Circle

• A two-dimensional figure where all points are 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.
• Chord of a circle is line segment on the interior of a circle.
• Diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle.  In the process industry, the diameter is typically used to describe the size pipe that the process is flowing through. Unless explictily specified, the diameter is assumed to mean the nominal pipe size (NPS). The inside diameter of a pipe is the longest distance between the two inside walls of the pipe. The outside diameter is the distance between the two outside walls. To find the thickness of the pipe, subtract the outside diameter from the inside diameter and divide by two.  When sizing flow meters or impact tees, a certain straight run maybe required. This is typically specified in terms of diameters. For example a 10" orifice meter with a 10 diameter upstream requirement will require 100" of unobstructed straight run upstream of the orifice plate.
• Radius of a circle is a line segment between the center point and a point on a circle or sphere.
• Sector of a circle is a fraction of the area of a circle with a radius on each side and an arc.
• Segment of a circle is the area of a sector of a circle minus a piece of that sector.

 

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• Geometric Properties of Structural Shapes

 

Circle area formula

\(\large{ A =\pi \; r^2 }\)

Where:

Units	English	Metric
\(\large{ A }\) = area	\(\large{ in^2 }\)	\(\large{ mm^2 }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)

 

Circle Circumference formula

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

Where:

Units	English	Metric
\(\large{ C }\) = circumference	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)

 

Circle Chord Arc Length formula

\(\large{ l =  \frac { \theta} { 180 } \; 2 \; \pi \;  r }\)

Where:

Units	English	Metric
\(\large{ l }\) = length	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \theta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Chord Length formulas

\(\large{ c = 2 \; r \; \sin \; \frac {\theta}{2}  }\)
\(\large{ c = 2 \; \sqrt{r^2-h^2}  }\)

Where:

Units	English	Metric
\(\large{ c }\) = chord	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \theta }\) = angle	\(\large{ deg }\)	\(\large{ rad }\)
\(\large{ h, h' }\) = height	\(\large{ in }\)	\(\large{mm  }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Diameter formulas

\(\large{ d = 2 \; r }\)
\(\large{ d = \frac {C} {\pi} }\)
\(\large{ d = \sqrt   {\frac {4 \; A} {\pi} }  }\)

Where:

Units	English	Metric
\(\large{ d }\) = diameter	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ A }\) = area	\(\large{ in^2 }\)	\(\large{ mm^2 }\)
\(\large{ C }\) = circumference	\(\large{ in }\)	\(\large{mm  }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Distance from Centroid formulas

\(\large{ C_x =  r}\)
\(\large{ C_y =  r}\)

Where:

Units	English	Metric
\(\large{ C_x, C_y }\) = distance from centroid	\(\large{ in }\)	\(\large{mm  }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Elastic Section Modulus formula

\(\large{ S =  \frac { \pi \; r^3 }  { 4  }  }\)

Where:

Units	English	Metric
\(\large{ S }\) = elastic section modulus	\(\large{\frac{lbf}{in^2}}\)	\(\large{Pa}\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Plastic Section Modulus formula

\(\large{ Z =  \frac { d^3 }  { 6  }  }\)

Where:

Units	English	Metric
\(\large{ Z }\) = plastic section modulus	\(\large{ in^4 }\)	\(\large{ mm^4 }\)
\(\large{ d }\) = diameter	\(\large{ in }\)	\(\large{ mm }\)

 

Circle Polar Moment of Inertia formulas

\(\large{ J_{z} = \frac { \pi \; r^4 }  {  2  } }\)
\(\large{ J_{z1} =  \frac { 5 \; \pi \; r^4 }  {  2 } }\)

Where:

Units	English	Metric
\(\large{ J }\) = torsional constant	\(\large{ in^4 }\)	\(\large{mm^4  }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Radius formulas

\(\large{ r = \frac{d}{2} }\)
\(\large{ r = \frac{C}{2 \; \pi} }\)
\(\large{ r = \sqrt{ \frac{A}{\pi} } }\)
\(\large{ r = \frac{ v_c \; t }{ 2 \; \pi } }\)

Where:

Units	English	Metric
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ A }\) = area	\(\large{ in^2 }\)	\(\large{ mm^2 }\)
\(\large{ v_c }\) = circular velocity	\(\large{\frac{ft}{sec}}\)	\(\large{\frac{m}{s}}\)
\(\large{ C }\) = circumference	\(\large{ in }\)	\(\large{mm  }\)
\(\large{ d }\) = diameter	\(\large{ in }\)	\(\large{mm  }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ t }\) = time	\(\large{ sec }\)	\(\large{ s }\)

 

Circle Sector Area formulas

\(\large{ A =  \frac { \theta } { 360 } \; \pi \; r^2  \;\;  }\)
\(\large{ A =  \frac { \theta \; \pi } { 360 } \; r^2  \;\;  }\)

Where:

Units	English	Metric
\(\large{ A }\) = area	\(\large{ in^2 }\)	\(\large{ mm^2 }\)
\(\large{ \theta }\) = angle	\(\large{ deg }\)	\(\large{ rad  }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Segment Area formulas

\(\large{ A =  \frac { 1 } { 2 } \; r^2 \; \left( \;  \frac {\pi} {180}  \theta \;-\; sin \; \theta \; \right)   \;\;  }\)
\(\large{ A =  \left(  \frac { \theta \; \pi } { 360 }   \;-\;    \frac { sin \; \theta } { 2 }  \right)   r^2   \;\; }\)

Where:

Units	English	Metric
\(\large{ A }\) = area	\(\large{ in^2 }\)	\(\large{ mm^2 }\)
\(\large{ \theta }\) = angle	\(\large{ deg }\)	\(\large{ rad  }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Radius of Gyration formulas

\(\large{ k_{x} =    \frac { r }  {  2  }    }\)
\(\large{ k_{y} =   \frac { r }  {  2 }  }\)
\(\large{ k_{z} =   \frac {  \sqrt {2}  }  { 2  } \;  r  }\)
\(\large{ k_{x1} =   \frac {  \sqrt {5}  }  { 2  } \; r  }\)
\(\large{ k_{y1} =   \frac {  \sqrt {5}  }  { 2  } \; r }\)
\(\large{ k_{z1} =   \frac {  \sqrt {10}  }  { 2  } \; r   }\)

Where:

Units	English	Metric
\(\large{ k }\) = radius of gyration	\(\large{ in }\)	\(\large{mm  }\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)

 

Circle Second Moment of Area formulas

\(\large{ I_{x} =  \frac { \pi \; r^4}{ 4 }  }\)
\(\large{ I_{y} = \frac { \pi \; r^4}{ 4 }  }\)
\(\large{ I_{x1} =   \frac { 5 \; \pi \; r^4}{ 4 }  }\)
\(\large{ I_{y1} =  \frac {5 \; \pi \; r^4}{ 4 }  }\)

Where:

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

 

Circle Torsional Constant formulas

\(\large{ J  =  \frac {  \pi \; r^4  } {  2  }   }\)
\(\large{ J  =  \frac {  \pi \; d^4  } {  32  }   }\)

Where:

Units	English	Metric
\(\large{ J }\) = torsional constant	\(\large{ in^4 }\)	\(\large{ mm^4 }\)
\(\large{ d }\) = diameter	\(\large{ in }\)	\(\large{ mm }\)
\(\large{ \pi }\) = Pi	\(\large{3.141 592 653 ...}\)
\(\large{ r }\) = radius	\(\large{ in }\)	\(\large{mm  }\)