Through radius and central angle

The area of an annulus sector is the area of a region between two concentric circles, cut by a central angle θ, and can be calculated using the following formula:

Area = 0.5 × r² × (θ - sinθ)

where:

• r is the radius of the larger circle (outer circle)
• θ is the central angle of the sector in radians.

To understand this formula, first consider that the area of a full circle with radius r is given by the formula A = πr². If we cut the circle with a central angle θ, the resulting area will be a fraction of the total area of the circle. The fraction of the area can be calculated as θ/2π.

Now, let's consider the annulus formed by two concentric circles with radii r1 and r2 (r2 > r1). We can calculate the area of the annulus by subtracting the area of the smaller circle (A1 = πr1²) from the area of the larger circle (A2 = πr2²). This gives us:

A = A2 - A1 = πr2² - πr1² = π(r2² - r1²)

To calculate the area of the annulus sector, we need to cut this annulus by a central angle θ. The area of the resulting sector can be found by multiplying the fraction of the area of the annulus (θ/2π) by the total area of the annulus (π(r2² - r1²)):

Area of annulus sector = (θ/2π) × π(r2² - r1²)

Simplifying this equation, we get:

Area of annulus sector = (θ/2) × (r2² - r1²)

Now, we can substitute r1 with 0 to get the formula for the area of an annulus sector with radius r and central angle θ:

Area = 0.5 × r² × (θ - sinθ)