# Hollow Circle

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.
• See Article Link  -  Geometric Properties of Structural Shapes

## area of a Hollow Circle formula

$$\large{ A = \pi \; \left( R^2 - r^2 \right) }$$     (Area of a Hollow Circle)

$$\large{ R = \sqrt{ \frac{ A \;-\; \pi \; r^2 }{ \pi } } }$$

$$\large{ r = \sqrt{ R^2 - \frac{ A}{\pi} } }$$

### Solve for R

 area of a hollow circle, A inside radius, r

### Solve for r

 area of a hollow circle, A outside radius, R

Symbol English Metric
$$\large{ A }$$ = area  $$\large{ in^2 }$$ $$\large{ mm^2 }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$

## Circumference of a Hollow Circle formulas

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

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

### Solve for r

 circumference, C

Symbol English Metric
$$\large{ C }$$ = circumference $$\large{ in }$$ $$\large{ mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$

## Distance from Centroid of a Hollow Circle formulas

$$\large{ C_x = r}$$

$$\large{ C_y = r}$$

Symbol English Metric
$$\large{ C_x, C_y }$$ = distance from centroid $$\large{ in }$$ $$\large{ mm }$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$

## Elastic Section Modulus of a Hollow Circle formula

$$\large{ S = \frac{ \pi \; \left( R^4 \;-\; r^4 \right) }{ 4\;R } }$$
Symbol English Metric
$$\large{ S }$$ = elastic section modulus $$\large{in^3}$$ $$\large{mm^3}$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## Plastic Section Modulus of a Hollow Circle formula

$$\large{ Z = \frac { 4 \; \left( R^3 \;-\; r^3 \right) } { 3 } }$$
Symbol English Metric
$$\large{ Z }$$ = plastic section modulus $$\large{ in^3 }$$ $$\large{mm^3 }$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$

## Polar Moment of Inertia of a Hollow Circle formulas

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

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

Symbol English Metric
$$\large{ J }$$ = torsional constant $$\large{ in^4 }$$ $$\large{mm^4 }$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## Radius of a Hollow Circle formula

$$\large{ r = \frac{C}{2 \; \pi} }$$

$$\large{ R = \frac{C}{2 \; \pi} }$$

### Solve for r

 circumference, C

### Solve for C

Symbol English Metric
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ C }$$ = circumference $$\large{ in }$$ $$\large{mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## Radius of Gyration of a Hollow Circle formulas

$$\large{ k_{x} = \frac {1}{2} \; \sqrt { R^2 + r^2 } }$$

$$\large{ k_{y} = \frac {1}{2} \; \sqrt { R^2 + r^2 } }$$

$$\large{ k_{z} = \frac { \sqrt { 2 } }{2} \; \sqrt { R^2 + r^2 } }$$

$$\large{ k_{x1} = \frac {1}{2} \; \sqrt { 5 \; R^2 + r^2 } }$$

$$\large{ k_{y1} = \frac {1}{2} \; \sqrt { 5 \; R^2 + r^2 } }$$

$$\large{ k_{z1} = \frac { \sqrt { 2 } }{2} \; \sqrt { 5 \; R^2 + r^2 } }$$

Symbol English Metric
$$\large{ k }$$ = radius of gyration $$\large{ in }$$ $$\large{mm }$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$

## Second Moment of Area of a Hollow circle formulas

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

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

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

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

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

## Sector of a Hollow Circle formula

$$\large{ A = \frac{\pi \; \Delta \; \left( r^2 \;-\; R^2 \right) }{360} }$$
Symbol English Metric
$$\large{ A }$$ = sector area $$\large{ in^2 }$$ $$\large{mm^2 }$$
$$\large{ \Delta }$$ = angle $$\large{ deg }$$ $$\large{rad }$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$

## Torsional Constant of a Hollow Circle formulas

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

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

Symbol English Metric
$$\large{ J }$$ = torsional constant $$\large{ in^4 }$$ $$\large{mm^4 }$$
$$\large{ d }$$ =  inside diameter $$\large{ in }$$ $$\large{mm }$$
$$\large{ D }$$ =  outside diameter $$\large{ in }$$ $$\large{mm }$$
$$\large{ r }$$ = inside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ R }$$ = outside radius $$\large{ in }$$ $$\large{mm }$$
$$\large{ \pi }$$ = Pi $$\large{3.141 592 653 ...}$$