees = (π/3) * (180/π) = 60 degrees.

Therefore, the central angle θ that forms a sector with an area of 6 m² in a 6-meter radius circle is 60 degrees.