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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