# Hollow Circle

Written by Jerry Ratzlaff on . Posted in Plane Geometry

• Two circles each having all points on each circle at a fixed equal distance from a center point.
• Center of a circle having all points on the line circumference are at equal distance from the center point.
• A hollow circle is a structural shape used in construction.

## Formulas that use area of a Hollow Circle

 $$\large{ A_{area} = \pi \; \left( R^2 - r^2 \right) }$$

### Where:

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

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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

## Formulas that use Circumference of a Hollow Circle

 $$\large{ C = 2 \; \pi \; R }$$ (outside) $$\large{ C = 2 \; \pi \; r }$$ (inside)

### Where:

$$\large{ C }$$ = circumference

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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

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

 $$\large{ C_x = r}$$ $$\large{ C_y = r}$$

### Where:

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

$$\large{ r }$$ = inside radius

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

 $$\large{ S = \frac{ \pi \; \left( R^4 \;-\; r^4 \right) }{ 4\;R } }$$

### Where:

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

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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

## Formulas that use Plastic Section Modulus of a Hollow Circle

 $$\large{ Z = \frac { 4 \; \left( R^3 \;-\; r^3 \right) } { 3 } }$$

### Where:

$$\large{ Z }$$ = plastic section modulus

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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

 $$\large{ J_{z} = \frac { \pi }{2} \; \left( R^4 - r^4 \right) }$$ $$\large{ J_{z1} = \frac { \pi }{2} \; \left( R^4 - r^4 \right) + 2\; \pi \; R^2 \left( R^2 - r^2 \right) }$$

### Where:

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

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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

## Formulas that use Radius of a Hollow Circle

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

### Where:

$$\large{ r }$$ = inside radius

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

$$\large{ r }$$ = inside radius

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

 $$\large{ k_{x} = \frac {1}{2} \; \sqrt { R^2 + r^2 } }$$ $$\large{ k_{y} = \frac {1}{2} \; \sqrt { R^2 + r^2 } }$$ $$\large{ k_{z} = \frac { \sqrt { 2 } }{2} \; \sqrt { R^2 + r^2 } }$$ $$\large{ k_{x1} = \frac {1}{2} \; \sqrt { 5 \; R^2 + r^2 } }$$ $$\large{ k_{y1} = \frac {1}{2} \; \sqrt { 5 \; R^2 + r^2 } }$$ $$\large{ k_{z1} = \frac { \sqrt { 2 } }{2} \; \sqrt { 5 \; R^2 + r^2 } }$$

### Where:

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

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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

 $$\large{ I_{x} = \frac { \pi }{4} \; \left( R^4 - r^4 \right) }$$ $$\large{ I_{y} = \frac { \pi }{4} \; \left( R^4 - r^4 \right) }$$ $$\large{ I_{x1} = \frac { \pi }{4} \; \left( R^4 - r^4 \right) + \pi \; R^2 \left( R^2 - r^2 \right) }$$ $$\large{ I_{y1} = \frac { \pi }{4} \; \left( R^4 - r^4 \right) + \pi \; R^2 \left( R^2 - r^2 \right) }$$

### Where:

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

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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

## Formulas that use Sector of a Hollow Circle

 $$\large{ A = \frac{\pi \; \theta \; \left( r^2 \;-\; R^2 \right) }{360} }$$

### Where:

$$\large{ A }$$ = sector area

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

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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

## Formulas that use Torsional Constant of a Hollow Circle

 $$\large{ J = \frac { \pi \; \left( R^4 \;-\; r^4 \right) } { 2 } }$$ $$\large{ J = \frac { \pi \; \left( D^4 \;-\; d^4 \right) } { 32 } }$$

### Where:

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

$$\large{ d }$$ =  inside diameter

$$\large{ D }$$ =  outside diameter

$$\large{ r }$$ = inside radius

$$\large{ R }$$ = outside radius

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