Surface Area & Volumes Chapter Notes - Mathematics (Maths) Class 10

Three dimensions can be measured, length, width, and height, for any object that you can see or touch. There are certain dimensions of our home that we live in. The rectangular display screen/Monitor you’re looking at has the width and breadth of its own length.
For every three-dimensional geometrical structure, surface area and volume are measured. 

Surface Area & Volume

1. Surface Area of a Combination of Solids

Surface area refers to the measurement of the space taken up by a flat two-dimensional surface. This measurement is expressed in terms of square units. For three-dimensional objects, the surface area represents the space occupied by their outer surfaces. Similar to the flat surface area, it is also measured in square units.
In general, there are two types of surface area

(a) Total Surface Area

Total surface area is the combined measurement of the base(s) and the curved portion of an object. It represents the sum of the object's surface coverage. In cases where the object's shape includes both a curved surface and a base, the total surface area is calculated by summing up the measurements of these two areas.

(b) Curved Surface Area or Lateral Surface Area

Curved surface area pertains to the measurement of only the curved portion of a shape, excluding any base(s). This aspect is also referred to as the lateral surface area, particularly for shapes like a cylinder.

1.1 Formulae for Curved Surface Area and lateral Surface Area

(i) Surface Area of a Cube

Cube is a solid shape having 6 equal square faces of length l.

Lateral surface area of a cube =4l²

 Total surface area of cube=6l ²

Cuboid is a solid shape having 6 rectangular faces at a right angle with length l, breadth b and height h.

Lateral surface area of a cuboid= 2h(l+b)

Total surface area of a cuboid= 2(lb+bh+lh)

(iii) Surface Area of a Right Circular Cylinder

If we fold a rectangular sheet with one side as its axis then it forms a cylinder. It is the curved surface of the cylinder. And if this curved surface is covered by two parallel circular bases then it forms a right circular cylinder. For a cylinder of height h and with base radius r.

Curved surface area of a Right circular cylinder=2πrh

Total surface area of a Right circular cylinder=2πr²+2πrh=2πr(r+h)

(iv) Surface Area of a Hollow Right Circular Cylinder

If a right circular cylinder is hollow from inside then it has a different curved surface.

Curved surface area of a Right circular cylinder=2πh(R+r)

Total surface area of a Right circular cylinder=2πh(R+r)+2π(R² −r²⁾

(v) Surface Area of a Right Circular Cone

If we revolve a right-angled triangle about one of its sides by taking other as its axis then the solid shape formed is known as a Right Circular Cone.

Curved surface area of a Right Circular Cone = πrl=πrh²+r²

 Total surface area of a Right Circular Cone= πr²+πrl=πr(r+l)

(vi) Surface Area of a Sphere

A sphere is a solid shape which is completely round like a ball. It has the same curved and total surface area.
Curved or Lateral surface area of a Sphere=4πr²

 Total surface area of a Sphere=4πr²

(vii) Surface Area of a Hemisphere

If we cut the sphere in two parts then it is said to be a hemisphere.

Curved or Lateral surface area of a Hemisphere=2πr²

 Total surface area of a Hemisphere=3πr²

2. Volume of Combination of Solids 

l=length, b= breadth, h=height

• Curved/lateral Surface Area = 2h(l +b)
• Surface Area = (lb+bh+lh)
• Volume = lbh

(i) Volume of Cube

Side of cube = a

• Curved/lateral surface area = 4a²
• Total Surface Area = 6a²
• Volume = a3

(ii) Volume of Cylinder

r = Radius of Cylinder

h = Height of Cylinder

• Curved Surface Area = 2πrh
• Total Surface Area = 2πrh +2 πr²
• Volume = πr²h

(iii) Volume of Cone

h= height of cone

r = radius of cone

l = slant height of cone = (h²+r²)

• Curved Surface Area = πrl
• Total Surface Area = πrl + πr²h
• Volume = 1/3πr²h

(iv) Volume of Sphere

          r= radius

• Volume =(V)= 4/3πr3

(v) Volume of Hemi-Sphere

• Volume =4/3πr³

Radius = r

• Curved Surface Area = πr²
• Total Surface Area = 2πr²
• Volume = 2/3πr³