Surface Areas And Volumes Class 9th Notes - Free NCERT Class 9 Maths Chapter 13 Notes - Download PDF

NCERT Notes Class 9 Maths Chapter 13 Surface Areas and Volumes

Surface Area: The entire space which the external surface of an object takes up is known as the surface area of that object.

(Measured in square units, e.g., cm², m²)

Volume: The measurement of the space occupied by an object is referred to as volume.

(Measured in cubic units, e.g., cm³, m³)

Surface Area and Volume of a Cube:

A cube is a cuboid with the same length, width, and height. Six surfaces, twelve edges, eight corners, and four diagonals make up a cube. Example: a dice.

Total Surface Area: 6 × (side)²

Lateral Surface Area: 4 × (side)²

Volume: (side)³

Example:

If side = 3 cm, then Volume = (side)³ = 3³ = 27 cm³

Surface Area and Volume of a Cuboid:

A cuboid is a figure that is surrounded by six rectangular surfaces, with opposite surfaces equal and parallel. In a Cuboid, 12 edges and 8 corners are there. The vertex of a cuboid is each of its four corners. The diagonal of a cuboid is the line segment connecting the opposite vertices. In a cuboid, there are four diagonals. Example: a brick or a matchbox.

L = Length, B = Breadth and H = Height

Total Surface Area: 2 × (lb + bh + hl)

Lateral Surface Area: 2 × (l + b) × h

Volume: l × b × h

Example:

If l = 5 cm, b = 4 cm, h = 3 cm, then Volume = l × b × h = 5 × 4 × 3 = 60 cm³

Surface Area and Volume of a Right Circular Cylinder:

A solid generated by the revolution of a rectangle around one of its sides is known as a right circular cylinder. It is a three-dimensional shape having two congruent and parallel sides. Thus, two circular and one lateral face, when combined, form a cylinder as shown in the figure. Example: a cold drink can or a water tank.

R = Radius and H = Height

Total Surface Area: 2πr(h + r)

Curved Surface Area: 2πrh

Volume: πr²h

Example:

If r = 7 cm, h = 14 cm, π = $\frac{22}{7}$, then Curved surface area = 2πrh = 2 × $\frac{22}{7}$ × 7 × 14 = 616 cm²

Surface Area and Volume of a Right Circular Cone:

A right circular cone is a solid formed when a right triangle is rotated around one of its sides (other than the hypotenuse). Thus, it contains a circular base and a curved surface that gradually decreases to a single point called the vertex. Example: an ice cream cone.

R = Radius, L = Slant Height and H = Height

Total Surface Area: Curved surface area + Base area

⇒ πrl + πr²

Curved Surface Area: πrl

Volume: $\frac{1}{3}$ × πr²h

Example:

If r = 7 cm, h = 12 cm, π = $\frac{22}{7}$,

then Volume = $\frac{1}{3}$ × πr²h

= $\frac{1}{3}$ × $\frac{22}{7}$ × 7 × 7 × 12 = 616 cm³

Surface Area and Volume of a Sphere:

A sphere is the spatial area of a solid that touches a curved surface by which every point maintains an equal distance to its fixed centre. The centre point of the sphere serves as its fixed position. A sphere possesses a radius that represents the straight distance from its centre point to the surface. Example: ball

R = Radius

Surface Area: 4πr²

Volume: $\frac{4}{3}$ × πr³

Example:

If r = 21 cm, π = $\frac{22}{7}$,

then Volume = $\frac{4}{3}$ × πr³

= $\frac{4}{3}$ × $\frac{22}{7}$ × 21 × 21 × 21 = 38808 cm³

Surface Area and Volume of a Hemisphere:

A plane which passes through the sphere's centre produces equal halves which correspond to each other as two separate hemispheres. Each divided part of the sphere is named a hemisphere. Example: a dome

R = Radius

Total Surface Area: 3πr²

Curved Surface Area (CSA) = 2πr²

Volume: $\frac{2}{3}$ × πr³

Example:

If r = 21 cm, π = $\frac{22}{7}$,

then Volume = $\frac{2}{3}$ × πr³

= $\frac{2}{3}$ × $\frac{22}{7}$ × 21 × 21 × 21 = 19404 cm³

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