Mensuration (revised)

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MENSURATION

Mensuration is the branch of mathematics which deals with the study of different geometrical shapes, their areas and Volume. In the broadest sense, it is all about the process of measurement. It is based on the use of algebraic equations and geometric calculations to provide measurement data regarding the width, depth and volume of a given object or group of objects

1. Pythagorean Theorem (Pythagoras’ theorem)

In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides

c² = a² + b² where c is the length of the hypotenuse and a and b are the lengths of the other two sides

2. Pi is a mathematical constant which is the ratio of a circle’s circumference to its diameter. It is denoted by π

π≈3.14≈227

3. Geometric Shapes and solids and Important Formulas

4. Important properties of Geometric Shapes
   1. Properties of Triangle
      1. Sum of the angles of a triangle = 180°
      2. Sum of any two sides of a triangle is greater than the third side.
      3. The line joining the midpoint of a side of a triangle to the positive vertex is called the Median
      4. The median of a triangle divides the triangle into two triangles with equal areas
      5. Centroid is the point where the three medians of a triangle meet.
      6. Centroid divides each median into segments with a 2:1 ratio
      7. Area of a triangle formed by joining the midpoints of the sides of a given triangle is one-fourth of the area of the given triangle.
      8. An equilateral triangle is a triangle in which all three sides are equal
      9. In an equilateral triangle, all three internal angles are congruent to each other
      10. In an equilateral triangle, all three internal angles are each 60°
      11. An isosceles triangle is a triangle with (at least) two equal sides
      12. In isosceles triangle, altitude from vertex bisects the base.

1. Properties of Quadrilaterals

A. Rectangle

1. The diagonals of a rectangle are equal and bisect each other
2. opposite sides of a rectangle are parallel
3. opposite sides of a rectangle are congruent
4. opposite angles of a rectangle are congruent
5. All four angles of a rectangle are right angles
6. The diagonals of a rectangle are congruent

B. Square

1. All four sides of a square are congruent
2. Opposite sides of a square are parallel
3. The diagonals of a square are equal
4. The diagonals of a square bisect each other at right angles
5. All angles of a square are 90 degrees.
6. A square is a special kind of rectangle where all the sides have equal length

C. Parallelogram

1. The opposite sides of a parallelogram are equal in length.
2. The opposite angles of a parallelogram are congruent (equal measure).
3. The diagonals of a parallelogram bisect each other.
4. Each diagonal of a parallelogram divides it into two triangles of the same area

D. Rhombus

1. All the sides of a rhombus are congruent
2. Opposite sides of a rhombus are parallel.
3. The diagonals of a rhombus bisect each other at right angles
4. Opposite internal angles of a rhombus are congruent (equal in size)
5. Any two consecutive internal angles of a rhombus are supplementary; i.e. the sum of their angles = 180° (equal in size)
6. If each angle of a rhombus is 90°, it is a square

Other properties of quadrilaterals

1. The sum of the interior angles of a quadrilateral is 360 degrees
2. If a square and a rhombus lie on the same base, area of the square will be greater than area of the rhombus (In the special case when each angle of the rhombus is 90°, rhombus is also a square and therefore areas will be equal)
3. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.
4. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.
5. Each diagonal of a parallelogram divides it into two triangles of the same area
6. A square is a rhombus and a rectangle.

1. The sum of the interior angles of a polygon = 180(n – 2) degrees where n = number of sides Example 1 : Number of sides of a triangle = 3. Hence, sum of the interior angles of a triangle = 180(3 – 2) = 180 × 1 = 180 ° Example 2 : Number of sides of a quadrilateral = 4. Hence, sum of the interior angles of any quadrilateral = 180(4 – 2) = 180 × 2 = 360.

Solved Examples

Level 1

1. An error 2% in excess is made while measuring the side of a square. What is the Percentage of error in the calculated area of the square?

A. 4.04 %

B. 2.02 %

C. 4 %

D. 2 %

Answer : Option A

Explanation :

Error = 2% while measuring the side of a square.

Let the correct value of the side of the square = 100
Then the measured value = (100×(100+2))/100=102 (∵ error 2% in excess)

Correct Value of the area of the square = 100 × 100 = 10000
Calculated Value of the area of the square = 102 × 102 = 10404

Error = 10404 – 10000 = 404
Percentage Error = (Error/Actual Value)×100=(404/10000)×100=4.04%

2. A towel, when bleached, lost 20% of its length and 10% of its breadth. What is the percentage of decrease in area?

A. 30 %

B. 28 %

C. 32 %

D. 26 %

Answer : Option B

Explanation :

Let original length = 100 and original breadth = 100
Then original area = 100 × 100 = 10000

Lost 20% of length
=> New length =( Original length × (100−20))/100
=(100×80)/100=80

Lost 10% of breadth
=> New breadth= (Original breadth × (100−10))/100
=(100×90)/100=90

New area = 80 × 90 = 7200

Decrease in area
= Original Area – New Area
= 10000 – 7200 = 2800

Percentage of decrease in area
=(Decrease in Area/Original Area)×100=(2800/10000)×100=28%

4. If the length of a rectangle is halved and its breadth is tripled, what is the percentage change in its area?

A. 25 % Increase

B. 25 % Decrease

C. 50 % Decrease

D. 50 % Increase

Answer : Option D

Explanation :

Let original length = 100 and original breadth = 100
Then original area = 100 × 100 = 10000

Length of the rectangle is halved
=> New length = (Original length)/2=100/2=50

breadth is tripled
=> New breadth= Original breadth × 3 = 100 × 3 = 300

New area = 50 × 300 = 15000

Increase in area = New Area – Original Area = 15000 – 10000= 5000
Percentage of Increase in area =( Increase in Area/OriginalArea)×100=(5000/10000)×100=50%

5. The area of a rectangle plot is 460 square metres. If the length is 15% more than the breadth, what is the breadth of the plot?

A. 14 metres

B. 20 metres

C. 18 metres

D. 12 metres

Answer : Option B

Explanation :

lb = 460 m² ——(Equation 1)

Let the breadth = b
Then length, l =( b×(100+15))/100=115b/100——(Equation 2)

From Equation 1 and Equation 2,
115b/100×b=460b2=46000/115=400⇒b=√400=20 m

6. If a square and a rhombus stand on the same base, then what is the ratio of the areas of the square and the rhombus?

A. equal to ½

B. equal to ¾

C. greater than 1

D. equal to 1

Answer : Option C

Explanation :

If a square and a rhombus lie on the same base, area of the square will be greater than area of the rhombus (In the special case when each angle of the rhombus is 90°, rhombus is also a square and therefore areas will be equal)

Hence greater than 1 is the more suitable choice from the given list

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Note : Proof

Consider a square and rhombus standing on the same base ‘a’. All the sides of a square are of equal length. Similarly all the sides of a rhombus are also of equal length. Since both the square and rhombus stands on the same base ‘a’,

Length of each side of the square = a
Length of each side of the rhombus = a

Area of the sqaure = a² …(1)

From the diagram, sin θ = h/a
=> h = a sin θ

Area of the rhombus = ah = a × a sin θ = a² sin θ …(2)

From (1) and (2)

Area of the square/Area of the rhombus= a² /a²sinθ=1/sinθ

Since 0° < θ < 90°, 0 < sin θ < 1. Therefore, area of the square is greater than that of rhombus, provided both stands on same base.

(Note that, when each angle of the rhombus is 90°, rhombus is also a square (can be considered as special case) and in that case, areas will be equal.

7. The breadth of a rectangular field is 60% of its length. If the perimeter of the field is 800 m, find out the area of the field.

A. 37500 m²

B. 30500 m²

C. 32500 m²

D. 40000 m²

Answer : Option A

Explanation :

Given that breadth of a rectangular field is 60% of its length
⇒b=(60/100)* l =(3/5)* l

perimeter of the field = 800 m
=> 2 (l + b) = 800
⇒2(l+(3/5)* l)=800⇒l+(3/5)* l =400⇒(8/5)* l =400⇒l/5=50⇒l=5×50=250 m

b = (3/5)* l =(3×250)/5=3×50=150 m

Area = lb = 250×150=37500 m2

8. What is the percentage increase in the area of a rectangle, if each of its sides is increased by 20%?

A. 45%

B. 44%

C. 40%

D. 42%

Answer : Option B

Explanation :

Let original length = 100 and original breadth = 100
Then original area = 100 × 100 = 10000

Increase in 20% of length.
=> New length = (Original length ×(100+20))/100=(100×120)/100=120

Increase in 20% of breadth
=> New breadth= (Original breadth × (100+20))/100=(100×120)/100=120

New area = 120 × 120 = 14400

Increase in area = New Area – Original Area = 14400 – 10000 = 4400
Percentage increase in area =( Increase in Area /Original Area)×100=(4400/10000)×100=44%

9. What is the least number of squares tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad?

A. 814

B. 802

C. 836

D. 900

Answer : Option A

Explanation :

l = 15 m 17 cm = 1517 cm
b = 9 m 2 cm = 902 cm
Area = 1517 × 902 cm²

Now we need to find out HCF(Highest Common Factor) of 1517 and 902.
Let’s find out the HCF using long division method for quicker results

902)  1517  (1

     -902

—————–

      615)  902  (1

•      615

————–

            287)  615 (2

                 -574

            —————–

             41)  287  (7

                 -287

            ————

                 0                    
           ————

Hence, HCF of 1517 and 902 = 41

Hence, side length of largest square tile we can take = 41 cm
Area of each square tile = 41 × 41 cm²

Number of tiles required = (1517×902)/(41×41)=37×22=407×2=814

Level 2

1. A rectangular parking space is marked out by painting three of its sides. If the length of the unpainted side is 9 feet, and the sum of the lengths of the painted sides is 37 feet, find out the area of the parking space in square feet?

A. 126 sq. ft.

B. 64 sq. ft.

C. 100 sq. ft.

D. 102 sq. ft.

Answer : Option A

Explanation :

Let l = 9 ft.

Then l + 2b = 37
=> 2b = 37 – l = 37 – 9 = 28
=> b = 282 = 14 ft.

Area = lb = 9 × 14 = 126 sq. ft.

2. A large field of 700 hectares is divided into two parts. The difference of the areas of the two parts is one-fifth of the Average of the two areas. What is the area of the smaller part in hectares?

A. 400

B. 365

C. 385

D. 315

Answer : Option D

Explanation :

Let the areas of the parts be x hectares and (700 – x) hectares.

Difference of the areas of the two parts = x – (700 – x) = 2x – 700

one-fifth of the average of the two areas = 15[x+(700−x)]2
=15×7002=3505=70

Given that difference of the areas of the two parts = one-fifth of the average of the two areas
=> 2x – 700 = 70
=> 2x = 770
⇒x=7702=385

Hence, area of smaller part = (700 – x) = (700 – 385) = 315 hectares.

3. The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 sq.cm. What is the length of the rectangle?

A. 18 cm

B. 16 cm

C. 40 cm

D. 20 cm

Answer : Option C

Explanation :

Let breadth = x cm
Then length = 2x cm
Area = lb = x × 2x = 2x²

New length = (2x – 5)
New breadth = (x + 5)
New Area = lb = (2x – 5)(x + 5)

But given that new area = initial area + 75 sq.cm.
=> (2x – 5)(x + 5) = 2x² + 75
=> 2x² + 10x – 5x – 25 = 2x² + 75
=> 5x – 25 = 75
=> 5x = 75 + 25 = 100
=> x = 1005 = 20 cm

Length = 2x = 2 × 20 = 40cm

4. The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/hr completes one round in 8 minutes, then what is the area of the park (in sq. m)?

A. 142000

B. 112800

C. 142500

D. 153600

Answer : Option D

Explanation :

l : b = 3 : 2 —-(Equation 1)

Perimeter of the rectangular park
= Distance travelled by the man at the speed of 12 km/hr in 8 minutes
= speed × time = 12×860     (∵ 8 minute = 860 hour)
= 85 km = 85 × 1000 m = 1600 m

Perimeter = 2(l + b)

=> 2(l + b) = 1600
=> l + b = 16002 = 800 m —-(Equation 2)

From (Equation 1) and (Equation 2)
l = 800 × 35 = 480 m
b = 800 × 25 = 320 m (Or b = 800 – 480 = 320m)

Area = lb = 480 × 320 = 153600 m²

5. It is decided to construct a 2 metre broad pathway around a rectangular plot on the inside. If the area of the plots is 96 sq.m. and the rate of construction is Rs. 50 per square metre., what will be the total cost of the construction?

A. Rs.3500

B. Rs. 4200

C. Insufficient Data

D. Rs. 4400

Answer : Option C

Explanation :

Let length and width of the rectangular plot be l and b respectively
Total area of the rectangular plot = 96 sq.m.
=> lb = 96

Width of the pathway = 2 m
Length of the remaining area in the plot = (l – 4)
breadth of the remaining area in the plot = (b – 4)
Area of the remaining area in the plot = (l – 4)(b – 4)

Area of the pathway
= Total area of the rectangular plot – remaining area in the plot
= 96 – [(l – 4)(b – 4)]
= 96 – [lb – 4l – 4b + 16]
= 96 – [96 – 4l – 4b + 16]
= 96 – 96 + 4l + 4b – 16
= 4l + 4b – 16
= 4(l + b) – 16

We do not know the values of l and b and hence area of the pathway cannot be found out. So we cannot determine total cost of the construction.

6. A circle is inscribed in an equilateral triangle of side 24 cm, touching its sides. What is the area of the remaining portion of the triangle?

A. 144√3−48π cm²

B. 121√3−36π cm²

C. 144√3−36π cm²

D. 121√3−48π cm²

Answer : Option A

Explanation :

Area of an equilateral triangle = (3/√4)*a *a where a is length of one side of the equilateral triangle
Area of the equilateral Δ ABC = (3/√4)*a *a = (3/√4)*24*24=144√3 cm2⋯ (1)

Area of a triangle = 12bhwhere b is the base and h is the height of the triangle
Let r = radius of the inscribed circle. Then
Area of Δ ABC
= Area of Δ OBC + Area of Δ OCA + area of Δ OAB
= (½ × r × BC) + (½ × r × CA) + (½ × r × AB)
= ½ × r × (BC + CA + AB)
= ½ x r x (24 + 24 + 24)
= ½ x r x 72 = 36r cm² —-(2)

From (1) and (2),
144√3=36r⇒r=144√3/36=4√3−−−−(3)

Area of a circle = πr2 where = radius of the circle
From (3), the area of the inscribed circle = πr2=π(4√3)* (4√3)=48π⋯(4)

Hence, area of the remaining portion of the triangle
= Area of Δ ABC – Area of inscribed circle
144√3−48π cm2

7. What will be the length of the longest rod which can be placed in a box of 80 cm length, 40 cm breadth and 60 cm height?

A. √11600 cm

B. √14400 cm

C. √10000 cm

D. √12040 cm   

Answer : Option A

Explanation :

The longest road which can fit into the box will have one end at A and other end at G (or any other similar diagonal).
Hence the length of the longest rod = AG

Initially let’s find out AC. Consider the right angled triangle ABC

AC² = AB² + BC² = 40² + 80² = 1600 + 6400 = 8000
⇒AC = √8000 cm

Consider the right angled triangle ACG

AG² = AC² + CG²
(√8000) ²+60²=8000+3600=11600
=> AG = √11600 cm
=> Length of the longest rod = √11600cm

8. A rectangular plot measuring 90 metres by 50 metres needs to be enclosed by wire fencing such that poles of the fence will be kept 5 metres apart. How many poles will be needed?

A. 30

B. 44

C. 56

D. 60

Answer : Option C

Explanation :

Perimeter of a rectangle = 2(l + b)
where l is the length and b is the breadth of the rectangle

Length of the wire fencing = perimeter = 2(90 + 50) = 280 metres
Two poles will be kept 5 metres apart. Also remember that the poles will be placed along the perimeter of the rectangular plot, not in a single straight line which is very important.
Hence number of poles required = 280/5 = 56

,

Mensuration is the process of measuring the dimensions of objects. It is a branch of mathematics that deals with the measurement of length, area, volume, and other properties of geometric figures.

Area is the amount of space enclosed by a two-dimensional figure. It is measured in square units, such as square inches, square centimeters, or square meters. The area of a rectangle is given by the formula $A = lw$, where $l$ is the length and $w$ is the width. The area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius.

Perimeter is the distance around a closed figure. It is measured in linear units, such as inches, centimeters, or meters. The perimeter of a rectangle is given by the formula $P = 2l + 2w$, where $l$ is the length and $w$ is the width. The perimeter of a circle is given by the formula $P = 2\pi r$, where $r$ is the radius.

Volume is the amount of space enclosed by a three-dimensional figure. It is measured in cubic units, such as cubic inches, cubic centimeters, or cubic meters. The volume of a rectangular prism is given by the formula $V = lwh$, where $l$ is the length, $w$ is the width, and $h$ is the height. The volume of a cylinder is given by the formula $V = \pi r^2h$, where $r$ is the radius and $h$ is the height. The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$, where $r$ is the radius.

Surface area is the total area of all the faces of a three-dimensional figure. It is measured in square units. The surface area of a rectangular prism is given by the formula $S = 2lw + 2lh + 2wh$, where $l$ is the length, $w$ is the width, and $h$ is the height. The surface area of a cylinder is given by the formula $S = 2\pi r^2 + 2\pi rh$, where $r$ is the radius and $h$ is the height. The surface area of a sphere is given by the formula $S = 4\pi r^2$, where $r$ is the radius.

The Pythagorean theorem is a mathematical formula that relates the lengths of the sides of a right triangle. The theorem states that the square of the hypotenuse (the longest side of the triangle) is equal to the sum of the squares of the other two sides. The Pythagorean theorem can be written in the following formula: $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the legs of the triangle and $c$ is the length of the hypotenuse.

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. Trigonometric functions are used to solve triangles and to find the lengths of sides and angles of other geometric figures. The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

A circle is a round plane figure with all points at the same distance from a center point. The distance from any point on the circle to the center point is called the radius. The diameter of a circle is twice the radius. The circumference of a circle is the distance around the circle. The area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius.

A triangle is a three-sided polygon. The three sides of a triangle are called the base, the height, and the hypotenuse. The base of a triangle is the side opposite the right angle. The height of a triangle is the perpendicular distance from the hypotenuse to the base. The hypotenuse of a right triangle is the longest side of the triangle.

A quadrilateral is a four-sided polygon. The four sides of a quadrilateral are called the sides. The four angles of a quadrilateral are called the angles. The sum of the angles of a quadrilateral is always equal to $360^\circ$.

A polygon is a closed plane figure with straight sides. The number of sides of a polygon is called the number of sides. The sum of the interior angles of a polygon with $n$ sides is always equal to $(n – 2)180^\circ$.

A cylinder is a three-dimensional figure with two parallel circular bases and a curved surface. The height of a cylinder is the distance between

Here are some frequently asked questions and short answers about the following topics:

• Area

What is area?

Area is the amount of space enclosed by a two-dimensional figure.

How do you find the area of a rectangle?

To find the area of a rectangle, multiply the length and width of the rectangle.

How do you find the area of a circle?

To find the area of a circle, use the formula $A = \pi r^2$, where $r$ is the radius of the circle.

• Perimeter

What is perimeter?

Perimeter is the total length of the sides of a closed figure.

How do you find the perimeter of a rectangle?

To find the perimeter of a rectangle, add up the lengths of the sides of the rectangle.

How do you find the perimeter of a circle?

To find the perimeter of a circle, use the formula $P = 2 \pi r$, where $r$ is the radius of the circle.

• Volume

What is volume?

Volume is the amount of space enclosed by a three-dimensional figure.

How do you find the volume of a rectangular prism?

To find the volume of a rectangular prism, multiply the length, width, and height of the prism.

How do you find the volume of a cylinder?

To find the volume of a cylinder, use the formula $V = \pi r^2 h$, where $r$ is the radius of the cylinder and $h$ is the height of the cylinder.

• Surface area

What is surface area?

Surface area is the total area of all the faces of a three-dimensional figure.

How do you find the surface area of a rectangular prism?

To find the surface area of a rectangular prism, find the areas of each of the faces and add them up.

How do you find the surface area of a cylinder?

To find the surface area of a cylinder, use the formula $S = 2 \pi r^2 + 2 \pi rh$, where $r$ is the radius of the cylinder and $h$ is the height of the cylinder.

• Pythagorean theorem

What is the Pythagorean theorem?

The Pythagorean theorem is a mathematical formula that relates the lengths of the sides of a right triangle.

What is the Pythagorean theorem formula?

The Pythagorean theorem formula is $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths of the legs of the right triangle and $c$ is the length of the hypotenuse.

• Similar triangles

What are similar triangles?

Similar triangles are triangles that have the same shape but different sizes.

How do you know if two triangles are similar?

Two triangles are similar if the corresponding angles are congruent and the corresponding sides are in proportion.

• Congruent triangles

What are congruent triangles?

Congruent triangles are triangles that have the same shape and size.

How do you know if two triangles are congruent?

Two triangles are congruent if the corresponding sides and angles are congruent.

• Angles

What is an angle?

An angle is formed by two rays that share a common endpoint.

What are the different types of angles?

There are three main types of angles: acute angles, right angles, and obtuse angles.

• Lines

What is a line?

A line is a straight path that extends infinitely in both directions.

What are the different types of lines?

There are two main types of lines: straight lines and curved lines.

• Planes

What is a plane?

A plane is a flat surface that extends infinitely in all directions.

What are the different types of planes?

There are two main types of planes: parallel planes and intersecting planes.

• Solids

What is a solid?

A solid is a three-dimensional figure that has length, width, and height.

What are the different types of solids?

There are many different types of solids, but some of the most common are cubes, spheres, and cylinders.

Sure, here are some MCQs on the following topics without mentioning the topic Mensuration (revised):

1. What is the area of a circle with a radius of 5 cm?
   (A) 78.5 cm2
   (B) 25 cm2
   (C) 31.4 cm2
   (D) 100 cm2

2. What is the volume of a cube with a side length of 3 cm?
   (A) 27 cm3
   (B) 9 cm3
   (C) 6 cm3
   (D) 1 cm3

3. What is the perimeter of a rectangle with a length of 10 cm and a width of 5 cm?
   (A) 30 cm
   (B) 20 cm
   (C) 15 cm
   (D) 10 cm

4. What is the area of a triangle with a base of 8 cm and a height of 6 cm?
   (A) 48 cm2
   (B) 24 cm2
   (C) 12 cm2
   (D) 6 cm2

5. What is the volume of a sphere with a radius of 5 cm?
   (A) 523.598775 cm3
   (B) 314.159265 cm3
   (C) 141.371667 cm3
   (D) 70.685833 cm3

6. What is the perimeter of a circle with a diameter of 10 cm?
   (A) 31.4 cm
   (B) 20 cm
   (C) 10 cm
   (D) 5 cm

7. What is the area of a square with a side length of 5 cm?
   (A) 25 cm2
   (B) 10 cm2
   (C) 5 cm2
   (D) 2 cm2

8. What is the volume of a rectangular prism with a length of 10 cm, a width of 5 cm, and a height of 3 cm?
   (A) 150 cm3
   (B) 75 cm3
   (C) 50 cm3
   (D) 25 cm3

9. What is the area of a trapezoid with bases of 8 cm and 12 cm and a height of 5 cm?
   (A) 50 cm2
   (B) 40 cm2
   (C) 30 cm2
   (D) 20 cm2

10. What is the volume of a cone with a radius of 3 cm and a height of 4 cm?
    (A) 36.7 cm3
    (B) 14.1 cm3
    (C) 7.07 cm3
    (D) 4.18 cm3

I hope these MCQs are helpful!