Class 10 Chapter 13 (Surface Area and Volumes) Class notes

Notes on Surface Areas and Volumes

Surface Areas and Volumes

Surface Area is the area of the outer part of any 3D figure and Volume is the capacity of the figure i.e. the space inside the solid. To find the surface areas and volumes of the combination of solids, we must know the surface area and volume of the solids separately. Some of the formulas of solids are -

Surface Area of a Combination of Solids

If a solid is molded by two or more than two solids then we need to divide it in separate solids to calculate its surface area.

Cone on a Cylinder.

r = radius of cone & cylinder;
h₁ = height of cone
h₂ = height of cylinder
Total Surface area = Curved surface area of cone + Curved surface area of cylinder + area of circular base
= πrl + 2πrh₂ +πr²;
Slant height, l = r2+h21−−−−−−√
Total Volume = Volume of cone + Volume of cylinder
= 13πr2h1+πr2h2

Cone on a Hemisphere:

h = height of cone;
l = slant height of cone = r2+h2−−−−−−√
r = radius of cone and hemisphere
Total Surface area = Curved surface area of cone + Curved surface area of hemisphere = πrl + 2πr²
Volume = Volume of cone + Volume of hemisphere = 13πr2h+23πr3

Conical Cavity in a Cylinder

r = radius of cone and cylinder;
h = height of cylinder and conical cavity;
l = Slant height
Total Surface area = Curved surface area of cylinder + Area of bottom face of cylinder + Curved surface area of cone = 2πrh + πr² + πrl
Volume = Volume of cylinder – Volume of cone = πr2h−13πr2h=23πr2h

Cones on Either Side of Cylinder.

r = radius of cylinder and cone;
h₁ = height of cylinder
h₂ = height of cones
Slant height of cone, l = h22+r2−−−−−−√
Surface area = Curved surface area of 2 cones + Curved surface area of cylinder = 2πrl + 2πrh₁
Volume = 2(Volume of cone) + Volume of cylinder = 23πr2h2+πr2h1

Cylinder with Hemispherical Ends.

r = radius of cylinder and hemispherical ends;
h = height of cylinder
Total surface area= Curved surface area of cylinder + Curved surface area of 2 hemispheres = 2πrh + 4πr²
Volume = Volume of cylinder + Volume of 2 hemispheres = πr2h+43πr3

Hemisphere on Cube or Hemispherical Cavity on Cube

a = side of cube;
r = radius of hemisphere.
Surface area = Surface area of cube – Area of hemisphere face + Curved surface area of hemisphere
= 6a² – πr² + 2πr² = 6a² + πr²
Volume = Volume of cube + Volume of hemisphere = a3+43πr3

Hemispherical Cavity in a Cylinder

r = radius of hemisphere;
h = height of cylinder
Total surface area = Curved surface area of cylinder + Surface area of base + Curved surface area of hemisphere
= 2πrh + πr² + 2πr² = 2πrh + 3πr²
Volume = Volume of cylinder – Volume of hemisphere = πr2h−23πr3

Example

Find the total surface area of the given figure.

Solution

This solid is the combination of three solids i.e.cone, cylinder and hemisphere.

Total surface area of the solid = Curved surface area of cone + Curved surface area of cylinder + Curved surface area of hemisphere

Curved surface area of cone

Given, h = 5cm, r = 3cm (half of the diameter of hemisphere)

Curved surface area of cylinder = 2πrh

Given, h = 8cm (Total height – height of cone – height of hemisphere), r = 3cm

Curved surface area of hemisphere = 2πr²

Given, r = 3 cm

Total surface area of the solid

Volume of a combination of solids

Find the volume of the given solid.

Solution

The given solid is made up of two solids i.e. Pyramid and cuboid.

Total volume of the solid = Volume of pyramid + Volume of cuboid

 Volume of pyramid = 1/3 Area of base x height

Given, height = 6 in. and length of side = 4 in.

Volume of cuboid = lbh

Given, l = 4 in., b = 4 in, h = 5 in.

Total volume of the solid = 1/3 Area of base x height + lbh

= 1/3 x 4 x 4 x 6 + (4) (4) (5)

= 32 + 80

= 112 in³

Conversion of Solid from One Shape to Another

When we convert a solid of any shape into another shape by melting or remoulding then the volume of the solid remains the same even after the conversion of shape.

Example

If we transfer the water from a cuboid-shaped container of 20 m x 22 m into a cylindrical container having a diameter of 2 m and height of 3.5 m. then what will be the height of the water level in the cuboid container if the cylindrical tank gets filled after transferring the water. 

Solution

We know that the volume of the cuboid is equal to the volume of the cylinder.

Volume of cuboid = volume of cylinder

l x b x h = πr²h

20 x 22 x h = 22/7 x 1 x 3.5

440 × h =11

H = 2.5 cm

Frustum of a Cone

If we cut the cone with a plane which is parallel to its base and remove the cone then the remaining piece will be the Frustum of a Cone.

Volume of the frustum of the cone
The curved or Lateral surface area of the frustum of the cone
Total surface area of the frustum of the cone	Area of the base + Area of the top + Lateral surface area
Slant height of the frustum

Example

Find the lateral surface area of the given frustum of a right circular cone. 

Solution

Given, r =1.8 in.

R = 4 in.

l = 4.5 in.

The lateral surface area of the frustum of the cone = πl (R + r)

= π x 4.5 (4 +1.8)

=3.14 x 4.5 x 5.8

= 81.95 sq. in. 

 

 

 

Name	Figure	Lateral or Curved Surface Area	Total Surface Area	Volume	Length of diagonal and nomenclature
Cube		4l2	6l2	l3	√3 l = edge of the cube
Cuboid		2h(l +b)	2(lb + bh + hl)	lbh	l = length b = breadth h = height
Cylinder		2πrh	2πr2 + 2πh = 2πr(r + h)	πr2h	r = radius h = height
Hollow cylinder		2πh (R + r)	2πh (R + r) + 2πh (R2 - r2)	-	R = outer radius r = inner radius
Cone			πr2 + πrl = πr(r + l)	1/3 πr2h	r = radius h = height l = slant height
Sphere		4πr2	4πr2	4/3 πr3	r = radius
Hemisphere		2πr2	3πr2	2/3 πr3	r = radius
Spherical shell		4πR2 (Surface area of outer)	4πr2 (Surface area of outer)	4/3 π(R3 – r3)	R = outer radius r = inner radius
Prism		Perimeter of base × height	Lateteral surface area + 2(Area of the end surface)	Area of base × height	-
pyramid		1/2 (Perimeter of base) × slant height	Lateral surface area + Area of the base	1/3 area of base × height	-

cbse class 10th surface area and volume

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