Mensuration
*********** is an important ******* that is asked in *** Quantitative Aptitude ******* if you are preparing *** exams like SSC CGL, SBI PO, SBI Clerk, IBPS SO, IBPS PO, RBI ***** B and **** more.
** is a ********** easy topic ** the questions are based on formulas. Since, there ** a time constraint in almost all government exams, *** questions will be quite direct *** will not require long-drawn calculations.Therefore, it is essential ** prepare this topic well so you can score very well ** the actual exam.
What ** Mensuration?
Mensuration was discovered ** the Greek mathematician Archimedes. It is considered as a part of geometry dealing **** ascertaining lengths, areas and volumes of various geometrical ******* like cone, cylinder, cube, cuboid, sphere, square, rectangle, circle etc. ** is applicable to both 2D *** 3D geometrical shapes.
Formulas for 2D Shapes
2D ******* have *** ********** i.e. height and breadth but no thickness (volume).
For example- square, rectangle,triangle, circle, etc. What this means ** that ***** objects do *** exist ** the real world. They can be represented by using plain surfaces.
Let us look at some 2D Shapes-
Square
A ****** has four equal sides and all *** four angles are ***** to 90 degrees. ******** lines/measurements in a square are equal to *** length.
Square |
Formulas |
Area of Square |
A= a2 (a=side) |
********* ** Square |
P= 4a |
Let’s **** at ** example ** understand this-
Q) Calculate the area of a square. The side length of the square is 20 m.
Answer:
The **** length, a = 20 m.
Hence, the area of *** ****** = a2
= (20)2 m2
= 4** m2
Rectangle
A rectangle *** four sides and four corners. It is a quadrilateral and has four right angles which are equal to 90 degrees. The opposite sides of the rectangle are parallel and equal to each other. Diagonals have the same length.
Rectangle |
Formulas |
Area of Rectangle |
Length x Breadth (a x b) |
Perimeter of Rectangle |
2 (Length + Breadth) (a + b) |
Let’s understand **** the help of ** example-
Q) ********* the perimeter ** a rectangle. The length is 5 m, and the width is 3 m.
Answer:
The length, L = 5 m.
The width, B = 3 m.
Therefore, the perimeter of the ********* is
2 (L+B)
= 2 (5+3) m
= 16 m.
Parallelogram
In a parallelogram, there are four sides and the opposite sides are parallel to each other. The ******** sides are ***** in length.
Parallelogram |
Formulas |
Area of Parallelogram |
b x h (******* x height) |
********* of Parallelogram |
2 (a + b) |
Circle
It is a closed 2D figure and it is a set of points in a plane **** are equidistant from the centre. The distance between the centre and any point on the circumference is called the radius.
Circle |
Formulas |
Diameter ** Circle |
2 x radius |
Circumference of a Circle |
3.14 x Diameter or 2 x 3.14 x radius |
Area of Circle |
3.14 x radius2 |
Let’s look at an example-
Q) A circle has a radius of 21 cm. Find its circumference and area. (*** π = 22/7)
Answer: ** know,
Circumference of ****** = 2πr = 2 x (22/7) x 21 = 2 x 22 x 3 = 132 cm
Area of circle = πr2 = (22/7) x 212 = 22/7 x 21 x 21 = 22 x 3 x 21
Area of ****** **** radius, 21cm = 1386 cm2
Triangle
A triangle *** three sides *** three ********* angles. All the ***** angles always *** up ** 180 degrees.
Triangle |
Formula |
Area of a Triangle |
½ x b x h |
Perimeter of a Triangle |
a + b + c |
Semi- Perimeter of a Triangle |
a + b + c / 2 |
Let us look at an example-
Q) The perimeter ** a triangular garden is given ** 26 feet. If two of its sides measure 7 feet and 11 feet respectively, what is the measure of the third side?
Answer:
We **** that the perimeter of a triangle ** the sum of *** three sides.
⇒ 26 = 7 + 11 + ******* side
Therefore, *** unknown side is given by:
=26−(7+11) = 8 feet
∴ The ***** **** of the given triangle measures 8 feet.
Properties ** Triangle
Triangle ********** property provides that the *** of the ****** of two ***** of a triangle is greater than *** third side.
According to the Pythagorean theorem, in a right angled triangle, the square of the hypotenuse is equal to the sum ** the square of *** other two sides. (Hypotenuse2= Base2 + Height2)
The **** opposite ** the greater ***** is the longest side.
Exterior angle of a triangle is always equal to *** sum of the interior opposite angles.
Types of Triangles
Equilateral Triangle
A triangle with three equal sides is called an equilateral triangle. They have three ***** angles which measure 60 degrees each and ∴ add up to 180 degrees.
Isosceles Triangle
Isosceles triangle ** *** where the length of two sides ** equal and the measure of the ************* opposite angles is also equal.
Scalene Triangle
A triangle where all *** three sides have different lengths is called a Scalene triangle.
All angles of a scalene triangle are also unequal. The ***** opposite to the longest side will ** the greatest angle.
For example, a triangle with side lengths ** 7 cm, 12 cm, and 15 ** would be a scalene triangle.
Right Angled Triangle
It refers to a triangle where one angle is equal ** 90 ******* (right angle). There are different names *** all the ***** ***** of a right angled triangle.
The longest side or the side opposite ** the 90 degree angle is called the ‘hypotenuse’. Conversely, the hypotenuse is always opposite the longest side of the triangle.
The base
The perpendicular (height)
Formulas of 3D Shapes
Unlike 2D shapes, 3D shapes have ****** (depth), three dimensional length and breadth. The 3 measures of 3D shapes are:
Volume, Total Surface **** (TSA)
Lateral ******* Area (LSA)
Curved Surface Area (CSA)
Let’s look ** some 3D shapes-
Cube
A **** has 6 square faces, 8 vertices and 12 edges. The three edges meet at one ****** point.
Cube |
Formulas |
Volume of Cube |
Side3 cubic units |
Lateral Surface **** of Cube |
4 x side2 square units |
***** Surface Area of Cube |
6 x side2 square units |
Let’s understand with the help of an example-
Q) If a cube has its side-****** equal to 5cm, **** its area is?
Answer: Given,
**** = 5 cm
Area = 6 x side2 = 6 x 5 x5 = 150 sq.cm
Cuboid
A cuboid also has three sides. However, unlike a cube, all *** sides ** a cuboid *** unequal. *** of its faces are ********** ****** a total of 6 faces, 8 vertices and 12 edges.
Cuboid |
Formulas |
Volume ** Cuboid |
(****** + breadth + height) cubic units |
Lateral Surface Area of Cuboid |
2 x height (length + breadth) square units |
Total Surface Area ** Cuboid |
2(lb + bh + lh) square units |
Diagonal Length of a Cuboid |
length2 + breadth2 + height2 |
Let’s look at an example-
Q) The height, length and width of a cuboidal box are 20 cm, 15 cm and 10 cm, respectively. Find its total surface area.
Answer: Total ******* area = 2 (20 × 15 + 20 × 10 + 10 × 15)
TSA = 2 ( 300 + 200 + 150) = 13** cm2
Q) **** the ****** of a cuboid whose volume is 275 cm3 and base area is 25 cm2.
Answer: Volume of cuboid = l × b × h
Base area = l × b = 25 cm2
Hence,
275 = 25 × h
h = 275/25 = 11 cm
Cone
A cone is formed by using a set of lines which connect to a common point called apex/vertex. The base of *** cone is circular, from which the radius can be calculated. The length of the cone from vertex to any point on the circumference of the base is the slant height.
Cone |
Formulas |
****** of Cone |
⅓ x 3.14 x radius2 x h cubic units |
****** Surface Area of Cone |
3.14 x radius x height square units |
***** Surface Area ** Cone |
3.14 x r (length + height) square units |
Let’s look at an example-
Q) Find the volume of the **** of radius, 5 cm, and height, 10 cm.
Answer:
By the volume ** a cone formula, we have,
⇒V = ⅓ πr2h
⇒V = ⅓ x 3.14 x 5 x 5 x 10
= 262 cm3
Cylinder
A cylinder is composed of two ********* circles ** parallel ****** and all the **** segments are parallel to *** segment containing the centres of both circles with ********* on the circular regions. *** circles and *** interiors are the bases. ****** of the cylinder is the ****** of the base.
Cylinder |
Formulas |
Volume of Cylinder |
3.14 x radius2 x height cubic units |
****** Surface **** of Cylinder |
2 x 3.14 x radius x height square units (excluding the areas of the top and bottom ******** regions) |
Total Surface **** ** Cylinder |
(2 x 3.14 x ****** + 2 x 3.14 x radius2) square units |
Let’s look ** an example-
Q) A rectangular piece of paper 11 cm × 4 cm is folded without overlapping to make a ******** of height 4 cm. Find the volume of the cylinder.
Answer: Length of *** paper will be the perimeter of the base of the cylinder and width will be its height.
Circumference of **** of cylinder = 2πr = 11 cm
2 x 22/7 x r = 11 cm
r = 7/4 cm
Volume ** cylinder = πr2h = (22/7) x (7/4)2 x 4
= 38.5 cm3
Sphere
A ****** is an absolutely round shape. It is a set of all points in an *********** space from a given point called the center of the sphere. The ******** between any ***** of the sphere *** its center is the radius. A hemisphere is ..
Sphere |
Formulas |
Volume of Sphere |
4/3 x 3.14 x radius3 cubic units |
Surface **** of Sphere |
4 x 3.14 x radius2 square units |
Hemisphere |
Formulas |
Volume ** Hemisphere |
⅔ x 3.14 x radius3 cubic units |
Surface Area of Hemisphere |
3 x 3.14 x radius2 square units |
Let’s understand with the help of an example-
Q) **** the volume of a sphere whose radius is 5 cm.
Answer:
By, *** ****** ** a sphere formula, ** have
V = 4/3 πr3
= (4/3) x 3.14 x 53
= (4/3) x 3.14 x 5 x 5 x 5
= 523.3 cm3
Q) The volume of a ******* baseball is 230 cm3. Find *** radius of the ball.
Answer:
Volume of a sphere = 4/3 πr3
230 = 4/3 x 3.14 x r3
230 = 4.19r3
r3 = 54.9
r = 3√54.9
r = 3.8
Thus, the radius of the baseball is 3.8 cm