This document discusses the formulas for calculating the surface areas and volumes of various 3D shapes. It covers cubes, cuboids, cylinders, cones, spheres, and hemispheres. For each shape, it provides the relevant formulas to calculate surface area (total and lateral) and volume based on attributes like length, width, height, radius, diameter, and slant height.

1. MATHS NCERT
CHAPTER- 13
SURFACE AREA AND
VOLUME
CLASS- IX

2. CUBE
A
A
A
A CUBE IS A SYMMETRICAL THREE-
DIMENSIONAL SHAPE, EITHER SOLID OR
HOLLOW, CONTAINED BY SIX EQUAL
SQUARES.

3. LOOKING AT THE CUBE TEMPLATE, IT IS EASY TO SEE THAT
THE CUBE HAS SIX SIDES AND EACH SIDE IS A SQUARE
THE AREA OF ONE SQUARE IS A × A = A2
SINCE THERE ARE SIX SIDES, THE TOTAL SURFACE AREA IS-
SA = A2 + A2 + A2 + A2 + A2 + A2
SA = 6 × A2

4. THE VOLUME OF A CUBE IS MEASURED BY MULTIPLYING THE
LENGTH, WIDTH AND HEIGHT OF THE CUBE.
THE FORMULA FOR THE VOLUME OF A CUBE =
LENGTH × WIDTH × HEIGHT.
IN A CUBE, THE LENGTH, WIDTH, AND HEIGHT ARE ALL SAME.
EQUATION FOR VOLUME OF A CUBE, V = S
VOLUME OF A CUBE = S CUBIC UNITS
WHERE, 'S' IS THE LENGTH OF ANY EDGE OF THE CUBE.
3
3
VOLUME

5. CUBOID
A CUBOID IS A BOX-SHAPED SOLID OBJECT.
IT HAS SIX FLAT SIDES AND ALL ANGLES
ARE RIGHT ANGLES.
AND ALL OF ITS FACES ARE RECTANGLES.

6. To calculate the surface area of the cuboid we need to first
calculate the area of each face and the add up all the areas to
get the total surface area.
Total area of top and bottom surfaces is lw + lw = 2lw
Total area of front and back surfaces is lh + lh = 2lh
Total area of the two side surfaces is wh + wh = 2wh
Surface area of cuboid = 2lw + 2lh + 2wh = 2(lw + lh + wh)
SURFACE AREA

7. VOLUME
Look at this shape.
There are 3 different
measurements:
Length, Width,
Height
The volume is found using the formula:
Volume = Length × Width × Height
Which is usually shortened to:
V = l × w × h
Or more simply:
V = lwh

8. CYLINDER
A solid object with:
•two identical flat ends that are circular or
elliptical
•one curved side.

9. SURFACE AREA
(LATERAL)
SURFACE AREA OF CURVE = 2 × π × r × h
= 2πrh
WHERE,
π = pi
r = RADIUS OF THE BASE
h = HEIGHT OF THE CYLINDER

10. SURFACE AREA
(TOTAL)
THE SURFACE AREA HAS THESE PARTS:
SURFACE AREA OF BOTH ENDS = 2 × Π × R2
SURFACE AREA OF SIDE = 2 × Π × R × H
WHICH TOGETHER MAKE:
SURFACE AREA = 2 × π × r × (r+h)
AND IN SHORT- 2πr(r+h)
WHERE,
π = pi
r = RADIUS OF THE BASE
h = HEIGHT OF THE CYLINDER

11. VOLUME
To calculate the volume we multiply the area of the
base by the height of the cylinder:
Area of the base: π × r2
Height: h
And we get:
Volume = π × r2 × h
Or,
Volume= πr2 h

12. RIGHT
CIRCULAR
CONE 900
A CONE WHOSE SURFACE IS GENERATED BY LIN
ES JOINING A FIXED POINT TO THEPOINTS OF A
CIRCLE, THE FIXED POINT LYING ON A PERPENDI
CULAR THROUGH THECENTER OF THE CIRCLE.

13. SLANT HEIGHT
IN A RIGHT CIRCULAR CONE, WE
CAND FIND SLANT HEIGHT USING
PATHAGORAS THEOREM, i.e.-
a + b = c
HERE, a = radius (r)
b = height (h)
c = slant height (l)
SO, r + h = l
OR,
l= √(h + r )
2 22
2 2 2
2 2

14. SURFACE AREA
(LATERAL)
THE AREA OF THE CURVED (LATERAL)
SURFACE OF A CONE = πrl

15. SURFACE AREA
(TOTAL)
THE SURFACE AREA OF THE CONE EQUALS THE AREA OF
THE CIRCLE PLUS THE AREA OF THE CONE AND THE FINAL
FORMULA IS GIVEN BY:
SA = πR2 + πRL
WHERE,
R IS THE RADIUS
π IS pi
L IS THE SLANT HEIGHT

16. VOLUME
The volume of a cylinder is: π × r2 × h
The volume of a cone is: 1/3 π × r2 × h
The volume formulas for cones and cylinders are very
similar:
So a cone's volume is exactly one third ( 1/3 ) of a
cylinder's volume.

17. SPHERE
A ROUND SOLID FIGURE, OR ITS SURFACE, WITH
EVERY POINT ON ITS SURFACE EQUIDISTANT FROM
ITS CENTRE.

18. SURFACE AREA
The surface area of a sphere is given by the
formula:-
Area=4πr
Where r is the radius of the sphere.
2

19. VOLUME
The volume enclosed by a sphere is given by the
formula:-
volume=4/3πr
Where r is the radius of the sphere.
3

20. HEMISPHERE
IN GEOMETRY IT IS AN EXACT HALF OF
A SPHERE.

21. SURFACE AREA
(LATERAL)
THE SURFACE AREA OF A SPHERE OF
RADIUS r IS 4πr2. HALF OF THIS IS 2πr2.
SO, LATERAL SURFACE IS 2πr2

22. SURFACE AREA
(TOTAL)
THE AREA OF A CIRCLE OF RADIUS R IS Π R2 AND
THUS IF THE HEMISPHERE IS MEANT TO
INCLUDE THE BASE THEN THE SURFACE AREA
IS:-
2πR2 + πR2 = 3πR2.

23. VOLUME
The volume of a sphere is 4/3 π r3. So
the volume of a hemisphere is half of
that:
V = (2 / 3) π r3

26. THE
END