Volumes
This page provides formulas for the volumes of solid shapes.
Polyhedron
A 3D object with flat faces and straight edges.
Cube
A polyhedron with six equal square faces.
Volume = Side ^ 3
Square Prism
A polyhedron with all rectangular faces.
Volume = Length * Width * Height
Prism
A polyhedron having two parallel polygon faces connected by rectangular faces.
Volume = AreaOfEndFace * Height
Pyramid
A polyhedron with a polygon base and a triangular face for each segment of the base.
Volume = BaseArea * Height / 3
Wedge
A polyhedron with a rectangular side and faces connecting the rectangle to a line parallel to an edge of the rectangle.
Let Height be the distance from the rectangular surface to the opposing edge of length EdgeLength.
Let Side1 be the side of the rectangle parallel to the wedge edge and Side2 be the other side of the rectangle.
Volume = ((2 * Side1) + EdgeLength) * Side2 * Height / 6
Cylinder
The shape formed by the straight lines joining corresponding points of two parallel circles, the lines being perpendicular to the surfaces of the circles.
Volume = Pi * (Radius ^ 2) * Height
Cylinder with one ellipsoid face
A cylinder cutoff at an angle providing one ellipse face.
Volume = 1.5708 * (Radius ^ 2) * (Height1 + Height2)
Cone
A solid shape formed by straight lines joining a point to a circle.
Volume = Pi * (Radius ^ 2) * Height / 3
Sphere
A solid shape formed by all points equidistant from a fixed point.
Volume = 4 * Pi * (Radius ^ 3) / 3
Spherical Sector
A pie shaped portion of a sphere.
Let Gap be the distance from the enclosed cone base center to the sphere edge.
Volume = 2 * Pi * (Radius ^ 2) * Gap / 3
Spherical Segment
A dome shaped portion of a sphere cutoff by a plane.
Let Gap be the distance from the circular disc center to the sphere edge.
Volume = Pi * (Gap ^ 2) * (SphereRadius - (Gap / 3))
Volume = Pi * Gap * ( ((CircularDiscDiameter ^ 2) / 8) + ((Gap ^ 2) / 6) )
Ellipsoid
A solid shape having ellipses or circles for all intersections of a plane.
Volume = 4 * Pi * Radius1 * Radius2 * Radius3 / 3
Torus
A donut-shaped solid generated by rotating a circle around an axis external to the circle.
Let SmallRadius be the radius of the circle that is rotated to form the torus.
Let HoleRadius be the radius of the donut hole.
Volume = 2 * (Pi ^ 2) * (HoleRadius + SmallRadius) * (SmallRadius ^ 2)
Barrel
A barrel-shaped solid having two parallel circular sides with curved walls bowed outwards.
Let MinorDiameter be the diameter of the circular sides.
Let MajorDiameter be the largest diameter of the bowed walls.
If the sides are arcs of a circle:
Volume = Pi * Height * (2 * (MajorDiameter ^ 2) + (MinorDiameter ^ 2)) / 12
If the sides are arcs of a parabola:
Volume = 0.209 * Height * (2 * (MajorDiameter ^ 2) + (MajorDiameter * MinorDiameter) + (MinorDiameter ^ 2) * 3 / 4)
