# Quarter Circle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• A part of the interior of a circle having two radius boundries at a 90° angle and an arc.
• Center of a circle having all points on the line circumference are at equal distance from the center point.
• A quarter circle is a structural shape used in construction.

## Formulas that use arc Length of a Quarter Circle

 $$\large{ l = \frac {2 \; \pi \; r} {4} }$$

### Where:

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

$$\large{ r }$$ = radius

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

## Formulas that use area of a Quarter Circle

 $$\large{ A_{area} = \frac{ \pi \; r^2 }{4} }$$

### Where:

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

$$\large{ r }$$ = radius

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

## Formulas that use Distance from Centroid of a Quarter Circle

 $$\large{ C_x = \frac {4 \; r} {3 \; \pi} }$$ $$\large{ C_y = \frac {4 \; r} {3 \; \pi} }$$

### Where:

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

$$\large{ r }$$ = radius

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

## Formulas that use Elastic Section Modulus of a Quarter Circle

 $$\large{ S = \frac { I_x } { C_y } }$$

### Where:

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

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

## Formulas that use Perimeter of a Quarter Circle

 $$\large{ P = \frac {2 \; \pi \; r} {4} + 2 \; r }$$ $$\large{ P = l + 2 \; r }$$

### Where:

$$\large{ P }$$ = perimeter

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

$$\large{ r }$$ = radius

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

## Formulas that use Polar Moment of Inertia of a Circle

 $$\large{ J_{z} = \left( \frac { \pi}{ 8 } - \frac { 8 }{ 9 \; \pi } \right) r^4 }$$ $$\large{ J_{z1} = \frac { \pi \; r^4 } { 8 } }$$

### Where:

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

$$\large{ r }$$ = radius

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

## Formulas that use Radius of a Quarter Circle

 $$\large{ r = \sqrt {\frac {2 \; A_{area}} {\pi} } }$$

### Where:

$$\large{ r }$$ = radius

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

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

## Formulas that use Radius of Gyration of a Half Circle

 $$\large{ k_{x} = r \; \sqrt { \frac { 1 } { 4 } - \frac { 16 } { 9 \; \pi^2 } } }$$ $$\large{ k_{y} = r \; \sqrt { \frac { 1 } { 4 } - \frac { 16 } { 9 \; \pi^2 } } }$$ $$\large{ k_{z} = r \; \sqrt { \frac { 1 } { 2 } - \frac { 16 } { 9 \; \pi^2 } } }$$ $$\large{ k_{x1} = \frac { r } { 2 } }$$ $$\large{ k_{y1} = \frac { r } { 2 } }$$ $$\large{ k_{z1} = \frac { \sqrt {2} } { 2 } \; r }$$

### Where:

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

$$\large{ r }$$ = radius

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

## Formulas that use Second Moment of Area of a Half circle

 $$\large{ I_{x} = \left( \frac { \pi}{ 16 } - \frac { 4 }{ 9 \; \pi } \right) \; r^4 }$$ $$\large{ I_{y} = \left( \frac { \pi}{ 16 } - \frac { 4 }{ 9 \; \pi } \right) \; r^4 }$$ $$\large{ I_{x1} = \frac { \pi \; r^4}{ 16 } }$$ $$\large{ I_{y1} = \frac { \pi \; r^4}{ 8 } }$$

### Where:

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

$$\large{ r }$$ = radius

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