Shape Calculations

Updated 18 October 2012

Note on Units: Remember to keep them constant, and don't mix different units in the formulas!

2D Circles

(Area) Basic Formula

Area = Pi * Radius2

(Circumference) Basic Formula using Diameter

Circumference = Pi * Diameter

(Circumference) Basic Formula using Radius

Circumference = 2 * Pi * Radius

(Diameter) Calculate the Diameter of a circle from it’s Area

  1. Divide the Area in Square Units by Pi

  2. Take the Square Root of the Result. This is the radius.

  3. Double that to get the diameter.

2D Ellipses

(Area) Basic Formula

Area = Pi * (AxisMajor * AxisMinor) / 4
or
Area = Pi * AxisMajor * AxisMinor

2D Rectangles/Squares

(Area) Basic Formula

Area = Length * Width

(Perimeter) Basic Formula

Perimeter = 2 * (Length + Width)
or
Perimeter = [2 * Length] + [2 * Width]

2D Triangles

(Area) Basic Formula

Area = Length * Width

2D Trapezoids

(Area) Basic Formula

Area = [ (TopLength + BottomLength) * Height ] / 2
or
Area = 0.5 * (TopLength + BottomLength) * Height

2D Parallelograms

(Area) Basic Formula

Area = Base * Height

(Perimeter) Basic Formula

Perimeter = [2 * Base] + [ 2 * Side]
or
Perimeter = Base + Base + Side + Side

Basic Cubes

(Volume) Basic Formula Using Each Side’s Length

Volume = Length3

(Surface Area) Basic Formula Using Each Side’s Length

Surface Area = 6 * Length2

Basic Rectangular Prisms

(Volume) Basic Formula

Volume = Length * Width * Height

(Surface Area) Basic Formula Using Each Side’s Length

Surface Area = (2 * Width * Height) + (2 * Height * Length) + (2 * Width * Length)
or
Surface Area = 2 * ( [Width * Height] + [Length * Width] + [Length * Height] )

Square Pyramids

Notes: For this shape, the formula names are slightly different:

Base is the length/width of the base.
Side is the Length of the side (measured at an angle).

(Volume) Basic Formula

Volume = (1/3) * Base2 * Height

(Surface Area) Basic Formula

Surface Area = [ 2 * Base * Side ] + Base2

Isosceles Triangular Prism

Notes: For this shape, the formula names are slightly different:

Base is the width of the base.
Length: is the length of the prism.
Side is the length of the side angle (measured at an angle).

(Volume) Basic Formula

Volume = 0.5 * [ Base * Height ] * Length

(Surface Area) Basic Formula

Surface Area = [Base * Height] + [2 * Length * Side] + [Length * Base]

Basic Spheres

(Volume) Basic Formula using Diameter

Volume = [ Pi * diameter3 ] / 6

(Volume) Basic Formula Using Radius

Volume = (4/3) * Pi * Radius3

(Surface Area) Basic Formula Using Radius

Surface Area = 4 * Pi * Radius2

Straight Ended Cylinders

Optimum Cylinder Dimensions

This finds the cylinder with the smallest surface area for the largest volume enclosed:
Height = 2 * Radius

(Volume) Basic Formula using Radius

Volume = Pi * Radius2 * Height

(Volume) Basic Formula using Diameter

Volume = (Pi * Diameter2 * Height) / 4

(Volume) Basic Formula Worked Backwards for Height

Height = Volume / (Pi * Radius2)

(Volume) Basic Formula Worked Backwards for Radius

Radius = SQRT(Volume / (Pi * Height))

(Surface Area) Basic Formula

Surface Area = 2(Pi * Radius2) + (2 * Pi * Radius) * Height
or
Surface Area = 2(Pi * Radius2) + (2 * Pi * Radius * Height)

(Surface Area) Formula for Cylinder Sides Only

Surface Area = 2 * Pi * Radius * Height

Hemispherical Ended Cylinders

Note: Both ends must be perfect half-spheres.

(Volume) Basic Formula for Two Hemispherical Ended Cylinder

Volume = (Pi * Radius2 * Length) + (4 * Pi / 3 * Radius3)
or
Volume = Pi * Radius2 [ (4/3) * Radius + Length)
Example: 3m radius, 10m length = 395.841 m3.

(Surface Area) Basic Formula for Two Hemispherical Ended Cylinder

Surface Area = 2 * Pi * Radius * Height + 2 * (2 * Pi * Radius2)
or
Surface Area = 2 * Pi * Radius * [ (2 * Radius) + Length ]
Example: 3m radius, 10m length = 301.593 m2.

(Circumference) Basic Formula

Circumference = 2 * Pi * Radius
Example: 3m radius = 18.850 m2.

(Length) Formula to find Length given the Volume and Radius of a Hemispherical Capsule

Notes: Will give a negative number if numbers are impossible to meet.
Length = [ Volume / (Pi * Radius2) ] – [ (4 * Radius) / 3 ]
Example: 600 m3 volume and 5m radius = 0.973m length

(Length) Formula to find Length given the Surface Area and Radius of a Hemispherical Capsule

Notes: Will give a negative number if numbers are impossible to meet.
Length = [AreaSurface / (2 * Pi * Radius) ] – 2 * Radius
Example: 500 m2 Surface Area and 5m radius = 5.915m length.

Ellipsoidal Cylinders

(Volume) Basic Formula

Volume = Pi * AxisMajor * AxisMinor * Length / 4

Cylindrical Tubes

(Surface Area) Total Surface Area

Surface Area = [ 2 * Pi * (R2 – r2) ] + [ 2 * Pi * Height (R + r)
Where:
R: Outer Radius
r: Inner Radius

Basic Hemispheres

(Volume) Basic Formula

Volume = (2/3) * Pi * Radius3

(Surface Area) Curved Surface Area (Exterior Side Only!)

Surface Area = 2 * Pi * Radius2

(Surface Area) Total Surface Area

Surface Area = (2 * Pi * Radius2) + (Pi * Radius2)
or
Surface Area = 3 * Pi * Radius2
Example: 10m radius = 942.478 m2

(Circumference) Of Hemisphere Base

Circumference = 2 * Pi * Radius

(Radius) Calculate the Radius given the Volume of a hemisphere

Radius = (3 * Volume) / (2 * Pi)^(1/3)

(Radius) Calculate the Radius given the Total Surface Area of a hemisphere

Radius = SQRT( Area / [2 * Pi] )

Frustrums

(Volume) Basic Frustrum Formula using Diameter

Volume = (Pi * Height / 12) * (DiameterBottom2 + DiameterBottom * DiameterTop + DiameterTop2)
Example: Height is 10m, top diameter is 5m, bottom diameter is 2.5m = 114.537 m3.

(Volume) Basic Frustrum Formula using Radius

Volume = (1/3) * Pi * Height * (RadiusTop2 + RadiusBottom2 + [ RadiusTop * RadiusBottom ] )
Example: Height is 5m, top radius is 3m, bottom radius is 2m = 99.484 m3

(Surface Area) Lateral Surface Area of the Frustrum using Radius

Surface Area = Pi * (RadiusTop + RadiusBottom) * Slant
Example: Top Radius is 3m, bottom radius is 2m, slant is 5.09902m = 80.0952 m2

(Slant Height) Calculating the Slant Height of a Frustrum using Radius and Height

Slant = SQRT[ (RadiusTop – RadiusBottom)2 + Height2 ]
Example: Height is 5m, top radius is 3m, bottom radius is 2m = 5.09902m

Basic Cones

(Volume) Basic Cone Formula using Diameter

Volume = [ Pi * Diameter2 * Height ] / 12

(Volume) Basic Cone Formula using Radius

Volume = (1/3) * Pi * Radius2 * Height

Parabolic Cones

(Smooth Curved Surface, Sharp Pointed Nose)

(Volume) Basic Parabolic Cone Formula using Diameter

Volume = [ 2 * Pi * Diameter2 * Height ] / 15

Elliptical Cones

(Smooth Curved Surface, Blunted Curved Nose)

(Volume) Basic Elliptical Cone Formula using Diameter

Volume = [ Pi * Diameter2 * Height ] / 6