RBSE Class 10 Maths Notes Chapter 13 Surface Areas and Volumes

These comprehensive RBSE Class 10 Maths Notes Chapter 13 Surface Areas and Volumes will give a brief overview of all the concepts.

RBSE Class 10 Maths Chapter 13 Surface Areas and Volumes

Important Points

(A) [Surface Area and Volume of Cube and Cuboid]

(1) Total surface area of cuboid = 2 (length x breadth + breadth x height + height x length) sq units

(2) Total surface area of cube = 6 (side)²

(3) Area of four walls of cuboid
= 2 x height (length + breadth) sq units
Or
= (height x perimeter) sq units

Volume :
(1) Volume of cuboid = [Length x Breadth x Height] cubic units
(2) Volume of cube = (side)³ cubic units

Diagonal of Cube and Cuboid
(1) Length of the diagonal of the cuboid
\(=\sqrt{(\text { length })^{2}+(\text { breadth })^{2}+(\text { height })^{2}}\)
(2) Length of the diagonal of a cube = √3 × Side

(B) [Right Circular Cylinder : Surface Area and Volume]
A right circular cylinder is that solid figure in which there is a curved surface and congruent circular transverse section and the axis of the cylinder is perpendicular to the circular transverse section.

General –

• Here cylinder has been used in the meaning of a right circular cylinder.
• On placing the cylinder in vertical position the lower circular end is called the base of the cylinder and the length of the cylinder is said to be its height (h). The radius of the circular end (r) is called the radius of the cylinder.

• Both the ends of a hollow cylinder are open. In a solid cylinder both the ends are closed.

Area –

(1) Area of the curved surface of the solid cylinder = 2πrh sq units
(2) The circumference of an end of a cylinder whose radius is r = 2πr sq units
(3) Area of the base of the cylinder = πr² sq units
(4) Total surface area of a cylinder = Curved surface area + 2 × Area of base
= 2πrh + 2πr²
= 2πr (h + r) sq units
(5) Total surface area of the hollow cylinder
= 2πr₁h + 2πr₂h + 2πr₁² – 2πr₂²
= 2πh (r₁+ r₂) + 2π (r₁² – r₂²)
= 2πh (r₁+ r₂) + 2π (r₁ + r₂) (r₁ – r₂)
= 2π (r₁+ r₂) (h+ r₁ – r₂)

Volume –

• Volume of cylinder = πr²h cubic units
• Volume of a hollow cylinder = π (r₁² – r₂²) h cubic units

Note : In a hollow cylinder these are two radii (internal and external), which are assumed to be ‘r₁‘ and ‘r₂‘.

(C) [Cone] –

A right circular cone is that figure when any line segment revolves at a fixed point at a constant angle with a fixed line.

• The shape of a cone is as that of an ice cream cone or a jocker’s cap.
• Height of the right circular cone = h
• Radius of cone= r and its slant height is assumed to be = l

Area –

• The formula for curved surface (slant surface) area of a cone is = Krl
• The formula for total surface area of a cone is = Kr (r + l)

Volume –

• Volume of cone = \(\frac{1}{3}\) πr²h
• Slant height of cone l = \(\sqrt{r^{2}+h^{2}}\)

D. [ Sphere]

Meaning –

• The solid generated by a complete revolution or half revolution of a circle or a semi circle respectively about its diameter as axis is called the sphere.
  

• The set of all these points situated in space can be called to be a sphere which are at the same distance from a fixed point.
• The fixed point is called the centre of the sphere.
• The distance of any point on it from the centre is called radius.
• A line segment that passes through the centre of the sphere and whose both points are on the sphere is called a diameter of the sphere. The radius of a sphere is half of its diameter.
• The place occupied in space is called its volume.

Area:
(1) Surface area of sphere = 4πr²
(2) Curved surface area of hemisphere = 2πr²
(3) Total surface area of a hemisphere = 3πr²
(4) If the external radius of a spherical shell be r₁ and the internal radius be r₂, then the total surface area of the spherical shell = 4π(r₁² + r₂²)

(1) Volume of sphere = \(\frac{4}{3} \pi r^{3}\)
(2) Volume of hemisphere = \(\frac{2}{3} \pi r^{3}\)
(3) Volume of spherical shell =\(\frac{4}{3} \pi\left(r_{1}^{3}-r_{2}^{3}\right)\)

[E] When a coile is cut by a plane parallel to its base and a small cone is removed, then the solid that remains is called a frustum of cone. The formulae related with frustum of a cone are as follows :

(i) Volume of a frustum of a cone =\(\frac{1}{3} \pi r\left(r_{1}^{2}+r_{2}^{2}+r_{1} r_{2}\right)\)

(ii) Curved surface area of a frustum of a cone = itl (r₁ + r₂)
Where \(l=\sqrt{h^{2}+\left(r_{1}-r_{2}\right)^{2}}\)

(iii) Total surface area of a frustum of a cone
= πl (r₁ + r₂) + π (r₁² – r₂²)
In above formulae.
h = Vertical height of frustum
l = Slant height of frustum
and r₁ and r₂ are the radii of both the circular ends of the frustum.