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

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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		$$Side = a$$ $$Diagonal = d$$ $$d=a\sqrt{2}$$ $$Perimeter = 4a$$ $$Area=a^2\ or \frac{d^2}{2}$$
RECTANGLE		$$Length = l$$ $$Breadth = b$$ $$Diagonal = d$$ $$Area =lb$$ $$Perimeter = 2 (l + b)$$ $$d = \sqrt{(l^2 + b^2 )}$$
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		$$Perimeter=3a$$ $$Area=\frac{\sqrt{3}}{2}a^2$$ $$Attitude \ or \ height = \frac{\sqrt{3}}{2}a$$
RHOMBUS		$$Side = a$$ $$Diagonals = d1, d2$$ $$Side = \frac{\sqrt{d1^2+d2^2}}{2}$$ $$Perimeter = 4a$$ $$Area = \frac{1}{2}d1d2$$
PARALLELOGRAM		$$Base = b$$ $$Height = h$$ $$Area = bh$$
TRAPEZIUM		$$d = depth (or) height$$ $$a, b = length \ of\  two\  parallel\  sides$$ $$Area = \frac{1}{2}d(a + b)$$
CIRCLE		$$Radius = r$$ $$Diameter = d$$ $$D = 2r$$ $$Area = πr^2 or \frac{πd^2}{4}$$ $$Circumference = 2πr (or) πd$$
SEMI CIRCLE		$$Radius = r, \ diameter = d$$ $$d = 2r$$ $$Area = \frac{1}{2}πr^2 $$ $$circumference = πr + 2r$$
SECTOR		$$Radius = r$$ $$n = angle\ between\ radii$$ $$Area = \frac{n^0}{360}πr^2$$ $$Where \ l = length\ of\ the\ arc $$ $$l = \frac{n}{360}× 2πr $$ $$Perimeter\ of\ sector = l + 2r$$

Important Mensuration Formulas for 3-D Figures (Volumes)

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

 

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:

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:

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:

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:

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:

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:

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.

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:

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