VOLUME FORMULAS FOR 3D SHAPES
VOLUME FORMULAS FOR 3D SHAPES
Volume - refers to the amount of space occupied by a three-dimensional object. In geometry, calculating the volume is essential for understanding the capacity of a shape. It is used in various fields like engineering, architecture, and manufacturing to determine the amount of material or space an object can hold.
Volume formulas are mathematical tools used to calculate the space inside 3D geometric shapes. Each shape, such as a cube, sphere, or cone, has its own specific formula for determining its volume.
1.Volume of Cube
A cube is a 3D solid whose all sides are equal. Let us consider a cube of side 'a'.
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Formula for Volume of Cube:
Volume of Cube (V) = a3 ,
Where a is Side of Cube.
The volume of Cube Using Diagonal:
Volume of Cube(V) = (√3 × d3)/9 , where, d is Length of Diagonal of Cube
Let's consider some examples based on the above formulas.
Example: Find the volume of a cube if its side is 2 meters.
Given,
Side of Cube(a) = 2 mVolume of Cube(V) = a3
V = (2)3 = 8 m3
2.Volume of Cuboid
Cuboid is a 3D solid with all three sides length breadth and height are unequal. Consider a cuboid of height h, length l, breadth b.
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Formula for Volume of Cuboid:
Volume of Cuboid(V) = l × b × h
Where:
- l is Length of Cuboid
- b is Breadth of Cuboid
- h is Height of Cuboid
Example: Find the volume of a cuboid of length 10 m height 10 m breadth 20 m.
Solution:
Given,
- Length of Cuboid(l) = 10 m
- Breadth of Cuboid(b) = 10 m
- Height of Cuboid(h) = 20 m
Volume of Cuboid(V) = l.b.h
V = (10)(10)(20)
V = 2,000 m3
3.Volume of Cone
A cone is a 3D solid with a circular base and a pointy head. Let us consider a cone of height h and base of radius r.
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Formula for Volume of Cone:
Volume of Cone(V) = πr2h/3
Where:
- r is Radius of Cone
- h is Height of Cone
Let's consider an example for a better explanation.
Example: A cone with a radius of 30m and a height of 50 m is filled with water. What amount of water is stored in it?
Solution:
Given,
Radius of cone (r) = 30m
Height of the cone (h) = 50m
Volume is (V) = πr2h/3V = (3.14 × 30 × 30 × 50)/3
V = 47,100 m3
4.Volume of Cylinder
A cylinder is a 3D solid with 2 faces as circles and some height. Let us consider a cylinder of base radius r and height h.
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Volume of Cylinder(V) = πr2h
Where:
- r is Radius of Cylinder
- h is Height of Cylinder
Example: A cylindrical water tank is of a height of 20 meters and has a diameter of 10 meters how much water can we hold in this tank?
Solution:
Given,
- Height of Water Tank (h) = 20 m
- Diameter of Water Tank (d) = 10 m
Radius of Water Tank (r) = d/2 = 10/2 = 5 m
Amount of water it holds is equal to the volume of water tankVolume of Water Tank(V) = πr2h
V = 3.14 × (5)2 × (20)
V = 1570 m3
5.Volume of Sphere
A sphere is a 3D version of a circle and only has a radius. Let us the consider a sphere of radius r.
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Volume of Sphere = 4/3πr3
Where, r is the Radius of Sphere.
Example: A spherical balloon with a radius of 10 m is filled with water. What amount of water is stored in it?
Solution:
Given,
Radius (r) =10 mVolume of Sphere (V) = 4/3πr3
V = 4/3 × (3.14) × (10)3
V = 4186.6 m3
6.Volume of Hemisphere
A hemisphere is a 3D figure and is half of the sphere it has a radius for its dimension.
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Volume of a Hemisphere = 2/3πr3
Where, r is the Radius of Hemisphere
Example: A hemispherical bowl with a radius of 10 m is filled with water. What amount of water is stored in it?
Given,
Radius (r) =10 mVolume of Hemisphere (V) = 2/3πr3
V = 2/3 × (3.14) × (10)3
V = 2093.3 m3
7.Volume of Prism
A prism is a 3-D figure in which the base is a quadrilateral and its faces are triangular and rectangular.
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Volume of Prism (V) = (Area of Base) × (Height of Prism)
Example: Find the volume of a square prism in which the side of the square base is 8 cm and the height is 10 cm.
Solution:
Given,
- Side of Square Base (a) = 8 cm
- Height of Prism (H) = 10 cm
Area of Base = a2 = (8)2 = 64
Volume of Prism(V) = (Area of Base) × (Height of Prism)
V = 64 × 10 = 640 cm3
8.Volume of Pyramid
A pyramid is a 3-D figure in which the base is triangular or square and the faces are also triangle.
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Formula for Volume of Pyramid:
Volume of Pyramid (V) = 1/3× (Area of Base) × (Height of Pyramid)
Example: Find the volume of the square pyramid in which the side of the square base is 9 cm and the height is 10 cm.
Solution:
Given,
- Side of Square Base (a) = 9 cm
- Height of Pyramid (H) = 10 cm
Area of Base = a2 = (9)2 = 81
Volume of Pyramid(V) = 1/3 (Area of Base) × (Height of Prism)
V = 27 × 10 = 270 cm3
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