Mensuration Formulas

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) = $\sqrt{l^2 + b^2 + c^2}$

14.   Volume of cube(V) = $a^3$, where a = length of a side.

Surface area of cube(V) = $ 6 a^2$

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

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

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

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

16.   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) = $2 \times \text{area of base} + \text{lateral surface area}$

17.   Volume of pyramid(V) = $\frac{1}{3} \times \text{area of base} \times \text{height}$

18.   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$

19.   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

20.   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:

i.                     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.

ii.                   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.

iii.                  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.