## Maths Class 10 Notes for Surface Areas and Volumes

**(A) RIGHT CIRCULAR CYLINDER:**

A right circular cylinder is solid generated by the revolution of a rectangle about of its sides.

NOTE : If a paper, cylinder open at both the ends is cut along a vertical line on the curved surface and stretched on a plane surface, we obtain a rectangle of length i.e., 27πr and breadth= Height of cylinder h.

So, curved surface area (C.S.A) or lateral surface area = 2πr * height

**Important Formula For Cylinder**

1. C. S. A of cylinder = ( Perimeter of base) * Height = 2πrh

2. Area of each end of cylinder = 2πr^{2}

3. Total surface area (including both circular ends) = 2πrh + 2πr^{2} = 27πr(h + r)

4. Volume of cylinder — πr^{2}h = **[(Area of base) * height]**

**Hollow Cylinder’s formulae e.g., (Rubber tubes pipes, etc.)**

1. Volume of material = Exterior volume — Interior

volume = πR^{2}h — πr^{2}h = πh(R^{2} – r^{2})

2. C. S. A or L. S. A = external surface area + internal surface area

= 2πRh + 2πrh

3. T. S . A. of hollow cylinder = C. S. A+ 2 ( area of base ring )

= (2πRh + 2πrh) + 2(πR^{2} – πr^{2})

**NOTE:**

1. Two end faces of right circular cylinder are circles having each area = πr^{2}

2. Mass of cylinder = Volume * density

3. When rectangular sheet of paper is rolled along its length , we get a cylinder whose base circumference is length of sheet and height is same as breadth of sheet.

**(B) CONE**

From figure, AO = height of cone and is denoted by ‘h’

OB = radius of the base of cone, AB = slant height of a cone (l)

**Important Formula Of rt. Circular Cone :**

1. Volume of cone = 1 / 3 πr^{2}h

2 C. S. A or L. S. A=πrl where slant height

= l =√ r^{2} + hr^{2}

3. T. S. A of cone = πrl + πr^{2}

**(C) FRUSTUM OF A CONE**

**FRUSTUM :** A cone is cut by a plane parallel to the base of the cone,

then the portion between the plane and base is called frustum of the cone

**Important Formulae for Frustum :**

1. Volume of frustum of cone

= πh / 3[R^{2} + r^{2} + Rr] cubic unit

2. L. S. A or C. S. A = πl(R + r) Sq units where l^{2} = h^{2} + (R – r)^{2}

3. T. S. A = πR^{2} + πr^{2} + πl(R + r) Sq. units.

(Area of base + Area of top + Area of lateral )

4. Slant height (l) = √h^{2}2 + (R – r)^{2}

**(D) IMPORTANT FORMULA FOR SPHERE AND HEW-SPHERE**

(a) Surface area of sphere = 4πr^{2}

(b) Volume of sphere = 4 / 3 πr^{3}

(c) Volume of hemisphere = 2 / 3 πr^{3}

(d) C.S.A. of hemisphere = 2πr^{2}

(e) Total surface area of Hemi-sphere = 2πr^{2} + πr^{2} =3πr^{2}

**(E) IMPORTANT FORMULA FUR SPHERICAL SHELL/ HEMILSPHERICAL SHELL**

(a) Outer surface area of spherical shell =4πR^{2}

(b) Inner S.A. of spherical shell = 4πr^{2}

(c) Total surface area of spherical shell = 4π(R^{2}+ r^{2})

(d) Volume of spherical shell of external radius R and internal

radius ‘r’ = 4 / 3π(R^{3} – r^{3})

(e) Outer curved surface area hemispherical shell = 2πR^{2}

(f) Inner curved surface area of hemispherical shell = 2πr^{2}

(g) Thick hemispherical bowl of external and internal radii R and r,

Total S.A. = π(3R^{2}+ r^{2})

(h) Volume of hemispherical shell of external radius ‘R’ and internal radius ‘r’

= 2 / 3π(R^{3} — r^{2}).