These notes for CBSE Class 10 Mathematics, Chapter 13, focus on Surface Areas and Volumes. They provide essential formulas for various 3D shapes. Key topics include the right circular cylinder, detailing its curved surface area (CSA), total surface area (TSA), and volume. Formulas for hollow cylinders are also presented. The notes then cover the right circular cone, including its volume, CSA, and TSA, along with the formula for slant height. The concept of a frustum of a cone is explained, with formulas for its volume, CSA, TSA, and slant height. Finally, the notes list important formulas for spheres and hemispheres, covering their surface areas and volumes. Formulas for spherical and hemispherical shells are also included. These notes are designed to aid students in revising these geometric concepts for their exams.
Last optimized 10 Aug 2026
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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.,
27?r and breadth=
Height of cylinder h.
So, curved surface area (C.S.A) or lateral surface area = 2?r * height Important Formula For Cylinder
1. C. S. A of cylinder = ( Perimeter of base) * Height = 2?rh
2. Area of each end of cylinder = 2?r2
3. Total surface area (including both circular ends) = 2?rh + 2?r2 = 27?r(h + r)
4. Volume of cylinder — ?r2h = [(Area of base) * height]
Hollow Cylinder’s formulae e.g., (Rubber tubes pipes, etc.)
1. Volume of material = Exterior volume — Interior volume = ?R2h — ?R2h = ?h(R2 – r2)
2. C. S. A or L. S. A = external surface area + internal surface area
= 2?Rh + 2?rh
3. T. S . A. of hollow cylinder = C. S. A+ 2 ( area of base ring )
= (2?Rh + 2?rh) + 2(?R2 – ?R2)
1. Two end faces of right circular cylinder are circles having each area = ?R2 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’
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 ?R2h
2. C. S. A or L. S. A=?rl where slant height
= l =? R2 + hR2
3. T. S. A of cone = ?rl + ?R2
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
1. Volume of frustum of cone = ?h / 3[R2 + R2 + Rr] cubic unit 2. L. S. A or C. S. A = ?l(R + r) Sq units where l2 = h2 + (R – r)2 3. T. S. A = ?R2 + ?R2 + ?l(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 = 4?R2
(b) Volume of sphere = 4 / 3 ?R3
(c) Volume of hemisphere = 2 / 3 ?R3
(d) C.S.A. of hemisphere = 2?R2
(e) Total surface area of Hemi-sphere = 2?R2 + ?R2 =3?R2
(a) Outer surface area of spherical shell =4?R2
(b) Inner S.A. of spherical shell = 4?R2
(c) Total surface area of spherical shell = 4?(R2+ R2)
(d) Volume of spherical shell of external radius R and internal radius ‘r’ = 4 / 3?(R3 – R3)
(e) Outer curved surface area hemispherical shell = 2?R2
(f) Inner curved surface area of hemispherical shell = 2?R2
(g) Thick hemispherical bowl of external and internal radii R and r, Total S.A. = ?(3R2+ R2)
(h) Volume of hemispherical shell of external radius ‘R’ and internal radius ‘r’
= 2 / 3?(R3 — R2).
The notes cover right circular cylinders, hollow cylinders, cones, frustums of cones, spheres, hemispheres, spherical shells, and hemispherical shells.
The CSA of a right circular cylinder is given by 2?rh, where r is the radius and h is the height.
The slant height (l) of a cone is calculated using the formula l = ?(R² + h²), where R is the radius and h is the height.
The volume of a frustum of a cone is given by V = (1/3)?h(R² + r² + Rr), where h is the height, R is the radius of the larger base, and r is the radius of the smaller base.
The total surface area of a sphere is 4?R², where R is the radius of the sphere.
These notes provide all the essential formulas for surface areas and volumes of various 3D shapes, making them a quick and effective resource for revision before exams.
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