# NCERT Class 9 Maths Formula CBSE Board Sample Problems Part 12

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## 12. Surface Area and Volume

S.no | Term | Description |

1 |
| It is branch of mathematics which is concerned about the measurement of length, area and Volume of plane and Solid figure |

2 |
| a) The perimeter of plane figure is defined as the length of the boundary b) It units is same as that of length i.e. m, cm, km |

3 |
| a) The area of the plane figure is the surface enclosed by its boundary b) It unit is square of length unit. i.e. |

4 |
| Volume is the measure of the amount of space inside of a solid figure, like a cube, ball, cylinder or pyramid. Its units are always “cubic”, that is, the number of little element cubes that fit inside the figure. |

### Volume Unit Conversion

1 | 1 mL | 1000 |

1 Litre | 1000mI | 1000 |

1 | 1000 L | |

1 | 1000 | 1 L |

### Surface Area and Volume of Cube and Cuboid

Cube

Cuboid

Type | Measurement |

Surface Area of Cuboid of Length L, Breadth B and Height H | |

Lateral surface area of the cuboids | |

Diagonal of the cuboids | |

Volume of a cuboids | LBH |

Length of all 12 edges of the cuboids | 4 (L+B+H). |

Surface Area of Cube of side L | |

Lateral surface area of the cube | |

Diagonal of the cube | |

Volume of a cube |

### Surface Area and Volume of Right Circular Cylinder

**Radius**

The radius (r) of the circular base is called the radius of the cylinder

**Height**

The length of the axis of the cylinder is called the height (h) of the cylinder

**Lateral Surface**

The curved surface joining the two base of a right circular cylinder is called Lateral Surface.

Type | Measurement |

Curved or lateral Surface Area of cylinder | |

Total surface area of cylinder | |

Volume of Cylinder |

### Surface Area and Volume of Right Circular Cone

**Radius**

The radius (r) of the circular base is called the radius of the cone

**Height**

The length of the line segment joining the vertex to the center of base is called the height (h) of the cone.

**Slant Height**

The length of the segment joining the vertex to any point on the circular edge of the base is called the slant height (L) of the cone.

**Lateral surface Area**

The curved surface joining the base a’ uppermost point of a right circular cone is called Lateral Surface

Type | Measurement |

Curved or lateral Surface Area of cone | |

Total surface area of cone | |

Volume of Cone |

### Surface Area and Volume of Sphere and Hemisphere

**Sphere**

A sphere can also be considered as a solid obtained on rotating a circle about its diameter

**Hemisphere**

A plane through the centre of the sphere divides the sphere into two equal parts, each of which is called a hemisphere

**Radius**

The radius of the circle by which it is formed

**Spherical Shell**

The difference of two solid concentric spheres is called a spherical shell

**Lateral Surface Area for Sphere**

Total surface area of the sphere

**Lateral Surface area of Hemisphere**

It is the curved surface area leaving the circular base

Types | Measurement |

Surface area of Sphere | |

Volume of Sphere | |

Curved Surface area of hemisphere | |

Total Surface area of hemisphere | |

Volume of hemisphere | |

Volume of the spherical shell whose outer and inner radii and ‘R’ and ‘r’ respectively |

### How the Surface Area and Volume Are Determined

**Area of Circle**

The circumference of a circle is 2nr.

This is the definition of n (pi). Divide the circle into many triangular segments. The area of the triangles is ½ times the sum of their bases, 2nr (the circumference), times their height, r.

**Surface Area of cylinder**

This can be imagined as unwrapping the surface into a rectangle.

**Surface area of cone**

This can be achieved by divide the surface of the cone into its triangles, or the surface of the cone into many thin triangles. The area of the triangles is ½ times the sum of their bases, p, times their height,