Mensuration – Concept, Formulas & Practice Questions

Mensuration is a subject of geometry. Mensuration is associated with the size, area, and density of various two and three-dimensional structures.

mensuration

Table of Contents:

What is an area?
The area is defined as the space occupied by two-dimensional shapes. The area of different shapes varies with their dimensions. It’s measured in square units.

Two-dimensional shapes include circles, triangles, squares, rectangles, parallelograms, pentagons, hexagons, and so on. As a result, the areas of each of these forms differ.

What is volume?
Volume is a parameter that applies solely to three-dimensional objects. The volume of each three-dimensional shape, such as a sphere, cube, cuboid, cylinder, or cone, varies. It’s measured in cubic units.

What is surface area?
The area is the amount of space that a two-dimensional flat surface takes up. It’s measured in square units. The surface area refers to the area occupied by a three-dimensional object’s exterior surface. It’s also measured in square units.

In general, there are two types of areas:

(i) Total Surface Area.

(ii) Curved Surface Area (or) Lateral Surface Area

Important Terminologies Used in Mensuration 

Total Surface Area: The total surface area includes both the base(s) and the curving section. It refers to the whole area of an object’s surface. If the shape has a curved surface and base, the total area equals the sum of the two areas.

Curved Surface Area / Lateral Surface Area: Curved surface area is the area of  the curved section of a shape, excluding its base(s). It is also known as lateral surface area in shapes like cylinders.

Perimeter: The perimeter is measure of the continuous line along the boundary of a given figure.

Square Unit: A square unit is defined as the area covered by a square with sides of one unit.

Cube Unit: A cube unit is defined as the volume occupied by a cube with sides of one unit.

Difference Between Area and Volume

The table below lists outs the key differences between area and volume.

Area Volume
Area is always defined for two-dimensional objects or flat figures. Volume is always defined for three-dimensional objects and solid figures.
It is measured in two dimensions, namely length and width. It is measured in three dimensions, including height.
It is measured in square units. It’s measured in cubic units.
It covers the outer space of an object. Volume is an object’s capacity.

Mensuration Formulas PDF – Click Here

Important Mensuration Formulas for 2-D Figures (Areas)

Name Figure Formulas 
SQUARE square img $$Side = a$$
$$Diagonal = d$$
$$d=a\sqrt{2}$$
$$Perimeter = 4a$$
$$Area=a^2\ or \frac{d^2}{2}$$
RECTANGLE RECTANGLE img $$Length = l$$
$$Breadth = b$$
$$Diagonal = d$$
$$Area =lb$$
$$Perimeter = 2 (l + b)$$
$$d = \sqrt{(l^2 + b^2 )}$$
TRIANGLE Triangle

$$Perimeter=a+b+c$$

$$Area = \frac{1}{2}bh$$
$$Area = \sqrt{(S (S-a)(S-b)(S-c))}$$
$$Where\ S = \frac{(a+b+c)}{2}$$

EQUILATERAL TRIANGLE EQUILATERAL TRIANGLE

$$Perimeter=3a$$

$$Area=\frac{\sqrt{3}}{2}a^2$$

$$Attitude \ or \ height = \frac{\sqrt{3}}{2}a$$

RHOMBUS RHOMBUS img

$$Side = a$$
$$Diagonals = d1, d2$$
$$Side = \frac{\sqrt{d1^2+d2^2}}{2}$$
$$Perimeter = 4a$$
$$Area = \frac{1}{2}d1d2$$

PARALLELOGRAM PARALLELOGRAM img

$$Base = b$$
$$Height = h$$
$$Area = bh$$

TRAPEZIUM TRAPEZIUM img

$$d = depth (or) height$$
$$a, b = length \ of\  two\  parallel\  sides$$
$$Area = \frac{1}{2}d(a + b)$$

CIRCLE CIRCLE img

$$Radius = r$$
$$Diameter = d$$
$$D = 2r$$
$$Area = πr^2 or \frac{πd^2}{4}$$
$$Circumference = 2πr (or) πd$$

SEMI CIRCLE SEMI CIRCLE img

$$Radius = r, \ diameter = d$$
$$d = 2r$$
$$Area = \frac{1}{2}πr^2 $$
$$circumference = πr + 2r$$

SECTOR SECTOR img

$$Radius = r$$
$$n = angle\ between\ radii$$
$$Area = \frac{n^0}{360}πr^2$$
$$Where \ l = length\ of\ the\ arc $$
$$l = \frac{n}{360}× 2πr $$
$$Perimeter\ of\ sector = l + 2r$$

Important Mensuration Formulas for 3-D Figures (Volumes)

Name Figure Formulas 
CUBE  

cube img

$$Side = a$$
$$Diagonal = d$$
$$Lateral \ Surface\  Area (L.S.A) = 4a^2 (Area \ of\  four \ walls)$$
$$Total \ Surface \ Area (T. S. A) = 6a^2$$
$$Volume = a^3 = base\ area × height$$
$$Diagonal = a\sqrt{3}$$
CUBOID CUBOID img $$Length = l$$
$$Breadth = b$$
$$Height = h$$
$$LSA = 2h (I + b) (Area\ of\ four\ walls)$$
$$TSA = 2 (lb + bh + lh)$$
$$Volume = lbh$$
$$Diagonal = \sqrt{l^2 + b^2+ h^2}$$
CYLINDER CYLINDER img $$Radius = r$$
$$Height = h$$
$$LSA = 2πrh (Curved\ surface\ area)$$
$$TSA = 2πr (h + r)$$
$$Volume = πr^2h$$
SPHERE SPHERE img $$Radius = r$$
$$Diameter = d = 2r$$
$$Surface area = 4πr^2$$
$$Volume = \frac{4}{3}πr^3$$
HEMI SPHERE HEMI SPHERE img $$Radius = r$$
$$Diameter = 2r$$
$$LSA = 2πr^2 (Curved\ surface\ area)$$
$$TSA = 3πr^2$$
$$Volume =\frac{2}{2}πr^3$$
CONE cone img $$Height = h$$
$$Base radius = r$$
$$Slant height = S$$
$$S = \sqrt{r^2+ h^2}$$
$$LSA = πrs (Curved\ surface\ area)$$
$$TSA = πr (s + r)$$
$$Volume = \frac{1}{3}πr^2h$$

 

Example problems

Q1. Find the ratio of area of a circle to area of square if perimeter of circle and square is equal?

  1. 11 : 14
  2. 11 : 4
  3. 8 : 11
  4. 14 : 11
  5. 2 : 4

Solution:

Areas and Volumes sol1

Q2. The circumference of two circles is 88 m and 132 m respectively. What is difference between the area of larger circle and smaller circle?

  1. 1052
  2. 1128
  3. 1258
  4. 770
  5. 1528

Solution:

Areas and Volumes sol2

Q3. The ratio between the curved surface area and total surface area of right circular cylinder is 12: 19. If the volume of the cylinder is then find the height of the cylinder.

  1. 12 cm
  2. 24 cm
  3. 14 cm
  4. 21 cm
  5. 18 cm

Solution:

Areas and Volumes sol 3

Q4. The radius of a circle is 12.5% less than side of a square and the difference between the perimeter of the circle and that of the square is 24 cm. Find the area of the square?

  1. 576 cm²
  2. 196 cm²
  3. 144 cm²
  4. 256 cm²
  5. 64 cm²

Solution:

Areas and Volumes sol4

Q5. Area of rectangle is 144 cm² and the length of rectangle is 10 cm more than its breadth. Find the perimeter of the rectangle.

  1. 62 cm
  2. 54 cm
  3. 56 cm
  4. 52 cm
  5. None of these

Solution:

Areas and Volumes sol 5

Q6. What is the radius of circle which area is 124.74 cm²?

  1. 4.9 cm
  2. 6.3 cm
  3. 0.63 cm
  4. 0.49 cm
  5. 7.2 cm

Solution:

Areas and Volumes sol 6

Q7. If area of a rectangle is 375 cm2 and the ratio of perimeter of the rectangle to its length is 16 : 5, then find the breadth of the rectangle?

  1. 15 cm
  2. 12 cm
  3. 18 cm
  4. 20 cm
  5. 25 cm

Solution: Let length and breadth of the rectangle be l and b cm respectively.

Areas and Volumes sol 7

Q8. The curved surface area of a cylinder is 528 cm² and volume of the cylinder is 1848 cm³. Find the total surface area of the cylinder.

  1. 742 cm²
  2. 588 cm²
  3. 836 cm²
  4. 957 cm²
  5. 616 cm²

Solution:

Areas and Volumes sol 8

Frequently Asked Questions about Mensuration

Q1. What exactly are the definitions of area and volume?
Ans: The space occupied by a two-dimensional figure in the 2D plane is referred to as area, but the space filled by the figure in the 3D plane is referred to as volume.

Q2. How Are Area and Volume Related?
Ans: Area and Volume are connected in the sense that extending, expanding, or rotating two-dimensional Areas in another (third) dimension creates a solid figure with a Volume.

Q3. How can you determine volume by area?
Ans: Volume, expressed in cubic units, is a measure of capacity. To calculate the volume of a rectangular prism, multiply the base area (length × width) by its height.

Q4. What is mensuration?
Ans: Mensuration is a branch of mathematics that deals with the measuring of various geometric shapes and their characteristics, like length, volume, shape, surface area, and lateral surface area, among others.


Latest Job Notifications | Daily Job updates

Latest Job Notifications 2023

Latest Job Notifications

Use The Links Below To Practice Topic-Wise Quizzes.


[wptb id="1125" not found ]
[wptb id="1176" not found ]
[wptb id="1179" not found ]
[wptb id="1185" not found ]
[wptb id="1186" not found ]

Current Affairs Quiz

Daily Current Affairs Quiz

Weekly Current Affairs Quiz

Monthly Current Affairs Quiz


competitiveexamsindia.com is one of the websites in India that provides the latest Job notification Information across India, such as IT jobs, government jobs, bank jobs, railway jobs, work-from-home jobs, part-time jobs, online jobs, pharmacist jobs, software jobs, etc. We also provide practice quizzes, Video classes, and courses through our website & YouTube channels.

Leave a Reply

Your email address will not be published. Required fields are marked *