Collegedunia Team Content Curator
Content Curator
Mensuration Formula helps in calculation of various 2D or 3D figures. Mensuration deals with measurements of solid closed figures, like calculation of their area, perimeter, volume, length, surface area, lateral surface area, and other parameters. These figures can be in two dimensions as well as three dimensions. It defines the principles of calculation and at the same time, determines all the important equations and properties of various geometric shapes and figures.
Read more: Circumference of Circle
| Table of Content |
Key Terms: Mensuration problems, 2D shape, 3D shape, Square, Rectangle, Circle, Triangle
What is Mensuration?
[Click Here for Sample Questions]
Suppose, you are at a playground that is rectangular in shape, how will you get to know the area your playground covers, or the boundary of the playground? With mensuration, these calculations become very easy. The area of the playground covers can be easily calculated by using the formula, area of a rectangle = l x b, where l is the length of the ground and b is the breadth. The boundary covered by the ground is known as the perimeter and is calculated by the formula, perimeter of a rectangle = 2(l+b), where l is the length of the ground and b is the breadth.
Important Terms Related to Mensuration
[Click Here for Sample Questions]
Here are some important terms related to Mensuration:
| Terms | Abbreviation | Unit | Definition |
|---|---|---|---|
| Area | A | m2 or cm2 | The area is the surface that is covered by the closed shape. |
| Perimeter | P | cm or m | The measure of the continuous line along the boundary of the given figure is called a Perimeter. |
| Volume | V | cm3 or m3 | The space occupied by a 3D shape is called a Volume. |
| Curved Surface Area | CSA | m2 or cm2 | If there’s a curved surface, then the total area is called a Curved Surface area. Example: Sphere |
| Lateral Surface area | LSA | m2 or cm2 | The total area of all the lateral surfaces that surrounds the given figure is called the Lateral Surface area. |
| Total Surface Area | TSA | m2 or cm2 | The sum of all the curved and lateral surface areas is called the Total Surface area. |
| Square Unit | – | m2 or cm2 | The area covered by a square of side one unit is called a Square unit. |
| Cube Unit | – | m3 or cm3 | The volume occupied by a cube of one side one unit |
What is a 2D Shape?
[Click Here for Sample Questions]
A 2D diagram is referred to a shape on a plane by three or more straight lines or a closed segment. Such shapes do not have width or height, but have two dimensions-length and breadth and hence are called 2D shapes or figures. Of 2D forms, area (A) and perimeter (P) is to be determined.
What is a 3D Shape?
[Click Here for Sample Questions]
A 3D shape is referred to a form that is surrounded by a variety of surfaces or planes. These are also considered robust types. Unlike 2D shapes, These shapes have height or depth, have three-dimensional length, breadth and height or depth and are therefore called 3D figures. Moreover, 3D shapes are made up of several 2D shapes. Volume (V), curved surface area (CSA), lateral surface area (LSA) and total surface area (TSA) , which are often known as strong forms, are measured for 3D shapes.
Differences between 2D and 3D Shapes
[Click Here for Sample Questions]
| 2D shape | 3D shape |
|---|---|
| If a shape is surrounded by three or more straight lines present in a plane, then it is known as a 2D shape. | If a shape is surrounded by a number of surfaces then it is known as a 3D shape. |
| It is known as 2D-two dimensional because it is defined by two dimensions- length and breadth. These shapes have no height. | It is known as 3D-three dimensional because it is defined by three dimensions- length, breadth, and height. |
| Area and perimeter can be calculated for 2D shapes. | For 3D shapes, volume, Lateral Surface Area, Curved Surface Area, Total Surface Area, and other parameters can be found. |
Mensuration Formulae for 2D Shapes
[Click Here for Sample Questions]
The 2D shapes have a list of formulas of measurement which define a relationship between the various parameters. Given below are the estimation equations of some kinds.
Square
Area of the square, A = a2
Perimeter of the square, P=4xa
Diagonal = √2 x side units
Rectangle
Area of rectangle, A = l x b
Perimeter of the rectangle, P=2( l+b )
Diagonal, D = √(length2+breath2 ) units.
Circle
Area of circle, A = πr2
Circumference of the circle, P=2πr
Diameter, D = 2×radius units
Scalene Triangle
Area of triangle, A = √[s(s−x)(s−y)(s−z)] , where s = (x+y+z)/2
Perimeter of the triangle, P=x+y+z.
Isosceles Triangle
Where AB=AC=a, a is the sides of the triangle, BC=b=base of the triangle, AD=h=height of the triangle
Area of the triangle, A = 0.5 x b x h,
Perimeter of the triangle, P = 2a+b.
Equilateral Triangle
Where all the sides of a triangle are the same.
Area of the triangle, A = (√3/4) × a2,
The perimeter of the triangle, P = 3a.
Trapezium
Area of trapezium = ½ x h x (a+c),
Perimeter of trapezium = a + b + c + d.
Rhombus
Here, d1 and d2 are the lengths of diagonals of the rhombus.
Area of Rhombus, A = ½ (d1xd2)
The perimeter of Rhombus, P = 4a,
where a is the length of the sides of the rhombus.
Mensuration Formulae for 3D shapes
[Click Here for Sample Questions]
Cube
The volume of cube = a3
Lateral Surface Area of cube = 4a2
Total Surface Area of cube = 6a2
Cuboid
Volume of cuboid = l x b x h
Lateral Surface Area of cuboid = 2h(l+b)
Total Surface Area of cuboid = 2(lb + lh + bh)
Sphere
Volume of sphere = (4/3) πr3
Curved Surface Area of sphere = 4πr2
Total Surface Area of sphere = 4πr2
Hemisphere
Volume of hemisphere = (2/3) πr3
Curved Surface Area of hemisphere = 2πr2
Total Surface Area of hemisphere = 3πr2
Cone
Volume of cone = (1/3) πr2h
Curved Surface Area of cone = πrl
Total Surface Area of cone = πr(r+l)
Cylinder
Volume of cylinder = πr2h
Curved Surface Area of cone = 2πrh
Total Surface Area of cone = 2πrh + 2πr2
Things to Remember
- Mensuration deals with the calculation of a number of parameters like volume, surface area, perimeter and other measurements of closed solid two dimensional or three dimensional figures using various formulae.
- All the formulae for calculation of surface area, perimeter, volume, or area are important and are to be memorised thoroughly.
- Mensuration is a highly real-life based topic with its applications ranging in almost every field.
- A 2D diagram is referred to a shape on a plane by three or more straight lines or a closed segment. Such shapes do not have width or height, but have two dimensions-length and breadth and hence are called 2D shapes or figures.
- A 3D shape is referred to a form that is surrounded by a variety of surfaces or planes. These are also considered robust types.
- Unlike 2D shapes, These shapes have height or depth, have three-dimensional length, breadth and height or depth and are therefore called 3D figures.
Sample Questions
Ques: Find the area of a rhombus whose diagonals are of lengths 10 cm and 8.2 cm. (2 marks)
Ans: Now, d1=10 cm, d2=8.2 cm where d1 and d2 are the lengths of diagonals of a rhombus.
Area of rhombus formula is ½ (d1xd2)
Therefore, area of the given rhombus = ½ x 10 x 8.2 cm2 = 41 cm2
Ques: The area of a rhombus is 240 cm2 and one of the diagonals is 16 cm. Find the other diagonal. (2 marks)
Ans: Let one of the diagonal be d1 cm and the other is d2 = 16 cm.
Now, Area of rhombus = ½ (d1xd2) = 240
Or, ½ (d1x16) = 240
Or, d1 = 30 cm.
Ques: Find the surface area of cube A with a =10 cm and lateral surface area of cube B with a = 8 cm. (3 marks)
Ans: Cube A has sides with length a = 10 cm.
Surface Area of cube = 6a2
Therefore, Surface Area of cube A = 6 x (10x10) = 600 cm2
Cube B has sides with length a = 8 cm.
Lateral Surface Area of cube = 4a2
Therefore, Lateral Surface Area of cube A = 4 x (8x8) = 256 cm2
Ques: An aquarium is in the form of a cuboid whose external measures are 80 cm × 30 cm × 40 cm. The base, side faces, and back face are to be covered with a coloured paper. Find the area of the paper needed? (4 marks)
Ans: The length of the aquarium = l = 80 cm,
Breadth of the aquarium = b = 30 cm,
Height of the aquarium = h = 40 cm.
Area of base of the aquarium = l x b = 80 × 30 = 2400 cm2
Area of the side of the aquarium = b x h = 30 × 40 = 1200 cm2
Area of the back of the aquarium = l x h = 80 × 40 = 3200 cm2
Required area = Area of the base + area of the back of the aquarium +(2× area of the side of the aquarium) = 2400 + 3200 + (2 × 1200)
= 8000 cm2
Hence the area of the coloured paper required is 8000 cm2.
Ques: In a building, there are 24 cylindrical pillars. The radius of each pillar is 28 cm and its height is 4 m. Find the total cost of painting the curved surface area of all pillars at the rate of ∏ 8 per m2. (3 marks)
Ans: Radius of the cylindrical pillar, r =28cm=0.28m
Height of the cylindrical pillar, h = 4 m
Curved surface area of a cylinder = 2πrh
Curved surface area of one pillar= 2×(22/7)×0.28×4 =7.04m2
Curved surface area of 24 such pillars = 7.04 × 24 = 168.96m2
Cost of painting an area of 1 = ∏ 8
Therefore, cost of painting 1689.6m2 = 168.96 × 8 = ∏1351.68
Ques: The internal measures of a cuboidal room are 12 m × 8 m × 4 m. Find the total cost of whitewashing all four walls of a room, if the cost of whitewashing is ∏5 per m2. What will be the cost of whitewashing if the ceiling of the room is also whitewashed? (4 marks)
Ans: Now, Let the length of the room = l = 12 m
Width of the room= b = 8 m
Height of the room= h = 4 m
Area of the four walls of the room = Perimeter of the base x Height of the room = 2 ( l + b ) x h = 2 (12 + 8) x 4 = 2 x 20 x 4 = 160 m2. Cost of white washing per m2 = ∏ 5
Hence the total cost of white washing four walls of the room = ∏ (160 x 5)
= ∏ 800
Area of the ceiling = 12x8 = 96 m2
Cost of white washing the ceiling = ∏ (96 x 5) = ∏ 480
So the total cost of white washing = ∏ (800 + 480) = ∏ 1280
Ques: A rectangular paper of width 14 cm is rolled along its width and a cylinder of radius 20 cm is formed. Find the volume of the cylinder. (Take π=22/7) (3 marks)
Ans: A cylinder is formed by rolling a rectangle of 14 cm about its width. The radius of the cylinder is 20 cm and the width of the paper becomes the height.
Height of the cylinder, h = 14 cm.
Radius of the cylinder, r = 20 cm.
Now, volume of the cylinder, V = πr2h = (22/7) x 20 x 20 x 14
= 17600 cm3
Hence, the volume of the cylinder is 17600 cm3.
Read more:
Check out:
Comments