Maths Class 10 Notes for Surface Areas and Volumes


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

Maths Class 10 Notes - Surface Areas and Volumes

2. Area of each end of cylinder = 2πr2

3. Total surface area (including both circular ends) = 2πrh + 2πr2 = 27πr(h + r)

4. Volume of cylinder — πr2h = [(Area of base) * height]

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

1. Volume of material = Exterior volume — Interior
volume = πR2h — πr2h = πh(R2 – r2)

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

Maths Class 10 Notes - Surface Areas and Volumes

= 2πRh + 2πrh

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

= (2πRh + 2πrh) + 2(πR2 – πr2)


1. Two end faces of right circular cylinder are circles having each area = πr2

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.


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

Maths Class 10 Notes - Surface Areas and Volumes

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 πr2h

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

= l =√ r2 + hr2

3. T. S. A of cone = πrl + πr2


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

Maths Class 10 Notes - Surface Areas and Volumes

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[R2 + r2 + Rr] cubic unit

2. L. S. A or C. S. A = πl(R + r) Sq units where l2 = h2 + (R – r)2

3. T. S. A = πR2 + πr2 + πl(R + r) Sq. units.
(Area of base + Area of top + Area of lateral )

4. Slant height (l) = √h22 + (R – r)2


(a) Surface area of sphere = 4πr2

Maths Class 10 Notes - Surface Areas and Volumes

(b) Volume of sphere = 4 / 3 πr3

(c) Volume of hemisphere = 2 / 3 πr3

(d) C.S.A. of hemisphere = 2πr2

(e) Total surface area of Hemi-sphere = 2πr2 + πr2 =3πr2


(a) Outer surface area of spherical shell =4πR2

(b) Inner S.A. of spherical shell = 4πr2

(c) Total surface area of spherical shell = 4π(R2+ r2)

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

radius ‘r’ = 4 / 3π(R3 – r3)

(e) Outer curved surface area hemispherical shell = 2πR2

Maths Class 10 Notes - Surface Areas and Volumes

(f) Inner curved surface area of hemispherical shell = 2πr2

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

Total S.A. = π(3R2+ r2)

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

= 2 / 3π(R3 — r2).

Maths Class 10 Notes - Surface Areas and Volumes

Click Here for All Maths Class 10 Notes

You wish to report grammatical or factual errors within our online articles, you can let us know using the article feedback form.