Sector Calculator: Calculate Radius, Angle, Arc, Chord, and Area

What is a Sector?

A sector is a portion of a circle enclosed by two radii and the corresponding arc. It resembles a slice of a pie or pizza.

Formulas for Sectors

Let \(r\) be the radius, \(\theta\) be the angle in degrees, \(L\) be the arc length, \(C\) be the chord length, and \(A\) be the area. Then:

1. Arc Length: \(L = \frac{\theta}{360} \times 2\pi r\)
2. Chord Length: \(C = 2r \sin\left(\frac{\theta}{2}\right)\)
3. Area: \(A = \frac{\theta}{360} \times \pi r^2\)

Step-by-Step Calculations

Let's calculate these properties for a sector with radius \(r = 5\) units and angle \(\theta = 60^\circ\):

1. Radius: \[r = 5 \text{ units}\]
2. Angle: \[\theta = 60^\circ\]
3. Arc Length: \[L = \frac{60}{360} \times 2\pi \times 5 = 5.24 \text{ units}\]
4. Chord Length: \[C = 2 \times 5 \times \sin\left(\frac{60}{2}\right) = 5 \text{ units}\]
5. Area: \[A = \frac{60}{360} \times \pi \times 5^2 = 13.09 \text{ square units}\]

Visual Representation

This diagram illustrates the sector with the calculated dimensions and properties.