Mensuration Formulas for CAT SSC CGL and Other Exams

Mensuration formulas based questions regularly feature in CAT, SSC CGL and other exams. Being a part of geometry, this topic demands that the aspirant be familiar not only with the different formulas of mensuration but also with the other theorems and concepts of geometry.  However, the questions from mensuration are less difficult that other types of geometry questions.

An aspirant must be familiar with different mensuration concepts and geometrical concepts such as  triangle, circle, rectangle, square, cone , cuboid, cube, cylinder, sphere. In addition to this, he must correctly remember the area, perimeter and volume formulae of different geometrical shapes. This article contains advance mensuration formulae and geometry shortcut tricks.

Mensuration Formulas PDF

All Mensuration Formulas for Triangles:

It is the sum of all three sides of a triangle. Let a, b, and c be the sides of a triangles then:

1. Formula of Perimeter of triangle (P) = a + b + c
2. Formula of Semi Perimeter of triangle (s) =(frac{a+b+c}{2})

The two mostly used mensuration formulas for area of triangle are:

1. Area of triangle =\(\frac{1}{2} \times b \times h\)
2. Area of triangle =\(\sqrt {s\left( {s – a} \right)\left( {s – b} \right)\left( {s – c} \right)} \)

Some advance mensuration formulas for area of triangle are:

1. Area of triangle =\(r \times s\), where \(r\) is in-radius of the triangle.
2. Area of triangle = \(\frac{abc}{4R}\), where \(R\) is circum-radius of the triangle.
3. Area of triangle=\(\frac{4}{3}\sqrt {u\left( {u – d} \right)\left( {u – e} \right)\left( {u – f} \right)} \)
   ,where d, e and f are medians and \(u = \frac{{d + e + f}}{2}\)
4. Area of triangle=\(\frac{1}{2} \times ab\sin \theta \), where a and b are the two known sides and   is the included angle.

Some mensuration formulas for area of special triangle:

1. Area of equilateral triangle =\(\frac{{\sqrt 3 }}{4}{a^2}\), where a is the side of the equilateral triangle
2. Area of isosceles triangles =\(\sqrt {{a^2} – \frac{{{b^2}}}{4}} \), where a be the measure of the equal sides of an isosceles triangle and b be the base of the isosceles triangle.

Mensuration Formulas for Circles

Circumference of circle=\(2\pi r\)

Area of circle=\(\pi {r^2}\)

Length of arc=\(\frac{\theta }{{360}} \times 2\pi r\)

Area of Sector=\(\frac{\theta }{{360}} \times \pi {r^2}\)

Area of Segment=\(\frac{\theta }{{360}} \times \pi {r^2} – \frac{1}{2} \times {r^2} \times \sin \theta \)

Mensuration Formulas for Quadrilaterals

Mensuration Formulas for Polygons

The area of a regular polygon is given by the formula below.

Area =\(\frac{1}{2}\) (apothem)(perimeter)

Regular Polygon Formulas

n = number of sides
s = length of a side
r = apothem (radius of inscribed circle)
R = radius of circumcircle

Several other area formulas:

Sum of interior angles = (n – 2)·180°

Interior angle =\(\frac{\left( n-2 \right)}{n}\times {{180}^{0}}\) for regular polygons

Exterior angle =\(\frac{{{360}^{0}}}{n}\) for regular polygons

Mensuration Formulas for Solid (3D) Geometry

Total Surface Area of Cuboid =2(lw + lh + wh), where l = length, w = width, h = height

Volume of Cuboid=lwh, where l = length, w = width, h = height

Total Surface Area of Cube=\(6a^2\)

Volume of Cube=\(a^3\)

here \(a\) is edge length of the cube.

Lateral Surface Area of Prism=perimeter of base × heightTotal Surface Area of Prism = 2 × area of base + perimeter of base × height

Volume of Prism = area of base x height

See Also: Solid Geometry – Prism and Pyramid

Lateral Surface Area of Pyramid=\(\frac{1}{2}\times\) perimeter of base × slant height

Total Surface Area of Pyramid= area of base + area of base + \(\frac{1}{2}\times\) perimeter of base × slant height

Volume of Pyramid= \(\frac{1}{3}\times\)area of base x height

Curved Surface Area of Cylinder=\(\2 \times\ pi rh\)

Surface Area of Cylinder=\(2\pi r\left( {r + h} \right)\)

Volume of Cylinder=\( \pi {{r}^{2}}h\)

Curved Surface Area of Cone=\(\pi rl\) , whe l = slant height

Total Surface Area of Cone=\(\pi r\left( {r + l} \right)\)

Volume of Cone=\(\frac{1}{3}\pi {r^2}h\)

Total Surface Area of Sphere =\(4\pi {r^2}\)

Volume of Sphere=\(\frac{4}{3} \times \pi {r^3}\)

Curved Surface Area of Hemisphere=\(2\pi {r^2}\)

Total Surface Area of Hemisphere =\(3\pi {r^2}\)

Volume of Hemisphere=\(\frac{2}{3} \times \pi {r^3}\)