Mensuration is the branch of mathematics that talks about the length, volume, and area of different geometrical objects. The shape may be in 2-D or 3-D. We find the volume of 3-D objects and the area of 2-D objects.
Table of Content
What is Volume?
Volume is the measurement of the total space occupied by a given solid. Volume is defined in 3-D because to have the volume the object must have three variables i.e. length, breadth or thickness or width, and height.
The difference between the total amount of space left inside the hollow body and the space occupied by the body is the volume of the figure.
Now, let us take the example of a 3d shape called a cuboid, it is a solid figure bounded by six rectangles and thus has six rectangular faces. The dimensions of the cuboid are as follows: length l, breadth b, and height h.
Volume of the cuboid is given as:
Volume = length * height * width
Volume is for both solid and hollow objects. E.g. Cube, Cuboid, Cone, Cylinder, etc.
What is Capacity?
When the hollow object is filled with liquid or air and it takes the shape of that object or container. The total volume of the water and air which is filled inside the container is called the capacity of the container.
Note: Capacity is calculated only for hollow object.
Example: Cone, Cylinder, Hollow hemisphere, etc.
Example: Find the volume and capacity of cylinder whose radius is 14 cm and height is 21cm.
Solution: We have given, r = 14cm, h = 21cm;
Volume of cylinder = πr*r*r*h.
V = 22/7 * 14 * 14 * 21
V = 12,936cm3.
and capacity is = 12,936/1000 litre .......as (1000cm3= 1litre)
= 12.936 litre.
Note: The unit of capacity is litres(l) and millilitres(ml).
Use our calculator to convert litre to millilitre.
Mensuration of Some Enclosed Figures
Cube
Surface area of cube is defined as the area of every face of the cube. For e.g. as we know that every side of cube is equal. So area of one face is length*length or width*width or height*height out of these three we pick one according to question.
So total surface area of cube is 6 * length * length. Because cube has six faces and each face has an area of length*length.
Now the volume of cube is its hollow portion and can be written as length * length * length or width * width * width or height * height * height. And its unit will be cubic.
Example: Find the width of the cube whose volume is 625cm3.
Solution:
We know that volume of the cube is a * a * a =a3 where a can be the length, width, height.
So, a3 = 125 then
a = 5cm ( 5 is cube root of 125)
So width is 5 cm.
Cuboid
Surface area of cuboid is defined as the surface area of cube but the difference here is all the sides are not equal and in cuboid length width height are different. So its surface area will be the area of each face and add all the areas which we will get total surface area.
So total surface area of cuboid is length * width + width * height + height * length + length * width + width * height + height * length i.e. the addition of areas of every six face
So total surface area is
2 * (length * width + width * height + height * length)
The volume of cuboid is the hollow portion inside the cuboid and it is
Volume = length * width * height
Example: Find the height of cuboid volume is 625cm3 and its base area is 25 cm2.
Solution:
Base area of cube means width * length this forms base in cube
As we know volume of cube is
=> l * b * h = 625
=> h = 625 / b * l
=> h = 625 / 25
=> h = 25cm
So the height is 25 cm
Cylinder
Here we are talking about right circular cylinder for e.g. round pillar, tube lights, water pipes, etc. Volume of cylinder is the hollow portion inside the cylinder.
Volume of Cylinder = 22/7 * r * r * h
Here,
r = radius of cylinder and h = height of cylinder
Example: A rectangular sheet of paper having length 11cm and width 4cm cm is being rolled to form a cylinder of height 4 cm. What is the volume of the cylinder?
Solution:
Let the cylinder if radius = r and its height = h
Perimeter of base of cylinder = 2 * pi * r = 11cm
=> 2 * 22/7 * r = 11cm
=> r = 7/4 cm
Volume = 22/7 * r* r * h
=> 22/7 * 7/4 * 7/4 * 4
=> 38.5cm3
Hence, the volume of cylinder is 38.5cm3
Cone
Volume of cone is the hollow portion inside cone.
Volume = 1/3 * 22/7 * r * r * h
Here,
h = height of cone
r = radius of cone
Formulas for Common Geometric Figures
1. Cube
(a) Surface Area:
Surface Area=6×(side 2)
(b) Volume:
Volume=side 3
2. Cuboid
(a) Surface Area :
Surface Area=2×(lw+lh+wh)
(b) Volume :
Volume=l×w×h
3. Cylinder
(a) Volume :
Volume=πr2h
Where 𝜋 ≈ 22/7 OR 𝜋 ≈ 3.14159
4. Cone
(a) Volume :
Volume= 1/3 πr2h
5. Sphere
(a) Surface Area :
Surface Area=4πr2
(b) Volume :
Volume= 3/4πr3
Step-by-Step Examples of Volume Calculation
Example 1: Cube
Problem: Find the width of a cube with a volume of 625 cm³.
Solution :
1. Volume Formula:
Volume=side3
2. Solve for side :
side=∛625 = 5cm.
3. Width = 5 cm.
Example 2: Cuboid
Problem: Find the height of a cuboid with a volume of 625 cm³ and a base area of 25 cm².
Solution:
1. Volume Formula:
Volume=Base Area X Height
2. Solve for height:
Height = Volume / Base = 625/ 25 = 25cm.
Example 3: Cylinder
Problem: A cylindrical container has a radius of 10 cm and a height of 7 cm. Find its volume.
Solution :
1. Volume Formula:
Volume=πr2h
2. Substitute values:
Volume = 22/7 X 102 X 7 = 4400cm3
Example 4: Cone
Problem: Find the volume of a cone with a radius of 9 cm and height 14 cm.
Solution :
1. Volume Formula:
Volume =1/3 πr2 h
2. Substitute values:
Volume = 1/3 X 22/7 X 92 X 14 =1188cm3
Conversion Between Different Units of Measurement
Volume to Capacity Conversion
1. Cubic Centimetres to Litres:
1 litre=1,000 cm3
To convert cubic centimetres to litres:
Capacity (litres) = Volume(cm3 ) / 1,000
2. Litres to Millilitres:
1 litre=1,000 millilitres
To convert litres to millilitres:
Capacity (millilitres) = Capacity (litres)×1,000
Example Conversions
1. Convert 12,936 cm³ to litres:
Capacity = 12,936 / 1,000 = 12.936 litres
2. Convert 1.5 litres to millilitres:
Capacity = 1.5 X 1,000 =1,500 millilitres
Summary
Mensuration is essential for determining the dimensions and capacity of various geometric figures. Understanding formulas for volume and surface area helps in solving real-world problems related to space and capacity. Mastery of unit conversions ensures accurate measurements and calculations.
Practice Problems: Volume and Capacity - Mensuration
Problem 1: Find the capacity of a cubic tank with dimensions 1 m × 1 m × 1 m.
Problem 2: A tank measures 2 m × 1 m × 2 m. Find the capacity of the tank.
Problem 3: Find the volume of a cylinder with a radius of 10 cm and height 7 cm.
Problem 4: Find the volume of a cone with radius 9 cm and height 14 cm.