Mensuration Formulas

Mensuration is the branch of mathematics which deals with the study of Geometric shapes, their area, volume and related parameters.

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 formulas is also knows 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 vertex 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 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 side , b= length of base of 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 it’s 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 sphere, d= diameter of 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 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 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 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 angle of the sector , r= radius of the sector



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