Volume is the amount of space contained in a three-dimensional figure. Volume or capacity of a container is the amount of fluid that the container can possess. The SI unit of volume is cubic meter. However, it can be measured as cubic units.

While calculating the volume of a figure, one has to be sure of units. The units of all the dimensions must be same. There are different formulas for calculating volumes of different types of shapes.

## Volume of a Cone Formula

Cone is a three-dimensional figure that tapers smoothly from a circular base to a point at the top. This point is called apex of the cone. The straight line joining apex and center of base is known as axis of the cone. A cone whose axis is exactly perpendicular to the base is known as right-circular cone. If we take a right-angled triangle and rotate it about its perpendicular side, then a imaginary three-dimensional shape is formed which is called a right-circular cone. A right-circular cone is shown in the following figure:

The radius of the base of a cone is denoted by "r". The length of axis or the distance between apex and center of the base is called height of the cone and is denoted by "h". The lateral height of the cone is referred as its slant height and is denoted by "l".
Volume of a cone is the amount of fluid occupied in it. The volume of cone is derived by the following formula:
Where,
Area of base = Area of circle = $\pi$ r2
Hence, formula for volume of cone can be rewritten as follows:
Where,
h = Height of cone.

## Volume of a Sphere Formula

A sphere is a three-dimensional form of a circle. A sphere is a ball. All the points on the surface of a sphere are equidistant from a fixed point which is called center of the sphere and this distance is called its radius. A sphere can be solid or hollow. Sphere is shown in the following figure:

Volume of a sphere is the amount of fluid enclosed by it. Formula for volume of a sphere is given below:
Where, r = Radius of sphere.

## Volume of a Hemisphere Formula

Earth is divided into four hemispheres: eastern, western, northern and southern. But geometrically, a hemisphere is exactly half of a sphere. When a sphere is cut by a plane passing through its center, sphere is divided into two parts. Each part is called hemisphere. A hemisphere may be solid or hollow. Following are the figures showing solid and hollow hemispheres respectively:
Volume of a hemisphere is amount of fluid contained in it. Volume of hemisphere is just half of that of sphere.
Volume of hemisphere = $\frac{1}{2}$ $\times$ Volume of a sphere

Volume of hemisphere = $\frac{1}{2}$ $\times$ $\frac{4}{3}$$\pi r^{3} Therefore, volume of a hemisphere is given by the following formula: Where, r = Radius of hemisphere. ## Volume of a Cube Formula A cube is three-dimensional form of a square. A shape bounded by six squares with same sides is known as a cube. The angle between two adjacent sides of a cube is right angle. If a room has equal length, width and height, then it is an example of cube. A cube may also be called "regular hexahedron". The formula for volume of a cube is given below: Where, a = Side of cube. ## Volume of a Pyramid Formula A pyramid is a three-dimensional shape which has triangular lateral sides converging to a single point. This point is called apex of the pyramid. A pyramid must have at least three lateral triangles as it is a three-dimensional figure. The base of a pyramid can be any polygon like triangle, square, rectangle, pentagon, hexagon etc. A pyramid with circular base is called cone which has been described above. Following two figures are examples of hexagonal pyramid and triangular pyramid respectively: Formula for calculating volume of a pyramid is given below: Height of a pyramid is perpendicular distance of apex of pyramid to its base. ## Volume of a Square Pyramid Formula A pyramid with a square base is called a square pyramid. A square pyramid as four triangular lateral side. The perpendicular distance from apex of the pyramid to its base is called its height. Volume of a square pyramid is same as that of a general pyramid. The only difference is that in place of area of base, we use area of square. Volume\ of\ Pyramid=$$ \frac{1}{3}$$\times Area\ of\ Base \times Height Volume\ of\ square\ Pyramid=$$\frac{1}{3}$$\times Area\ of\ square\ base \times Height If each side of square base measures "a", then area of square base = a2. Therefore, volume of square pyramid is given by: Where, a = Side of square base h = Height of the pyramid. ## Volume of a Prism Formula Prism is a three-dimensional shape which has two end faces. These end faces are parallel to each other and are of same shape and size. These end faces may be any polygon. Prisms have same cross section perpendicular to its height. It means that if we cut a prism by a plane perpendicular to the height, we obtain a same shape. Prisms are of two types: • Regular Prism: A prism with regular base is called regular prism. It means that all the sides of base are equal. • Irregular Prisms: A prism with irregular base is called irregular prism. It means that all the sides of base are not equal. A prism is demonstrated in the following figure: The space enclosed by a prism is called its volume. Volume of a prism is given below: ## Volume of a Rectangular Prism Formula A prism whose end faces are rectangle is called rectangular prism. When we take cross section of a rectangular prism perpendicular to its height, we get a rectangle. A rectangular prism is simply a cuboid. Following figure is demonstrating a rectangular prism: Volume of the rectangular prism is same as volume of prism. The only difference is that we use area of base as area of rectangle. Volume of rectangular prism = Area of rectangular base \times height Therefore, volume of rectangular prism can be defined as follows: ## Volume of a Box Formula A box is a three-dimensional shape which has six rectangular faces. A box is also called a cuboid and a regular rectangular prism. Opposite faces of a cuboid or a box are parallel and congruent to each other. All the adjacent sides of a box are perpendicular to one another. Volume of a box is the amount of fluid held by it. The formula for volume of a box or cuboid is same as the formula for that of a rectangular prism: Where, l = Length of box b = Breadth of box h = Height of box ## Volume of a Cylinder Formula A cylinder has two end faces which are flat and circular. These two faces are parallel and congruent to each other. If we take a rectangular sheet of paper and roll it along its length, we get a cylinder. The line segment joining the centers of the end faces is called axis of a cylinder. A cylinder whose axis is perpendicular to its base is referred as right circular cylinder. When the cylinder is not right-circular cylinder, its height is referred as perpendicular distance between the two end faces. Following image is demonstrating a right circular cylinder: The formula for volume of a cylinder is given by: Volume of cylinder = Area of base * height If radius of circular base is denoted by "r" and height by "h". Therefore, formula of for volume of a cylinder is rewritten as: ## Tetrahedron Volume Formula A tetrahedron is a three-dimensional shape with four triangular faces, three of which meet at a point called vertex. Tetrahedron has 6 edges and 4 vertices. Tetrahedron is a type of pyramid. Volume of a tetrahedron is same as that of volume of pyramid which is given as follows: A regular tetrahedron is a tetrahedron in which all four triangles are equilateral triangles. Following is the image showing a regular tetrahedron: Let us assume that each side of regular tetrahedron is "a". Then, Area of base = Area of equilateral triangle ΔBDC Area of base = \frac{\sqrt{3}}{4}a^{2}.........1 In right triangle BED, using Pythagorean theorem, DE2 = BD2 - BE2 DE = \sqrt{a^{2}-\frac{a^{2}}{4}} DE = \frac{\sqrt{3}}{2}a Since DE is a median of ΔBDC, ∴ DO = \frac{2}{3} DE DO = \frac{2}{3} \times \frac{\sqrt{3}}{2}a DO = \frac{\sqrt{3}}{3}a In right triangle AOD, using Pythagorean theorem, AO2 = AD2 - DO2 AO = \frac{\sqrt{6}}{3}a AO = height = \frac{\sqrt{6}}{3}a .........2 Using equation 1 and 2, we obtain Volume of regular tetrahedron = \frac{1}{3} \times \frac{\sqrt{3}}{4}a^{2}$$\times$$\frac{\sqrt{6}}{3}a Therefore, the formula for volume of regular tetrahedron is given below: ## Volume Formula for a Triangular Prism A triangular prism is a kind of prism in which two end faces are congruent triangles and are parallel to each other. A triangular prism has five faces. Other three faces are parallelograms. Volume of a prism = area of base \times height and area of base = area of triangle Therefore, Formula for volume of triangular prism is given below: Where, b = Triangle base length h = Triangle base height l = Distance between two parallel end faces. ## Formula for Volume of a Square Prism A prism whose end faces are squares is called square prism. When we take cross section of a square prism perpendicular to its height, we get a square. A square prism is simply a cuboid with square base. A square prism which has square lateral surfaces is a cube. If each side of square base is "a" and height of the prism is "h", then we have Volume of square prism = Area of base \times height Therefore, formula for volume of a square prism can be expressed as follows: Where, a = Side of square base h = Height of prism. ## Finding Volume of Shapes Following are few problems based on finding volumes of different shapes: ### Solved Examples Question 1: Find the volume of a box whose dimensions are 0.2 m, 10 cm and 15 cm. Solution: Make all the dimensions of same units. l = 0.2 m = 20 cm b = 10 cm h = 15 cm Volume of a box = l \times b \times h = 20 \times 10 \times 15 = 3000 cm3 Question 2: Following figure shows a cone filled with ice-cream. The radius of hemispherical top is 4 cm and height of cone is 10 cm. Evaluate the amount of ice-cream it can hold. Solution: Volume of ice-cream = volume of cone + volume of hemisphere Volume of ice-cream = \frac{1}{3}$$\pi r^{2}h$ + $\frac{2}{3}$$\pi r^{3} = \frac{1}{3}$$\times$$\frac{22}{7} \times 4 \times 4 \times 10 + \frac{2}{3} \times \frac{22}{7} \times 4 \times 4 \times 4 = \frac{1}{3} \times \frac{22}{7} \times 16(10+8) = 301.14 cm3 Question 3: A ball of radius 5 cm is melted and is recast into small balls each of which having radius 1 cm. How many small balls are created. Solution: Volume of sphere = \frac{4}{3}$$\pi r^{3}$

Let n number of small balls are created.

Volume of big ball = n $\times$ volume of small ball

$\frac{4}{3}$$\pi \times 5^{3} = n \times \frac{4}{3}$$\pi$\times$1^{3}$

125 = n $\times$ 1

n = 125

Hence 125 small balls are created.

Question 4: A tent is in the form of cylinder surmounted by a cone as shown in the following figure:

Height of the tent is 13 m and that of cone is 2 m. Radius of cone as well as cylinder is 1 m. Find the volume of the tent.
Solution:

Height of cone = hcone = 2 m
Height of cone = hcylinder = 13 - 2
= 11 m
Volume of tent = volume of cone + volume of cylinder

Volume of tent = $\frac{1}{3}$$\pi r^{2}h_{co}+\pi r^{2}h_{cy}$

= $\frac{1}{3}$ $\times$ $\frac{22}{7}$ $\times$ $1^{2}$ $\times$ $2\ +\$ $\frac{22}{7}$ $\times$ $1^{2}$ $\times$ $11$

= $\frac{22}{7}$($\frac{2}{3}$+11)

= 36.67

Therefore, the volume of tent is 36.67 m3