# Circle Sector

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Sector of a circle (a two-dimensional figure) is a fraction of the area of a circle with a radius on each side and an arc.
• Center of a circle having all points on the line circumference are at equal distance from the center point.
• A half circle is a structural shape used in construction.

### Arc Length of a Circle Sector formula

$$\large{ l = \theta \; \frac{\pi}{180} \; r }$$

Where:

$$\large{ l }$$ = arc length

$$\large{ \theta }$$ = angle

$$\large{ \pi }$$ = Pi

$$\large{ r }$$ = radius

### area of a Circle Sector formula

$$\large{ A_{area} = \theta \;r^2 }$$

Where:

$$\large{ A_{area} }$$ = area

$$\large{ \theta }$$ = angle

$$\large{ \pi }$$ = Pi

$$\large{ r }$$ = radius

### Distance from Centroid of a Circle Sector formula

$$\large{ C_x = 2 \; r \; \frac{sin \; \theta}{3\; \theta} }$$

$$\large{ C_y = 0 }$$

Where:

$$\large{ C_x, C_y }$$ = distance from centroid

$$\large{ \theta }$$ = angle

$$\large{ r }$$ = radius

### Elastic Section Modulus of a Circle Sector formula

$$\large{ S = \frac{ I_x }{ sin \; \theta \; r } }$$

Where:

$$\large{ S }$$ = elastic section modulus

$$\large{ \theta }$$ = angle

$$\large{ I }$$ = moment of inertia

$$\large{ r }$$ = radius

### Perimeter of a Circle Sector formula

$$\large{ P = 2 \; r + 2 \; r \; \theta }$$

Where:

$$\large{ P }$$ = perimeter

$$\large{ \theta }$$ = angle

$$\large{ r }$$ = radius

### Polar Moment of Inertia of a Circle Sector formula

$$\large{ J_{z} = \frac {r^4}{18} \; \left( \frac {9 \; \theta^2 \;-\; 8 \; sin^2 \; \theta }{\theta} \right) }$$

$$\large{ J_{z1} = \frac {r^4 \; \theta}{2} }$$

Where:

$$\large{ J }$$ = torsional constant

$$\large{ \theta }$$ = angle

$$\large{ r }$$ = radius

### Radius of Gyration of a Circle Sector formula

$$\large{ k_{x} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \theta \;-\; sin \; \left(2 \; \theta \right) }{\theta} } }$$

$$\large{ k_{y} = \frac{1}{12} \; \sqrt { 2 \; r^2 \; \frac{180^2 \; + \; 9 \; \theta \; sin \; \left(2 \; \theta \right) \;-\; 32 \; + \; 32 \; cos^2 \; \theta }{\theta^2} } }$$

$$\large{ k_{z} = \frac{1}{6} \; \sqrt { 2 \; r^2 \; \frac{9 \; \theta^2 \;-\; 8 \; sin^2 \; \left(2\; \theta \right) }{\theta^2} } }$$

$$\large{ k_{x1} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \theta \;-\; sin \; \left(2 \; \theta \right) }{\theta} } }$$

$$\large{ k_{y1} = \frac{1}{4} \; \sqrt { 2 \; r^2 \; \frac{2 \; \theta \; + \; sin \; \left(2 \; \theta \right) }{\theta} } }$$

$$\large{ k_{x1} = \frac{r}{ \sqrt{2} } }$$

Where:

$$\large{ k }$$ = radius of gyration

$$\large{ \theta }$$ = angle

$$\large{ r }$$ = radius

### Second Moment of Area of a Circle Sector formula

$$\large{ I_{x} = \frac{r^4}{4} \; \left[ \theta \;-\; \frac{1}{2} \; sin \left( 2 \; \theta \right) \right] }$$

$$\large{ I_{y} = \frac{r^4}{4} \; \left[ \theta + \frac{1}{2} \; sin \left( 2 \; \theta \right) \right] \;-\; \frac{4r^4}{9 \theta} \; sin^2 \; \theta }$$

$$\large{ I_{x1} = I_x + r^4 \; \theta \; sin^2 \; \theta }$$

$$\large{ I_{y1} = \frac{r^4}{4} \left[ \theta + \frac{1}{2} \; sin \; \left( 2 \; \theta \right) \right] }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ \theta }$$ = angle

$$\large{ r }$$ = radius