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In mathematics, solid geometry was the traditional name for the geometry of three-dimensional solids – for practical purposes the kind of space we live in. It was developed following the development of plane geometry. Stereometry deals with the measurements of volumes of various three dimensional solid figures including cylinder, circular cone, truncated cone, sphere, and prisms. (Source: From Wikipedia).

Here we are going to learn about the volume and surface area three dimensional solids.

Formulas to Find the Volume and Surface Area of three Dimensional Solids

Here we are going to see some arithmetic formulas to find the volume and surface area of simple three dimensional solids such as cube, cone, cylinder, and sphere.

Cube

Volume of cube = a3 cubic units.

Surface area of a cube = 6a2 square units

Cone

Volume of cone = ‘1/3 pi r^2 h’ cubic units.

Surface area of a cone = ‘pi r(r + s)’ square units.

Cylinder

Volume of cylinder = ‘pi r^2 h’ cubic units.

Surface area of cylinder = ‘2 pi r(r + h)’ square units

Sphere

Volume of sphere = ‘4/3 pi r^3’ cubic units.

Surface area of a sphere = ‘4 pi r^2’ square units.

Example Problems to Find the Volume and Surface Area of three Dimensional Solids

Example 1

Find the volume of a three dimensional solid with all sides equal to 3.5 feet.

Solution

A three dimensional solid with equal sides is a cube.

Volume of a cube = a3 cubic units

= 3.53

= 3.5 * 3.5 * 3.5

= 42.88

So, the volume of the given three dimensional shape is 42.88 cubic feet.

Example 2

Find the volume of the cone, whose radius is 3.5 cm and height is 4.8 cm.

Solution

Volume of a cone = ‘1/3 pi r^2 h’ cubic units

= ‘1/3’ * 3.14 * 3.5 * 3.5 * 4.8

= 61.544

So, the volume of the given cone is 61.544 cubic cm.

Example 3

Find the surface area of a cylinder with radius 4 cm and height 8 cm.

Solution

The surface area of a cylinder = ‘2 pi r(r + h)’ square units

= 2 * 3.14 * 4(4 + 8)

= 2 * 3.14 * 4 * 12

= 301.44

The surface area of the given cylinder is 301.44 square cm.

Example 4

Find the surface area of a sphere, whose radius is 1.5 m.

Solution

The surface area of a sphere = ‘4 pi r^2’ square units

= 4 * 3.14 * 1.5 * 1.5

= 28.26

So, the surface area of the given sphere is 28.26 square meter.