These notes for CBSE Class 9 Mathematics, Chapter 13, cover the concepts of Surface Areas and Volumes. They define solids and explain volume as the space occupied by a solid. The notes detail the properties of cuboids and cubes, including their faces, edges, and vertices. Formulas for the total and lateral surface areas of cuboids and cubes are provided, along with the formula for the diagonal of a cuboid and cube. The chapter also explains the surface area and volume formulas for right circular cylinders, cones, and spheres. For cylinders, both curved and total surface areas are given, along with the volume formula. For cones, curved surface area, total surface area, and volume are explained, including the relationship between radius, height, and slant height. Sphere and hemisphere surface area and volume formulas are also included. These notes are a valuable resource for students to revise these important geometric concepts for their exams.
Last optimized 10 Aug 2026
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The bodies occupying space (i.e. have 3-dimension) are called solids such as a cuboid, a cube, a cylinder, a cone, a sphere etc.
The measure of space occupied by a solid-body is
called its volume. The units of volume are cubic centimeters (written as cm3) or cubic meters
(written as m3).
CUBOID: A solid bounded by six rectangular faces is called a cuboid.
In the given figure, ABCDEFGH is a cuboid whose
(i) 6 faces are : ABCD, EFGH, ABFE, CDHQ ADHE, and BCGF Out of these, the four faces namely ABFE,
DCGH, ADHE and BCGF are called lateral faces of the cuboid.
(ii) 12 edges are : AB, BC, CD, DA, EF, FG GH, HE, CG BF, AE and DH
(iii) 8 vertices are : A, B, C, D, E, F, and H.
Remark : A rectangular room is in the form of a cuboid and its 4 walls are its lateral surfaces.
Cube : A cuboid whose length, breadth and height are all equal, is called a cube.
A cube has 6 faces, each face is square, 12 edges, all edges are of equal lengths and 8 vertices.
Let us consider a cuboid of length = 1 units
Breadth = b units and height = h units
Then we have :
(i) Total surface area of the cuboid
=2(l * b + b * h + h * l) sq. units
(ii) Lateral surface area of the cuboid
= [2 (1 + b)* h] sq. units
(iii) Area of four walls of a room = [2 (1 + b)* h] sq. units.
= (Perimeter of the base * height) sq. units
(iv) Surface area of four walls and ceiling of a room
= lateral surface area of the room + surface area of ceiling
=2(1+b)*h+l*b
(v) Diagonal of the cuboid = ?l2 + b2 + h2
Consider a cube of edge a unit. (i) The Total surface area of the cube = 6a2 sq. units (ii) Lateral surface area of the cube = 4a2 sq. units. (iii) The diagonal of the cube = ?3 a units.
Cylinder: Solids like circular pillars, circular pipes, circular pencils, road rollers and gas
cylinders etc. are said to be in cylindrical shapes.
Curved surface area of the cylinder
= Area of the rectangular sheet
= length * breadth
= Perimeter of the base of the cylinder * height
= 2?r * h
Therefore, curved surface area of a cylinder = 2?rh
Total surface area of the cylinder =2?rh + 2?r2
So total area of the cylinder=2?r(r + h)
Remark : Value of TE approximately equal to 22 / 7 or 3.14.
If a cylinder is a hollow cylinder whose inner radius is r1 and outer radius r2 and height h then
Total surface area of the cylinder
= 2?r1h + 2?r2h + 2?(r2
2 – r2 1) = 2?(r1 + r2)h + 2? (r2 + r1) (r2 – r1)
= 2?(r1 + r2) [h + r2 – r1
A figure generated by rotating a right triangle about a perpendicular side is called the right circular cone. SURF
curved surface area of a cone = 1 / 2 * l * 2?r = ?rl
where r is base radius and l its slant height
Total surface area of the right circular cone
= curved surface area + Area of the base
= ?rl + ?r2 = ?r(l + r)
Note : l2 = r2 + h2
By applying Pythagorus
Theorem, here h is the height of the cone.
Thus l = ?r2 + h2 and r
= ?l2 – h2 h = ?l2 + r2
Sphere: A sphere is a three dimensional figure (solid figure) which is made up of all points in
the space which lie at a constant distance called the radius, from a fixed point called the centre
of the sphere.
Note : A sphere is like the surface of a ball. The word solid sphere is used for the solid whose
surface is a sphere.
Surface area of a sphere: The surface area of a sphere of radius r
= 4 x area of a circle of radius r = 4 * ?r2
= 4?r2
Surface area ofa hemisphere
= 2?r2
Total surface area of a hemisphere
= 2?r2 + ?r2
= 3?r2
Total surface area of a hollow hemisphere with inner and outer radius r1 and r2 respectively
= 2?r2
1 + 2?r2
2 + ?(r2
2 — r2 1) = 2?(r2
1 + r2
2) + ?(r2
2 —r2 1)
Volume : Solid objects occupy space.
The measure of this occupied space is called volume of the object.
Capacity of a container : The capacity of an object is the volume of the substance its interior
can accommodate.
The unit of measurement of either of the two is cubic unit.
Volume of a cuboid : Volume of a cuboid =Area of the base * height V=l * b * h
So, volume of a cuboid = base area * height = length * breadth * height
Volume of a cube : Volume of a cube = edge * edge * edge
= a3
where a
= edge of the cube
Volume of a cylinder = ?r2h
volume of the hollow cylinder ?r2
2h — ?r2 1h = ?(r2
2 – r2 1)h
volume of a cone = 1 / 3 ?r2h,
where r is the base radius
and h is the height of the cone.
VOLUME OF A SPHERE
volume of a sphere the sphere = 4 / 3 ?r3, where r is the radius of the sphere.
Volume of a hemisphere = 2 / 3 ?r3
Volume of the material of a hollow sphere with inner and outer radii r1 and
r2 respectively
= 4 / 3 ?r3
2 – 4 / 3 ?r3 1 = 4 / 3?(r3
2 – r3 1) Volume of the material of a hemisphere with inner and
outer radius r1 and r2 respectively
= 2 / 3?(r3
2 – r3 1)
Solids are three-dimensional bodies that occupy space, such as cuboids, cubes, cylinders, cones, and spheres. The volume of a cuboid is calculated by multiplying its length, breadth, and height: V = l * b * h. The total surface area of a cube with edge 'a' is 6a² square units. The curved surface area of a cylinder is given by the formula 2?rh, where 'r' is the radius and 'h' is the height. The slant height (l) of a cone is the distance from the apex to any point on the circumference of the base. It can be calculated using the Pythagorean theorem: l = ?(r² + h²). The surface area of a sphere with radius 'r' is 4?r² square units. The volume of a sphere with radius 'r' is given by the formula (4/3)?r³ cubic units. These notes provide clear definitions and formulas for surface areas and volumes of various geometric shapes, making them ideal for quick revision before exams.
Frequently asked questions
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