Surface Area and Volume

Model three-dimensional objects and calculate surface area and volume for the shapes prescribed in the current course: cuboids, cubes, cylinders, cones, pyramids, spheres and hemispheres.

Formula guide

Solid	Surface area	Volume
Cube, side a	TSA=6a²	a³
Cuboid, l,b,h	TSA=2(lb+bh+hl)	lbh
Right circular cylinder, r,h	CSA=2πrh; TSA=2πr(h+r)	πr²h
Right circular cone, r,h, slant height l	CSA=πrl; TSA=πr(l+r)	(1/3)πr²h
Sphere, radius r	4πr²	(4/3)πr³
Hemisphere, radius r	Curved=2πr²; total=3πr²	(2/3)πr³
Pyramid	Base area + lateral area	(1/3)×base area×perpendicular height

For a cone, l=√(r²+h²). A pyramid’s lateral area depends on its face shapes; for a regular square pyramid of base side a and slant height l, lateral area=2al. A cube is a special cuboid with l=b=h=a.

Worked examples

Open-top cuboid box

A box is 8 cm×5 cm×3 cm. Volume=8×5×3=120 cm³. Its cardboard area without a lid is lb+2lh+2bh=40+48+30=118 cm².

Cone

A cone has radius 3 cm and height 4 cm, so l=5 cm. Volume=(1/3)π×9×4=12π cm³; curved surface area=π×3×5=15π cm².

Sphere and hemisphere

A sphere of radius 3 cm has surface area 36π cm² and volume 36π cm³. A hemisphere with the same radius has total area 27π cm² and volume 18π cm³.

Practice questions with answers

Find the volume of a cube of side 4 cm.

4³=64 cm³.

Find the volume of a cylinder with r=7 cm and h=10 cm. Use π=22/7.

πr²h=(22/7)×49×10=1,540 cm³.

A pyramid has base area 36 cm² and perpendicular height 9 cm. Find its volume.

(1/3)×36×9=108 cm³.

Practical investigation

Choose a clean box, can or ball and record its dimensions. Select the matching solid model, calculate volume and surface area, and state assumptions (for example, whether a can has two closed ends). Compare surface area when one dimension changes but volume is held approximately fixed.

Use square units for area and cubic units for volume. Convert 1 L=1000 cm³ only after the calculation.

These are original revision notes aligned to the CBSE Class IX 2026–27 curriculum. Use them with your prescribed NCERT book and your teacher’s directions. Official CBSE syllabus · Official NCERT textbook resource