Mensuration Formulas for Classs 8 and 10 – Toppr

Mensuration: Among the Most Practical Concepts of Maths!

Mensuration is a very important topic when it comes to the geometry of the universe. By definition, mensuration refers to the part of geometry concerned with ascertaining lengths, areas, and volumes. Hence, it is easy to see why mensuration is instrumental and plays a big part in real-world applications. Everything from discovering intra-cellular entities to launching a rocket into space, and from surgical equipment in the world of medicine to make sure all instruments work properly in order to get a 300-tonne aircraft from 37,00 feet back on the ground, mensuration’s applications everywhere. It is also an extremely important concept for classes 8th, 9th and 10th! Let’s now look at some important dimensions of mensuration.

1. Length

Length refers to the measurement or extent of something from end to end. It is usually the greater of two or the greatest of three dimensions of an object. The most simple device we use to measure everyday objects is a ruler, and in some cases, measuring tape. However, as scale shifts, we’ll need to change our tools – microscopic rulers need to be used to, for example, find out the length of a cell, whereas telescopic rulers are employed in order to find out the distance, say, between two star systems. Length measurements may be the most common of all measurement types. The SI units of measurement, which define a standard set of units with which to measure standard dimensions, defines the standard unit of length to be a meter, which is defined as the length of the path travelled by light in a vacuum in 1/299 792 458 seconds.

2. Area

It is the quantity that is used to express the extent of a two-dimensional figure or shape, or planar lamina, in the plane. A similar quantity known as the surface area can be calculated for three-dimensional objects. The standard unit for this would be m^2 (or square meter), and it is calculated as follows for various shapes and figures:-

Area of Rectangle: Length X Breadth
Area of Triangle: 0.5 X Base X Height
Area of Square: Side X Side (or Side^2)
Area of Circle: Pi X Radius X Radius
Surface Area of a Cylinder = 2 X Pi X Radius X (Radius + Height)
Surface Area of a Sphere = 4 X Pi X Radius X Radius
Surface Area of a Cube = 6 X Side X Side
Surface Area of a Cuboid = 2 X (Length X Breadth + Breadth X Height + Length X Height)

3. Volume

It refers to the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic meter. Given below are some standard definitions for volumes of recognizable objects:-

Volume of Cuboid: Length X Breadth X Height
Volume of Cube: Side X Side X Side
Volume of Sphere: 4/3 X Pi X Radius X Radius X Radius
Volume of Prism: Area of Base X Height
Volume of Cone: 1/3 X Pi X Radius X Radius X Height

4. Perimeter

A perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path or its length—it can be thought of as the length of the outline of a shape. The perimeter of a circle or ellipse is called its circumference. As mentioned before, since perimeter is nothing but a measure of length, it’s SI unit will be the meter itself. Given below are some formulae for perimeters of two-dimensional figures:-

Perimeter of Circle: 2 X Pi X Radius
Perimeter of Triangle: Side A + Side B + Side C
Perimeter of Square: 4 X Side
Perimeter of Rectangle: 2 X (Length + Breadth)

Above are just some of the many examples of the real-world applications of mensuration. To give you a broader sense, everything from measuring jugs to determine the volume of ingredients to be used in cooking, to checking the weather forecast before heading out, or to find out the cost of painting the Empire State Building, to everything in-between. It is an indispensable tool in the modern world.

All the Formulas

Some important mensuration formulas are:

1. Area of rectangle (A) = length(l) × Breath(b)

 A = l \times b

2. Perimeter of a rectangle (P) = 2 × (Length(l) + Breath(b))

 P = 2 \times(l + b)

3. Area of a square (A) = Length (l) × Length (l)

 A = l \times l

4. Perimeter of a square (P) = 4 × Length (l)

P = 4 \times l

5. Area of a parallelogram(A) = Length(l) × Height(h)

 A = l \times h

Parallelogram

6. Perimeter of a parallelogram (P) = 2 × (length(l) + Breadth(b))

 P = 2 \times (l + b)

7. Area of a triangle (A) = (Base(b) × Height(b)) / 2

 A = \frac{1}{2} \times b \times h

Triangle

And for a triangle with sides measuring “a”, “b” and “c”, Perimeter = a+b+c

and s = semi perimeter = perimeter / 2 = (a+b+c)/2

And also: Area of triangle =  A = \sqrt{s(s-a)(s-b)(s-c)}

This formula is also known as “Heron’s formula”.

8. Area of triangle(A) = \frac{1}{2} a \times b \times \angle C = \frac{1}{2} b \times c \times \angle A = \frac{1}{2} a \times c \times \angle B

Where A, B and C are the vertexes and angle A, B, C are respective angles of triangles and  a, b, c are the respective opposite sides of the angles as shown in the figure below:

area of triangle - mensuration

area of triangle – mensuration

9. Area of isosceles triangle = \frac{b}{4}\sqrt{4a^2 - b^2}

Where a = length of two equal sides, b= length of the base of an isosceles triangle.

10. Area of trapezium (A) = \frac{1}{2} (a+b) \times h

Where “a” and “b” are the length of parallel sides and “h” is the perpendicular distance between “a” and “b”.

Trapezium

11. Perimeter of a trapezium (P) = sum of all sides

12. Area of rhombus (A) =  Product of diagonals / 2

13. Perimeter of a rhombus (P) = 4 × l

where l = length of a side

14. Area of quadrilateral (A) = 1/2 × Diagonal × (Sum of offsets)

quadrilateral

15.  Area of a Kite (A) = 1/2 × product of its diagonals

16. Perimeter of a Kite (A) = 2 × Sum on non-adjacent sides

17.  Area of a Circle (A) =  \pi r^2 = \frac{\pi d^2}{4}

Where r = radius of the circle and d = diameter of the circle.

18. Circumference of a Circle =  2 \pi r = \pi d

r= radius of circle

d= diameter of circle

19. Total surface area of cuboid =  2 (lb + bh + lh)

where l= length , b=breadth , h=height

20. Total surface area of cuboid =  6 l^2

where l= length

21. length of diagonal of cuboid =  \sqrt{l^2+b^2+h^2}

22. length of diagonal of cube =  \sqrt{3 l}

23. Volume of cuboid = l × b × h

24. Volume of cube = l × l × l

25. Area of base of a cone = \pi r^2

26.  Curved surface area of a cone = C = \pi \times r \times l

Where r = radius of base, l = slanting height of cone

27. Total surface area of a cone =  \pi r (r+l)

28. Volume of right circular cone =  \frac{1}{3} \pi r^2 h

Where r = radius of base of cone, h= height of the cone (perpendicular to base)

29. Surface area of triangular prism = (P × height) + (2 × area of triangle)

Where p = perimeter of base

30. Surface area of polygonal prism = (Perimeter of base × height ) + (Area of polygonal base × 2)

31. Lateral surface area of prism = Perimeter of base × height

32. Volume of Triangular prism = Area of the triangular base × height

33. Curved surface area of a cylinder =  2 \pi r h

Where r = radius of base, h = height of cylinder

34. Total surface area of a cylinder =  2 \pi r(r + h)

35. Volume of a cylinder =  \pi r^2 h

36. Surface area of sphere =  4 \pi r^2 = \pi d^2

where r= radius of a sphere, d= diameter of a sphere

37. Volume of a sphere =  \frac{4}{3} \pi r^3 = \frac{1}{6} \pi d^3

38. Volume of hollow cylinder = \pi r h(R^2-r^2)

where , R = radius of cylinder , r= radius of hollow , h = height of cylinder

39. Right Square Pyramid:

If a = length of the base, b= length of equal side; of the isosceles triangle forming the slanting face, as shown in figure:

net diagram of right square pyramid

net diagram of the right square pyramid

39.a Surface area of a right square pyramid =  a \sqrt{4b^2 - a^2}

39.b Volume of a right square pyramid =  \frac{1}{2} \times base \, \, area \times height

40. Square Pyramid:

40.a. Johnson Pyramid:

net diagram of johnson pyramid
net diagram of johnson pyramid

Volume = (1+ \sqrt{3})\times a^2
Total Surface Area: \frac{\sqrt{2}}{6} \times a^3

40.b. Normal Square pyramid:

If a = length of the square base and h = height of the pyramid then:
Volume = V=\frac{1}{3}a^2h
Total Surface Area = a^2+a\sqrt{a^2+(2h)^2}

41. Area of a regular hexagon =  \frac{3\sqrt{3}a^2}{2}

42. area of equilateral triangle =  \frac{\sqrt{3}}{4} a^2

43. Curved surface area of a Frustums = \pi h (r_1 + r_2)

44. Total surface area of a Frustums = \pi (r_1^2 + h(r_1+r_2) + r_2^2)

45. Curved surface area of a Hemisphere =  2 \pi r^2

46. Total surface area of a Hemisphere =  3 \pi r^2

47. Volume of a Hemisphere =   \frac{2}{3} \pi r^3 = \frac{1}{12} \pi d^3

48. Area of sector of a circle =  \frac{\theta r^2 \pi}{360}

where  \theta  = measure of the angle of the sector, r= radius of the sector

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