In this lesson, we give

- a table of volume formulas and surface area formulas used to
calculate the volume and surface area of three-dimensional
geometrical shapes:
**cube, cuboid, prism, solid cylinder, hollow cylinder, cone, pyramid, sphere and hemisphere**. - a more detailed explanation (in text and videos) of each volume formula.

Related Topics: More Geometry Lessons, Volume Games

The following table gives the volume formulas for solid shapes or three-dimensional shapes. Scroll down the page if you need more explanations about the formulas, how to use them as well as worksheets.

A cube is a three-dimensional figure with six matching square sides.

The figure above shows a cube. The dotted lines indicate edges hidden from your view.

If *s* is the length of one of its sides,
then the volume of the cube is *s* × *s* × *s*

Volume of the cube = *s*^{3}

A rectangular solid is also called a rectangular prism or a cuboid.

In a rectangular solid, the length, width and height may be of different lengths.

The volume of the above rectangular solid would be the product of the length, width and height that is

Volume of rectangular solid =

lwh

A **prism** is a solid that has
two parallel faces which are congruent polygons at both ends.
These faces form the **bases** of
the prism. The other faces are in the shape of rectangles. They
are called **lateral faces**.A
prism is named after the shape of its base.

When we cut a prism parallel to the base, we get a **cross
section** of a prism. The cross section has the same size
and shape as the base.

The volume of a right prism is given by the formula:

Volume of prism = Area of base × length

V =Al

where *A* is the area of the base and *l* is the
length or height of the prism.

A cylinder is a solid that has two parallel faces which are **congruent circles**. These faces form
the **bases** of the cylinder.
The cylinder has one **curved surface**.
The **height** of the cylinder is
the perpendicular distance between the two bases.

Volume = Area of base × height

V = π

r^{2}h

where *r* = radius of cylinder and *h* is the
height or length of cylinder.

How to find the volume of a cylinder, given that the radius is 9 and the height is 12.

Sometimes you may be required to calculate the volume of a hollow cylinder or tube.

Volume of hollow cylinder

where *R* is the radius of the outer surface and *r*
is the radius of the inner surface.

How you can find the volume of a hollow cylinder and a cone using the formula of volume of prism and pyramid

A cone is a solid with a circular **base**.
It has a **curved surface** which
tapers (i.e. decreases in size) to a **vertex
** at the top. The **height **of
the cone is the perpendicular distance from the base to the
vertex.

The volume of a cone is given by the formula:

Volume of cone = Area of base × height

V = where *r* is the radius of the base and *h*
is the height of the prism.

How to find the volume of a cone, given that the radius is 12 and the height is 16?

A pyramid is a solid with a polygonal **base**
and several triangular **lateral faces**.
The lateral faces meet at a common **vertex**.
The **height** of the pyramid is
the perpendicular distance from the base to the vertex. The
pyramid is named after the shape of its base. For example a
rectangular pyramid or a triangular pyramid.

The volume of a pyramid is given by the formula:

Volume of pyramid = Area of base × height

V **= ** where *A* is the area
of the base and *h* is the height of the pyramid.

A **sphere** is a solid in
which all the points on the round surface are equidistant from a **fixed point**, known as
the center of the sphere. The distance from the center to the
surface is the **radius**.

** Volume of sphere** = where *r* is
the radius.

How to find the volume of a sphere? What is the volume of air in the ball?

A **hemisphere** is half a
sphere, with one flat circular face and one bowl-shaped face.

** Volume of hemisphere ** where *r* is
the radius

How to find the volume of a hemisphere? What is the volume of water in a bowl?

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