Sector Area Calculator - Circle Sector

Formula

A = πr²(θ/360°)

The area is the total area of the circle multiplied by the fraction of the circle covered by the angle.

Worked Examples

Example 1: Pizza Slice

Problem:12" pizza (r=6), 45° slice

Solution:Area = π(6²)(45/360) = 36π(0.125) = 4.5π

Result:14.14 sq in

Example 2: Pacman Shape

Problem:r=10, 300° angle

Solution:Area = π(100)(300/360) = 100π(0.833)

Result:261.80

Frequently Asked Questions

What is a Circle Sector?

A sector is a portion of a circle enclosed by two radii and an arc. Think of it as a "slice of pizza".

What is the formula for Sector Area?

Area = πr² × (θ/360) for degrees. For radians: Area = ½r²θ.

What is the difference between a Sector and a Segment?

A sector includes the center of the circle (the pointy part of the slice). A segment is the region between a chord and an arc (the crust part without the point).

What is a Quadrant?

A sector with a central angle of 90 degrees (1/4 of a circle).

What is a Semicircle?

A sector with a central angle of 180 degrees (1/2 of a circle).

How is it related to Arc Length?

Arc length is the distance around the curve. Sector Area is the space inside. They are related: Area = (Arc Length × Radius) / 2.

Can area be calculated without angle?

Yes, if you know the arc length (L) and radius (r), Area = Lr/2.

What are Major and Minor sectors?

A Minor Sector has an angle less than 180°. A Major Sector has an angle greater than 180°.

Applications of Sector Area?

Calculating material for circular parts, land surveying, pie charts, and architectural designs.

What is the perimeter of a sector?

Perimeter = Arc Length + 2 × Radius.

Background & Theory

**Sector Area (A):** The area of the "pie slice" region. **Formulas:** * **Degrees:** $A = \\pi r^2 \\left(\\frac{\\theta}{360}\\right)$ * **Radians:** $A = \\frac{1}{2} r^2 \\theta$ * **Using Arc Length (L):** $A = \\frac{Lr}{2}$ **Logic:** The area of a full circle is $\\pi r^2$. A sector is just a fraction of the circle. The fraction is the angle $\\theta$ divided by the full circle angle ($360^\\circ$ or $2\\pi$).

History

The study of circle areas dates back to ancient Egypt and Babylon. The Rhind Papyrus (c. 1650 BC) gives a formula for the area of a circle. Archimedes (c. 250 BC) rigorously proved the area of a circle relates to its radius squared. The concept of a sector naturally follows from dividing the circle into equal parts (degrees) or using the arc length.