Complete Mensuration Forumula and Tricks for UPSC CSAT 2024 and 2025

On this page, we have compiled the complete list of short tricks and formula for CSAT questions related to Mensuration.

Rectangle related formulae for CSAT

Rectangle: a rectangle is a plane whose opposite sides are equal and diagonals are equal. Each angle is equal to 90 degrees. [l = length; b = breadth]

$$\text{Perimeter} = 2 \times (l + b)$$

$$\text{Area} = l \times b$$

$$\text{length } = {area \over breadth}$$

$$\therefore length= \left({Perimeter \over 2} - breadth\right)$$

$$\text{breadth} = {area \over length}$$

$$\therefore breadth = \left({Perimeter \over 2} - length\right)$$

$$\text{diagonal} = {\sqrt{length^2 + breadth^2}}$$

Square related formulae for CSAT

SQUARE: A square is a plane figure bounded by four equal sides having all its angles at right angles.

In the below square, AB = BC = CD = DA = a (let)

1. $$\text{Perimeter of a square} = {4 \times side}$$
2. $$\text{Area of a square} = {side^2}$$
3. $$\text{Side of a square} = {\sqrt{area}}$$
4. $$\text{Diagonal of a square} = {\sqrt{2} \times side}$$
5. $$\text{side of a square} = {\text{diagonal} \over {\sqrt{2}}}$$

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Triangle

• Types of triangle:
  1. equilaterial: All three sides are equal in length, and all three angles are equal in measure. Each angle measures 60 degrees, and the shape is regular
  2. isosceles: Two sides are equal, and the third side is not equal to the other two. The angles opposite to the equal sides are also equal.
  3. scelene: All sides are different lengths, and all angles have different measures. However, the internal angles of the triangle will continue to add up to 180 degrees.
  4. right triangle: One angle is equal to 90 degrees, called a right angle. The side opposite to the right angle is the longest side and is called the hypotenuse.

$$\text{Area} \triangle = {{1 \over 2} \times base \times height}$$

$$\text{Semi Perimeter, s} \triangle = {{a + b + c} \over 2}$$

$$\text{Area} \triangle = {\sqrt{s(s-a)(s-b)(s-c)}}$$

$$\text{Perimeter} \triangle = {2s} = {a + b + c}$$

$$\text{Area, equilateral} \triangle = {{\sqrt{3} \over 4} \times (side)^2}$$

$$\text{Height, equilateral} \triangle = {{\sqrt{3} \over 2} \times side}$$

$$\text{Perimeter, equilateral} \triangle = {3 \times side}$$

Quadrilateral, Parallelogram, Rhombus and Trapezium

Parallelogram

$$\text{area} = base \times height$$

$$\text{perimeter} = 2 \times \left(a + b\right)$$

Rhombus

Area of rhombus is equal to half of the product of its diagonals.: $$\text{area} = {1 \over 2}{\left(AC \times BD\right)}$$

Perimeter of rhombus is the sum of all its sides, which becomes equal to four times length of any one side: $$\text{perimeter} = {4 \times side}$$

Trapezium

Area of Trapezium is half the product of its height (h) with the sum of its parallel sides (a + b).

$$\text{area} = {1 \over 2} \times {\left(a+b\right)} \times {height}$$

Regular Hexagon

$$\text{area} = {6 \times {\sqrt{3} \over 4} \times {side}^2}$$

$$\text{perimeter} = {6 \times side}$$

Circle

We have attached some of the most important circle related formula below.

Circumference (or, perimeter) of a circle is the product of its diameter (2r) with 3.142 (pi).

$$circumference = diameter \times \pi = 2 \pi r$$

$$radius = \frac{circumference}{2 \pi}$$

$$\text{area of circle} = \pi \times r^2$$

$$radius = \sqrt{area \over \pi}$$

$$\text{area of semi circle} = {1 \over 2}{\pi r^2} = {1 \over 8}{\pi d^2}$$

$$\text{circumference of semi circle} = \pi r$$

$$\text{perimeter of semi circle} = {\pi r + 2r}$$

$$\therefore \text{perimeter of semi circle} = r\left(\pi + 2\right)$$

$$\therefore \text{perimeter of semi circle} = \frac{d}{2}\left(\pi + 2\right)$$

$$\text{Area of a Sector OAB} = {\pi r^2} \times {\angle AOB \over \text{360 degree}}$$

$$\angle \text{AOB} = \frac{\text{area OAB}}{\text{Area of Circle}} \times \text{360 deg.}$$

$$\text{radius of circle} = \sqrt{\frac{\text{360 deg.}}{\angle AOB} \times \frac{\text{Area of OAB}}{\pi}}$$

$$\text{area of ring} = \pi \left(R^2-r^2\right)$$

$$\therefore \text{area of ring} = \pi \left(R+r\right)\left(R-r\right)$$

Cuboid and Cube

In the following formular: TSA = Total Surface Area; l = length; b = breadth; h = height; V = Volume;

$$V_{cuboid} = l \times b \times h$$

$$TSA_{cuboid} = 2 \left(lb + hb + hl\right)$$

$$\text{diagonal}_{cuboid} = \sqrt{l^2+b^2+h^2}$$

$$\text{length}_{cuboid} = \frac{V}{b \times h}$$

$$\text{breadth}_{cuboid} = \frac{V}{l \times h}$$

$$\text{height}_{cuboid} = \frac{V}{l \times b}$$

$$\text{volume}_{cube} = {side^3}$$

$$\text{diagonal}_{cube} = \sqrt{3} \times side$$

$$\text{length of side}_{cube} = \sqrt[3]{side}$$

$$\text{TSA}_{cube} = {6 \times side^2}$$

Right Circular Cylinder

Area of the Curved Surface Area (CSA) of a cylinder is equal to the product of its height (h) and the perimeter of its base.

$$\text{CSA} = 2 \pi r \times h$$

The Total Surface Area (TSA) of a right circular cylinder is equal to the sum of curved surface area and the area of its two circular ends.

$$\text{TSA} = CSA + \text{area of circular ends}$$

$$\therefore \text{TSA} = 2 \pi r h + 2 \pi r^2 = 2 \pi \left(r+h\right)$$

Volume of cylinder is equal to the product of area of base into height.

$$Volume = \pi r^2 \times h$$

Volume of Hollow Cylinder

Volume (V) of a Hollow Cylinder can be calculated using the following formula:

$$V = \pi R^2 h - \pi r^2 h = \pi h \left(R^2-r^2\right)$$ $$\text{or, V} = \pi h \left(R+r\right)\left(R-r\right)$$

Cone

In the cone below, in its right angled triangle OAC, we have: l = slant height; h = height; r = radius of base; TSA = Total Surface Area; CSA = Curved Surface Area; V= Volume (of cone).

$$l^2 = h^2 + r^2$$

$$\text{CSA} = {1 \over 2} \times \text{perimeter of base} \times l$$

$$\text{TSA} = {\text{area of circular base} + \text{CSA}}$$ $$\text{TSA} = {\pi r^2 + \pi r l} = {\pi r \left(r+l\right)}$$

$$V = {{1 \over 3} \times {\text{area of base}} \times {height}}$$

$$\therefore V = {{1 \over 3} \pi r^2 h}$$

More formula below .... 🔒LOCKED

Frustom of Cone

In the formulae below, R₁ (or, r) and R₂ (or, R) are the radius of top and bottom surface of frustum respectively; h and l are the vertical height and slant height respectively.

In the formula below: LSA = Lateral Surface Area; TSA = Total Surface Area.

$$\text{Volume} = \frac{1}{3}\pi h\left(R_1^2 + R_2^2 + R_1R_2\right)$$

$$\text{LSA} = {\pi l \left(R_1+R_2\right)}$$

$${l^2} = {h^2} + {\left(R_1-R_2\right)}^2$$

$$\text{TSA} = \pi[R_1^2+R_2^2+l\left(R_1+R_2\right)]$$

5. .... 🔒LOCKED

Sphere

V = Volume; A = area; h = height; b = base or breadth; d = diameter; R = outer radius; r = inner radius; a = side; CSA = Curved Surface Area; TSA = Total Surface Area

$$\pi = {22 \over 7} = 3.142$$

$$\text{surface area} = {4 \pi r^2}$$

$$\text{radius of sphere} = \sqrt{\text{surface area} \over {4 \pi}}$$

$$\text{diameter of sphere} = \sqrt{\text{surface} \over \pi}$$

$$\text{volume of sphere, V} = {{4 \over 3} \pi r^3} = {{4 \over 3} \pi {\left(d \over 2\right)}^3}$$

$$\therefore V = {{1 \over 6} \pi d^3}$$

$$\text{radius of sphere} = \sqrt{{3 \over 4 \pi} \times V}$$

$$\text{diameter} = \sqrt[3]{{6 \times V} \over \pi}$$

$$\text{volume of spherical ring} = {{4 \over 3} \pi \left(R^3-r^3\right)}$$

$$\text{CSA of hemisphere} = {2 \pi r^2}$$

$$\text{volume of hemisphere} = {{2 \over 3} \pi r^3}$$

$$\text{TSA of hemisphere} = {3 \pi r^2}$$