# 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.

## arc Length of a Quarter Circle formula

$$\large{ L = \frac {2 \; \pi \; r} {4} }$$
Symbol English Metric
$$\large{ L }$$ = arc length  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## area of a Quarter Circle formula

$$\large{ A = \frac{ \pi \; r^2 }{4} }$$
Symbol English Metric
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Distance from Centroid of a Quarter Circle formulas

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

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

Symbol English Metric
$$\large{ C_x, C_y }$$ = distance from centroid  $$\large{ in }$$  $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Elastic Section Modulus of a Quarter Circle formula

$$\large{ S = \frac { I_x } { C_y } }$$
Symbol English Metric
$$\large{ S }$$ = elastic section modulus $$\large{in^3}$$ $$\large{mm^3}$$
$$\large{ I }$$ = moment of inertia $$\large{\frac{lbm}{ft^2-sec} }$$ $$\large{\frac{kg}{m^2} }$$

## Perimeter of a Quarter Circle formulas

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

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

Symbol English Metric
$$\large{ P }$$ = perimeter $$\large{ in }$$ $$\large{ mm }$$
$$\large{ L }$$ = arc length  $$\large{ in }$$  $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Polar Moment of Inertia of a Circle formulas

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

$$\large{ J_{z1} = \frac { \pi \; r^4 } { 8 } }$$

Symbol English Metric
$$\large{ L }$$ = arc length  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Radius of a Quarter Circle formula

$$\large{ r = \sqrt {\frac {2 \; A } {\pi} } }$$
Symbol English Metric
$$\large{ r }$$ = radius  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ A }$$ = area $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## Radius of Gyration of a Half Circle formulas

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

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

$$\large{ k_{z} = r \; \sqrt { \frac { 1 } { 2 } - \frac { 16 } { 9 \; \pi^2 } } }$$

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

$$\large{ k_{y1} = \frac { r } { 2 } }$$

$$\large{ k_{z1} = \frac { \sqrt {2} } { 2 } \; r }$$

Symbol English Metric
$$\large{ k }$$ = radius of gyration  $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$

## Second Moment of Area of a Half circle formulas

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

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

$$\large{ I_{x1} = \frac { \pi \; r^4}{ 16 } }$$

$$\large{ I_{y1} = \frac { \pi \; r^4}{ 8 } }$$

Symbol English Metric
$$\large{ I }$$ = moment of inertia  $$\large{ in^4 }$$ $$\large{ mm^4 }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = radius $$\large{ in }$$ $$\large{ mm }$$