Aptitude Volume and Surface Area Theory

Formulas

Cuboid	Volume V = length × breadth × height Surfacearea = 2 (lw + wh + hl)
Cube	Volume V = $S^3$ Surfacearea = $6a^2 $ Diagonal = $\sqrt{3}$a
Cone	Volume V = $\dfrac{1}{3}Bh$   or   $V=\dfrac{1}{3}\pi r^{2}h$ Curved surface area = $πrl$ Total Surface area = $πrl + πr^2$   or   $πr (l+r)$
Cylinder	Volume V = $πr^{2}h$ Curved surface area = 2πrh Total Surface area = $2πrh+2πr^2$   or   2πr(r + h)
Sphere	Volume V = $\dfrac{4}{3} \pi r^{3}$ Surface area = $4π r^2$
Hemi Sphere	Volume V = $\dfrac{2}{3} \pi r^{3} $ Surface area = $2 π r^2$ Total Surface area = $3 π r^2$

Volume

Volume is the amount of space inside a three-dimensional (length, width and height.) object, or its capacity. measured in cubic units.

Surface Area

Total area of the surface of a 3D object, measured in square units.

Difference between Volume & Surfce Area with example

Surface area is the area of the thin sheet of aluminum required to make a soda can.
Volume is the amount of soda in the can.

CUBOID

A cuboid is a three-dimensional shape with a length, width, and a height.

Volume of a Cuboid

Formula

Volume of a Cuboid= length × breadth × height

Example

Find the volume of a cuboid of length 20 cm, breadth 15 cm and height 10 cm.

here,

length - 20 cm

breadth - 15 cm

height - 10 cm

Now calculating Volume of Cuboid

Volume of cuboid = length × breadth × height

Now applying this values to formula

Volume = 20cm x 15 cm x 10 cm.

= 3000 $cm^3$

Exercise

Surface area of a Cuboid

Formula

Surface area of a Cuboid = 2 (lw + wh + hl)

Diagonal = $\sqrt{l^{2}+ w^{2} + h^{2}}$

The Surface area of a cuboid is the sum of the areas of its 6 faces.

(i.e) Surface Area of a Cuboid = 2lw + 2wh +2 hl = 2 (lw + wh + hl)

Example

Find the total surface area and diagonal length of a cuboid with dimensions 8 cm by 6 cm by 5 cm

here,

l = 8 cm

w = 6 cm

h = 5 cm

First calculating Surface Area

Surfacearea of a Cuboid = 2 (lw + wh + hl)

= 2 [$(8 \times 6) + (6 \times 5) + (5 \times 8)$]

= 2 (48 + 30 + 40)

= 2(118)

= 236 $cm^2$

Next Calculating diagonal length

Diagonal = $\sqrt{l^{2}+ w^{2} + h^{2}}$

= $\sqrt{8^{2}+ 6^{2} + 5^{2}}$

= $\sqrt{64 + 36 + 25}$

= $\sqrt{125}$

= $\sqrt{125}$

= $\sqrt{5^3}$

= $\sqrt{5^2 \times 5^1}$

= 5 $\sqrt{5}cm$

Exercise

CUBE

A cube has Six sides (faces). Each face is a square of side length 'a'. which are all the same size.

Volume of a Cube

Formula

Volume of a Cube = $S^3$

So, the formalu of volume is = length × breadth × height

= S x S x S

Volume = $S^3$

Example

The length of a cubic shape box is 15 cm. What is the volume of the box?

here,
The length of cubic box = 15 cm

Now calculating Volume of Cube

Volume of Cube = $S^3$

= 15 x 15 x 15

= 3375 $cm^3$

Exercise

Surface Area of a Cube

Formula

Surfacearea of a Cube = $6a^2 $

DiagonalofaCube = $\sqrt{3}$a

The area of a square is $ a^2 $.

The total surface area is the sum of the area of each surface.

∴ Surface Area = $ a^2 + a^2 + a^2 + a^2 + a^2 + a^2 = 6a^2 $

The total surface area is the sum of the area of each surface.

The main diagonal of any cube is multiplying the length of one side by the square root of 3.

Example

Find the total surface area and diagonal of a box whose edges are all 5.5 cm long.

here,

Edge (face) = 4.5 cm

First , calculating Surface area

Now calculating Surface area of Cube

Surface area of Cube = $6a^2 $

= $6(5.5)^2 $

= 6 $\times$ 30.25

= 181.5 $cm^2$

Now calculating the diagonal of the cube

DiagonalofaCube = $\sqrt{3}$a

= $\sqrt{3}\times 5.5$

= 1.732 $\times 5.5$

= 9.526 cm

Exercise

CONE

A solid (3-dimensional) object with a circular flat base joined to a curved side that ends in an apex point.

Volume of a Cone

Formula

Volume of a Cone = V=$\dfrac{1}{3}Bh$   or   $V=\dfrac{1}{3}\pi r^{2}h$

Where B= $\pi r^2$

The volume V of a cone with radius r is one-third the area of the base B times the height h

where π is a number that is approximately equals to 3.14 or $\dfrac{22}{7}$

Example

Calculate the volume of a cone if the height is 12 cm and the radius is 7 cm.

here,

Height of the cone = 12 cm

Radius of the cone = 7 cm.

V = $\dfrac{1}{3} \pi r^{2}h$

V = $\dfrac{1}{3} \times \dfrac{22}{7} \times 7^{2} \times 12$

V = 616$ cm^{3}$

Exercise

Surface Area of the Cone

Formula

Curved surface area of the cone = $πrl$

Total Surface area of a Cone = $πrl + πr^2$   or   $πr (l+r)$

slant height l = $\sqrt{h^{2}+r^{2}}$

The Curved surface area of a Cone is the surface area of the outside of the cone

The Total surface area of a cone = The surface area of the outside of the cone + The surface area of the circle

r - is the radius, h - is the height , l - is the slant height

Example

The diameter of a cone is 16 cm and its height is 6 cm. Find the Curved surface area and Total surface area of cone.

here,

d = 16 => r = d/2 = 16/2 = 8cm

r= 8 cm

h = 6 cm

l = $\sqrt{h^{2}+r^{2}}$ = $\sqrt{6^{2} + 8^{2}}$ = $\sqrt{36+64}$ = $\sqrt{100}$ = 10 cm

l = 10 cm

First calculating Curved surface area of the cone

Curved surface area of the cone = $πrl$

= $\dfrac{22}{7} \times 8 \times 10$

= 3.142 $\times 8 \times 10$

= 251.36 $cm^2$

Now calculating Total Surface area of a cone

Total Surface area of a Cone = $πrl + πr^2$

= 251.36 + $\dfrac{22}{7} \times 8^2$

= 251.36 + (3.142 $\times $ 64)

= 251.36 + 201.08

= 452.44 $cm^2$

Exercise

CYLINDER

A cylinder is a closed solid that has two parallel (usually circular) bases connected by a curved surface.

Volume of a Cylinder

Formula

volume of a cylinder = $πr^{2}h$

where, r: Radius of a cylinder , h: Height of a cylinder and π is equal to 22/7 or 3.14

Example

Find the Volume of a cylinder whose height is 28 cm and radius 6 cm

here,

Height of the cylinder = 28 cm

Radius of the cylinder = 6 cm

Now calculating Volume of cylinder

volume of a cylinder = $πr^{2}h$

= $ \dfrac{22}{7} \times 6^{2} \times 28$

= 3165.12 $cm^3$

Exercise

Surface Area of a Cylinder

Formula

curved surface area of a cylinder = 2πrh

Total surface area of a cylinder = $2πrh+2πr^2$   or   2πr(r + h)

The curved surface area is defined as the area of only curved surface, leaving the circular top and base

The Total surface area of the curved surface as well as the bases.

The Total surface area = curved surface area + (2 × area of circle)

The area of a circle is $πr^2$, so the combined area of the two disks is twice that, or $2πr^2$.

Example

The diameter of the base of a cylinder is 10 cm and the height is 8 cm. Find the surface area of the solid cylinder

here,

Diameter (base) = 10 cm

Radius = Diameter /2 = 10/2 = 5 cm

r = 5m

First calculate the curved surface area

curved surface area of a cylinder = 2πrh

= $2 \times \dfrac{22}{7} \times 5 \times 8$

=251.36 $cm^2$

now find the total surface area of a cylinder

Total surface area of a cylinder = $2πrh+2πr^2 $

= 251.36 + 2 $\times 3.142 \times 5^2$

= 408.41 $cm^2$

Exercise

SPHERE

A 3-dimensional object shaped like a ball. Every point on the surface is the same distance from the centre.

Volume of a Sphere

Formula

volume of a sphere V = $\dfrac{4}{3} \pi r^{3}$

Where, r: Radius of a Sphere

The volume V of a sphere is four-thirds times pi times the radius cubed.

Example

Calculate the volume of sphere with radius 4 cm.

here,

Radius of the sphere = 4 cm

Now calculating Volume of sphere

volume of a sphere V = $\dfrac{4}{3} \pi r^{3}$

= $\dfrac{4}{3} \times \dfrac{22}{7} \times 4 \times 4 \times 4$

= 268.19 $cm^3$

Exercise

Surface Area of a Sphere

Formula

Surface area of a sphere = $4π r^2$

Where r is the radius of the sphere.

The surface area of a sphere is exactly four times the area of a circle with the same radius.

The area of a circle = $πr^2 $, and the surface area of a sphere = $4πr^2$

Example

Find the surface area of a sphere of radius 25cm.

here,

r = 25 cm

Surface area of a sphere = $4π r^2$

π = 3.142

= $4 \times 3.142 \times 25 \times 25$

= 7855 $cm^2$

Exercise

HEMISPHERE

A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.

Volume of a Hemisphere

Formula

Volume of a sphere V = $\dfrac{2}{3} \pi r^{3} $

The volume of a hemisphere is equal to two thrids of the cube of the radius by pi.

Where, r: Radius of a Sphere

Example

Find the volume of the hemisphere, whose radius is 10 cm.

here,

Radius of the sphere = 10 cm

Now calculating Volume of sphere

volume of a hemisphere V = $\dfrac{2}{3} \pi r^{3}$

=$\dfrac{2}{3} \times \dfrac{22}{7} \times 10^{3}$

= 23 x 3.14 x 1000

= 23 x 3140

= 2093.3 $cm^3$

Exercise

Surface Area of Hemisphere

Formula

Curved surface area of Hemisphere = $2 π r^2$

Total surface area of Hemisphere = $3 π r^2$

The hemisphere is to include the base then the surface area is 2 π $r^2$ + π $r^2$ = 3 π $r^2$

Example

Find the curved surface area and total surface area of a hemisphere having the radius of 7 cm?

here,

r = 7 cm

First calculate the curved surface area

Curved surface area of Hemisphere = $2 π r^2$

= 2 $\times 3.142 \times 7 \times 7

= 308 cm^2$

Now calculate the total surface area

Total surface area of Hemisphere = $3 π r^2$

= 3 $\times 3.142 \times 7 \times 7$

= 462 $cm^2$

Exercise

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