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)
- $$\text{Perimeter of a square} = {4 \times side}$$
- $$\text{Area of a square} = {side^2}$$
- $$\text{Side of a square} = {\sqrt{area}}$$
- $$\text{Diagonal of a square} = {\sqrt{2} \times side}$$
- $$\text{side of a square} = {\text{diagonal} \over {\sqrt{2}}}$$
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Triangle
- Types of triangle:
- 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
- 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.
- 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.
- 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).
$$\text{area} = {1 \over 2} \times {\left(a+b\right)} \times {height}$$
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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;
$$V_{cuboid} = l \times b \times h$$
$$TSA_{cuboid} = 2 \left(lb + hb + hl\right)$$
$$\text{diagonal}_{cuboid} = \sqrt{l^2+b^2+h^2}$$
$$\text{length}_{cuboid} = \frac{V}{b \times h}$$
$$\text{breadth}_{cuboid} = \frac{V}{l \times h}$$
$$\text{height}_{cuboid} = \frac{V}{l \times b}$$
$$\text{volume}_{cube} = {side^3}$$
$$\text{diagonal}_{cube} = \sqrt{3} \times side$$
$$\text{length of side}_{cube} = \sqrt[3]{side}$$
$$\text{TSA}_{cube} = {6 \times side^2}$$
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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:
$$V = \pi R^2 h - \pi r^2 h = \pi h \left(R^2-r^2\right)$$ $$\text{or, V} = \pi h \left(R+r\right)\left(R-r\right)$$
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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).
$$l^2 = h^2 + r^2$$
$$\text{CSA} = {1 \over 2} \times \text{perimeter of base} \times l$$
$$\text{TSA} = {\text{area of circular base} + \text{CSA}}$$ $$\text{TSA} = {\pi r^2 + \pi r l} = {\pi r \left(r+l\right)}$$
$$V = {{1 \over 3} \times {\text{area of base}} \times {height}}$$
$$\therefore V = {{1 \over 3} \pi r^2 h}$$
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Frustom of Cone
In the formulae below, R_{1} (or, r) and R_{2} (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.
$$\text{Volume} = \frac{1}{3}\pi h\left(R_1^2 + R_2^2 + R_1R_2\right)$$
$$\text{LSA} = {\pi l \left(R_1+R_2\right)}$$
$${l^2} = {h^2} + {\left(R_1-R_2\right)}^2$$
$$\text{TSA} = \pi[R_1^2+R_2^2+l\left(R_1+R_2\right)]$$
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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
$$\pi = {22 \over 7} = 3.142$$
$$\text{surface area} = {4 \pi r^2}$$
$$\text{radius of sphere} = \sqrt{\text{surface area} \over {4 \pi}}$$
$$\text{diameter of sphere} = \sqrt{\text{surface} \over \pi}$$
$$\text{volume of sphere, V} = {{4 \over 3} \pi r^3} = {{4 \over 3} \pi {\left(d \over 2\right)}^3}$$
$$\therefore V = {{1 \over 6} \pi d^3}$$
$$\text{radius of sphere} = \sqrt{{3 \over 4 \pi} \times V}$$
$$\text{diameter} = \sqrt[3]{{6 \times V} \over \pi}$$
$$\text{volume of spherical ring} = {{4 \over 3} \pi \left(R^3-r^3\right)}$$
$$\text{CSA of hemisphere} = {2 \pi r^2}$$
$$\text{volume of hemisphere} = {{2 \over 3} \pi r^3}$$
$$\text{TSA of hemisphere} = {3 \pi r^2}$$
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