Mensuration | Quantitative Aptitude

Mensuration

Mensuration is a branch of mathematics in which we deal with the calculation of length, breath, height, area, volume and so on. Mensuration is mainly divided into 3 parts: Volume, Area and Perimeter. A section of mathematics that communicates the length, volume or area of various geometric shapes is termed Mensuration. These shapes or patterns exist in two dimensions or three dimensions.

What is Mensuration 2D?

Mensuration 2D is an important topic in Quantitative Aptitude, which mainly deals with problems related to the perimeter and area of two-dimensional shapes such as triangles, squares, rectangles, circles, parallelograms, etc.

Mensuration 2D Important Terminologies

Let us discover a few more definitions linked to the various geometric shapes.

Terms	Abbreviation	Unit	Definition
Area	A	᠎᠎᠎᠎᠎᠎᠎᠎m2 or cm2$m^2\text{ or }c{m}^2$	The area is the surface that is covered by the closed shape.
Perimeter	P	cmorm$cmorm$	The measure of the endless line by the boundary of the presented figure is called a Perimeter.
Volume	V	m3 or cm3$m^3\text{ or }c{m}^3$	The space utilized by a 3D shape or object is termed a Volume.
Curved Surface Area	CSA	᠎᠎᠎᠎᠎᠎m2 or cm2$᠎᠎᠎᠎᠎᠎m^2\text{ or }c{m}^2$	If there is a curved surface/shape/object, then the total area obtained is called a curved surface area.
Lateral Surface area	LSA	m2 or cm2$m^2\text{ or }c{m}^2$	The total area of all the lateral surfaces that encloses the presented figure is termed the lateral surface area.
Total Surface Area	TSA	m2 or cm2$m^2\text{ or }c{m}^2$	The aggregate of all the curved and lateral surface areas is designated as the total surface area.
Square Unit	–	m2 or cm2$m^2\text{ or }c{m}^2$	The area incorporated by a square of the length of one unit is termed a square unit.
Cube Unit	–	m3 or cm3$m^3\text{ or }c{m}^3$	The volume involved by a cube of one side one is termed a cube unit.

Properties of Triangle with Formulas

A triangle is a three sided closed plane figure. The sum of three interior angles of any triangle is always equal to 180 degree. Triangle can be categorized on the basis of sides as well as on the basis on angles.

Important Relations of Triangle

Perimeter = a + b + c

Area of Triangle = [½ × base × height] or = s(s–a)(s–b)(s–c)−−−−−−−−−−−−−−√$\sqrt{s(s–a)(s–b)(s–c)}$

Here, s is a semi perimeter

Properties of Equilateral Triangle with Formulas

• All three sides are equal.
• All three angles are 60°

Formulas of Equilateral Triangles

• Perimeter = 3 × side = 3a
• Area of equilateral triangle = 34−−√a2$\sqrt{\frac{3}{4}}{a}^2$
• Height = Median = 32−−√a$\sqrt{\frac{3}{2}}a$
• Side of an equilateral triangle = The sum of distances of a point inside a triangle from all the sides × 2 / 3–√$\sqrt{3}$

Properties of Isosceles with Formulas

• Two sides are equal.
• AB = AC

Formula:

• Area of Δ = ½ × base × height
• Perpendicular on unequal side = (equalside)2–(unequalside2)2−−−−−−−−−−−−−−−−−−−−−√$\sqrt{(equalside{)}^2–(\frac{unequalside}{2}{)}^2}$
• AD = (AC)2–(BC2)2−−−−−−−−−−−√$\sqrt{(AC{)}^2–(\frac{BC}{2}{)}^2}$

Properties of Right Angle Triangle with Formulas

• Hypotenuse = (base)2+(height)2$(base{)}^2+(height{)}^2$
• Area = ½ × base × height = ½ × b × p
• Length of Perpendicular at hypotenuse = height × base/hypotenuse

Properties of Square with Formulas

Square is a four-sided closed plane figure. The sum of 4 interior angles of a square is 360 degrees.

• Opposite sides of a square are both parallel and equal in length.
• The diagonals of a square are equal. AC = BD
• The diagonals of a square bisect each other and meet at 90°. AO = OC, BO = OD and ∠BOC = 90°
• All four angles of a square are 90°.

Formula 

• Perimeter = 4 × side
• Area = (side)2$(side{)}^2$ = ½ × (diameter)
• Diameter = 2–√$\sqrt{2}$ side

Properties of Rectangle with Formulas

Rectangle is a four sided closed plane figure. The sum of four interior angles of a rectangle is 360 degree.

• Opposite sides of a rectangle are equal in length. AB = DC, AD = BC
• Opposite sides of a rectangle are parallel. AB ∥ CD, AD ∥ BC
• The diagonals of a rectangle bisect each other. AO = OC, BO = OD
• All four angles of a square are 90°.

Formula

• Area = l × b
• Perimeter = 2(l + b)
• Diameter = l2+b2−−−−−√$\sqrt{l^2+b^2}$
• Here, Length = l, Breadth = b

Properties of Parallelogram with Formulas

• A quadrilateral having opposite sides is parallel and equal.
  AB = DC and AD = BC
  AB ∥ DC and AD ∥ BC
• Area of parallelogram ABCD = Base × Altitude on corresponding base = DC × h1 = BC × h2

Properties of Rhombus with Formulas

• A quadrilateral having all sides equal and their diagonals intersect at 90°.
  AB = BC = CD = DA

• Area of rhombus
  = ½ × product of the lengths of its diagonals
  = ½ × d1 × d2

Properties of Trapezium with Formulas

• A quadrilateral having a pair of sides is parallel and unequal.
  AB ∥ DC

• Area of trapezium ABCD
  = ½ × (sum of its parallel sides) × height
  = ½ × (AB + DC) × h

Properties of Circle with Formulas

Circle is a round, two-dimensional shape. All points on the edge of the circle are at the same distance from the center.

• O is a center of this circle and OA is radius.
• Diameter = 2 × radius
• Area = πr2$\pi r^2$
• Perimeter = 2πr = πd

Properties of Semi Circle with Formulas

• Area of semi-circle = ½ πr2$\pi r^2$
• Perimeter of semi-circle = π r + 2r

Properties of Incircle

It is the circle inside the polygon whose all the sides are tangent to the circle.

• Radius of incircle = 2 × area of triangle / Perimeter of triangle
• Radius of incircle of equilateral triangle = side23√$\frac{side}{2\sqrt{3}}$

Properties of Circumcircle

Circle whose circumference touches all the vertices of the polygon is called a circumcircle.

• Circumradius of a triangle = Product of sides/(4 × area of triangle)
• Circumradius of a equilateral triangle = a3√

Formulas for Mensuration 2D Shapes

The important formulas for mensuration 2D shapes are given below:

Shape	Area (Square units)	Perimeter (units)
Square	a2$a^2$	4a$4a$
Rectangle	l×b$l\times b$	2(l+b)$2\left(l+b\right)$
Circle	πr2$\pi r^2$	2πr$2\pi r$
Scalene Triangle	[s(s−a)(s−b)(s−c)]−−−−−−−−−−−−−−−−−−−√$\sqrt{\left[s\left(s-a\right)\left(s-b\right)\left(s-c\right)\right]}$ ,Where,s=(a+b+c)2$s=\frac{\left(a+b+c\right)}{2}$	a+b+c$a+b+c$
Isosceles Triangle	12×b×h$\frac{1}{2}\times b\times h$	2a+b$2a+b$
Equilateral triangle	34−−√×a2$\sqrt{\frac{3}{4}}\times a^2$	3a$3a$
Right Angle Triangle	12×b×h$\frac{1}{2}\times b\times h$	base+hypotenuse+height$base+hypotenuse+height$
Rhombus	12×d1×d2$\frac{1}{2}\times d_1\times d_2$	4×side$4\times side$
Parallelogram	b×h$b\times h$	2(l+b)$2\left(l+b\right)$
Trapezium	12h(a+c)$\frac{1}{2}h\left(a+c\right)$ Here a+c=sum of the parallel sides of a Trapezium.	a+b+c+d$a+b+c+d$ Here a, b,c and d denote all four sides of a Trapezium.

Tips for Solving Mensuration 2D Questions

Students can find different tips and tricks for solving the questions related to Mensuration 2D from below:

• Tip 1: Make sure you remember all the properties and formulas mentioned above to solve the questions related to this section quickly.
• Tip 2: Attempt mock and quizzes to brush up on your concepts of Mensuration.

Difference Between Mensuration 2D and Mensuration 3D Shapes

The difference between mensuration 2D and mensuration 3D shapes are given below:

Mensuration 2D Shape	Mensuration 3D Shape
If a shape or pattern is enclosed by three or added straight lines in a plane, then such a shape is a 2D shape.	If a shape or pattern is enclosed by several surfaces or planes then it is a 3D shape.
Two-dimensional shapes hold no depth/height.	Three-dimensional shapes are also termed solid shapes and when compared to 2D they possess height or depth.
2D shapes, as the name suggests, have only two dimensions: length and breadth.	The three-dimensional shapes as compared to 2D have three dimensions namely depth/height, breadth and length.
In two dimensions, we can measure the area and perimeter of shapes.	In three dimensions, we can measure the volume, CSA(Curved Surface area), LSA( Lateral Surface area.) or TSA(Total Surface area.) of the given shapes.

Solved Examples of Mensuration 2D

Example 1: The area of an equilateral triangle is 493–√m2$49\sqrt{3}{m}^2$, find the value of its height and side.

Solution: The area of an equilateral triangle = 493–√m2$49\sqrt{3}{m}^2$ =34−−√a2$\sqrt{\frac{3}{4}}{a}^2$ = 493–√$49\sqrt{3}$
a2$a^2$= 49 × 4

⇒a = 14 m

⇒Height = 32−−√a$\sqrt{\frac{3}{2}}a$

⇒Height = 3√2$\frac{\sqrt{3}}{2}$ × 14 = 73–√m$7\sqrt{3}m$

Example 2: In an isosceles triangle, if the ratio of equal side and unequal side is 5 : 8 and its perimeter is 36 cm. Find the area of this triangle.

Solution: Let the equal and unequal side be 5x and 8x Perimeter of this triangle = 5x + 5x + 8x

⇒36 = 18x

⇒x = 2

Sides of this triangle are 10 cm, 10 cm and 16 cm Perpendicular on unequal side = 102–(162)2−−−−−−−−√$\sqrt{102–(\frac{16}{2})2}$
= 100−64−−−−−−−√$\sqrt{100-64}$
∴ Perpendicular on unequal side = 6 cm
∴ Area of isosceles triangle = ½ × unequal side × perpendicular on unequal side
∴ Area of isosceles triangle = ½ × 16 × 6 = 48 cm2$c{m}^2$

Example 3: Area of a square is 1352 m2$m^2$. Find the perimeter and the length of the diagonal.

Solution: Area of a square = 1352

(Side)2$(Side{)}^2$ = 1352

⇒Side = 262–√m$26\sqrt{2}m$

⇒Perimeter = 4 × side

⇒Perimeter = 4×262–√=1042–√m$4\times 26\sqrt{2}=104\sqrt{2}m$

⇒Length of diagonal = 2–√×side$\sqrt{2}\times side$

⇒Length of diagonal = 2–√×262–√=52m$\sqrt{2}\times 26\sqrt{2}=52m$

Example 4: In a rectangle diagonal is 9 times of its length. Find the ratio of length and breath.

Solution: Let the length of rectangle be x Diagonal = 9x

⇒Diameter =l2+b2−−−−−√$\sqrt{l^2+b^2}$

9x = x2+b2−−−−−−√$\sqrt{x^2+b^2}$

x2+b2−−−−−−√$\sqrt{x^2+b^2}$ = 81x280x2=b2$81x^{2}80x^2=b^2$

b = (4√5)x

⇒Length : breadth = x : 45–√$4\sqrt{5}$ x

⇒Length : breadth = 1 : 45–√$4\sqrt{5}$

Example 5: Find the area of parallelogram PQOS given in the figure:

Solution: 

⇒ ∠PSR = ∠QOR = 60

⇒ QR/OR = tan (60)

⇒OR = QR/tan(60)

⇒ OR = 103√3√$\frac{10\sqrt{3}}{\sqrt{3}}$ = 10 cm

⇒ OS = SR – OR = 20 – 10 = 10 cm

⇒ Area of parallelogram = Base × Altitude = 10∗103–√=10×17.32=173.2cm2$10\ast 10\sqrt{3}=10\times 17.32=173.2c{m}^2$

Example 6: The side of the rhombus is 20 cm and one of the diagonals of rhombus is 32 cm. Find the area of the rhombus.

Solution: Using Pythagoras theorem,

⇒ (side)2=(1/2×onediagonal)2+(1/2×otherdiagonal)2$(side{)}^2=(1/2\times onediagonal{)}^2+(1/2\times otherdiagonal{)}^2$

⇒ (1/2×otherdiagonal)2=202–(32/2)2$(1/2\times otherdiagonal{)}^2=202–(32/2{)}^2$

⇒ Otherdiagonal=24cm$Otherdiagonal=24cm$

Area of rhombus = 1/2 × product of diagonals

= 1/2 × 24 × 32

= 384 sq.cm

Example 7: If the area of a trapezium is 250 sq.m and the lengths of the two parallel sides are 15m and 10 m respectively then distance between the parallel sides of the trapezium is

Solution: Given,

Area of trapezium = ½ × sum of parallel sides × distance between them Given,

⇒ Area of trapezium = 250 sq.m

⇒ Sum of parallel sides = 15 + 10 = 25 m Then,

⇒ 250 = 1/2 × 25 × distance between them

⇒ Distance between them = 20 m

Example 8: The arc of a circle is 27.5 cm and the angle made by the arc at the center of the circle is 75°, then find the circumference of the circle.

Solution: The angle made by arc at the center of the circle = 75° = 75 × π/180 = 5π/12 The length of the arc of the circle = 27.5

Now, Angle = Arc/Radius

Radius = Arc/Angle = 27.5/(5π/12)

Radius = (27.5 × 12 × 7)/(5 × 22)

Radius = 21 cm

So, the circumference of the circle = 2 × (22/7) × 21 = 132 cm

Example 9: The area of a circle is 3850 cm2$c{m}^2$. What is its circumference (in cm).

Solution: Area of circle with radius ‘r’ = πr2$\pi r^2$

⇒ 22/7×r2=3850$22/7\times r^2=3850$

⇒ r2=1225$r^2=1225$

⇒ r=1225−−−−√=35cm$r=\sqrt{1225}=35cm$

⇒ Circumference of circle = 2πr = 2 × 22/7 × 35 = 220 cm

Example 10: What will be the area of a semi-circle of perimeter 72 metres?

Solution: As we know, perimeter of semicircle = πr + 2r = (π + 2)r Where r = radius of semicircle

(π + 2)r = 72 36/7 × r = 72

⇒ r = 14 m

⇒ Area of semi-circle = ½ × (22/7) × 14 × 14

⇒ Area of semi-circle = 308 m2$m^2$

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