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 triangle(A) = \(\frac{1}{2} \times \text{base} \times \text{altitude}\)
  2. Area of triangle(A) = \(\sqrt{s(s -a)( s – b)(s – c)}\) where, a, b, c are sides of \(\Delta ABC\) and \(s = \frac{a+b+c}{2}\)
  3. Area of equilateral triangle(A) = \(\frac{\sqrt{3}}{4}a^2\), where a is length of a side.
  4. Area of rectangle(A) = \(l \times b\) and area of square = \(l^2\)
  5. Area of parallelogram(A) = \(\text{base} \times \text{altitude}\)
  6. Area of trapezium(A) = \(\frac{1}{2} (\text{Sum of parallel sides}) \times \text{height}\)
  7.  Area of rhombus(A) = \(\frac{1}{2} ( \text{product of the diagonals})\)
  8. Area of regular hexagon(A) = \(6 \left( \frac{\sqrt{3}}{4}a^2 \right) = \frac{3\sqrt{3}}{2}a^2\), where a = length of a side of the hexagon.
  9. Area of quadrilateral(A) = \(\frac{1}{2} \times \text{diagonal} \times ( \text{sum of the perpendiculars from opposite} \\ \text{vertices to the diagonal})\)
  10. Area of circle(A) = \(\pi r^2\) where r = radius or, \(A = \frac{\pi d^2}{4}\) where d = 2r.
  11. Circumference of circle(C) = \(2 \pi r\) or \(C = \pi d\)
  12. Perimeter of rectangle(P) = 2(l + b) and perimeter of square(P) = 4l.
  13. Volume of cuboid(V) = \(l \times b \times h\)

Area of four walls of cuboid(A) = 2h(l + b).

Surface area of cuboid(A) = \(2(l \times b + b \times h + h \times l)\)

Length of diagonal of cuboi(L) = \(\sqrt{l^2 + b^2 + c^2}\)

  1. Volume of cube(V) = \(a^3\), where a = length of a side.

Surface area of cube(V) = \(6 a^2\)

  1. Volume of a cylinder(V) =  \(\pi r^2 h\), base area of cylinder(A) = $latex  \pi r^2$

Circumference of cylinder(C) = \(2 \pi r\)

Curved surface area of cylinder(C) = \(2 \pi rh\)

Total surface area of cylinder(C) = \(2 \pi r (r + h)\)

  1. Volume of a prism(V) = \(\text{Area of base} \times \text{height}\)

Lateral surface area of prism(A) = \(\text{perimeter of base} \times \text{height}\)

Total surface area of prism(A) = \(2 \times \text{area of base} + \text{lateral surface area}\)

  1. Volume of pyramid(V) = \(\frac{1}{3} \times \text{area of base} \times \text{height}\)
  2. Volume of cone(V) = \(\frac{1}{3} \times \text{area of base} \times \text{height} = \frac{1}{3} \pi r^2 h\)

Lateral surface area of cone(A) = \(\pi rl\), where l = slant height of cone.

Total surface area of cone(A) = \(\pi r (r + l)\)

Area of base of cone(A) = \(\pi r^2\)

  1. Surface area of sphere (A) = \(4 \pi r^2\) or \(\pi d^2\), where d = diameter

Volume of sphere(V) = \(\frac{4}{3} \pi r^3\) or, \(V = \frac{1}{6} \pi d^3\), where d = diameter

Circumference of great circle of sphere(C) = \(2 \pi r\), where d = diameter or \(C = \pi d\), where d = diameter

  1. Curved surface area of hemisphere(A) = \(2 \pi r^2\)

Total surface area of hemisphere(A) = \(3 \pi r^2\)

Volume of hemisphere(A) = \(\frac{2}{3} \pi r^3\) or, \(\frac{1}{12} \pi d^3\)

21. Area of pathways:

Area of path of fixed width ‘d’ running outside a rectangular filed \(A = 2d(l + b + 2d)\) units. Where ‘l’ and ‘d’ are length and breadth of the rectangular field.

Area of path of fixed width ‘b’ running inside a rectangular field. \(A= 2d(l + b -2d)\). Where l and b are length and breadth of the field.

Area of two intersecting paths of fixed width ‘d’ passing through the middle and parallel to the sides of the rectangular field. \(A = (l \times b + b \times d – d^2) = d(l + b -d)\) square units. Where l and b are length and breadth of rectangular field.

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