Sector Area (Radians)
Area of a circular sector with angle in radians.
Formula
Variables
Example
Half of r-squared times angle
How it works
This formula gives the area of a slice of a circle when the central angle is measured in radians. It works because a full circle has area \u03c0r^2 and contains 2\u03c0 radians, so each radian accounts for a proportion of the circle's area equal to (\u03c0r^2)/(2\u03c0) = r^2/2. Multiplying that area-per-radian by the angle \u03b8 gives A = (1/2)r^2\u03b8.
When to use it
Use this when you know the radius of a circle and the central angle in radians and need the area of the sector. It appears in geometry, physics, and engineering problems involving circular slices, rotating parts, or arc-based measurements.
Worked example, step by step
Find the area of a sector with radius 8 cm and angle 1.5 rad. A = (1/2)r^2\u03b8 A = (1/2)(8^2)(1.5) A = (1/2)(64)(1.5) A = 32 \u00d7 1.5 A = 48 Final answer: 48 cm^2
Common mistakes to avoid
- Using degrees instead of radians without converting first
- Forgetting to square the radius
- Using the full circle area formula instead of the sector formula
Did You Know?
A radian is the angle that makes an arc as long as the radius, so a full circle is exactly 2π radians.
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