Surface Areas and Volumes of the Basic Solids

The Five Basic Solids

Everything in this chapter is built from just five shapes you met in Class 9: the cuboid, cube, cylinder, cone, sphere and hemisphere. Master their formulas and the rest of the chapter is simply adding and subtracting.

Two words we use constantly:

• CSA = Curved (Lateral) Surface Area — the "side" surface only.
• TSA = Total Surface Area — every face, including flat tops and bottoms.

Key Point: For a solid with a flat base (cylinder, cone, hemisphere), TSA $=$ CSA $+$ the area of the flat face(s). A sphere has no flat face, so its "CSA" and "TSA" are the same.

Cuboid and Cube

Cuboid with length $l$, breadth $b$, height $h$:

• Volume $= l\,b\,h$
• Total surface area $= 2(lb + bh + hl)$
• Lateral surface area (4 walls) $= 2h(l+b)$
• Diagonal $= \sqrt{l^2+b^2+h^2}$

Cube with edge $a$ (a cuboid with all sides equal):

• Volume $= a^3$
• Total surface area $= 6a^2$
• Lateral surface area $= 4a^2$
• Diagonal $= a\sqrt3$

Cylinder and Cone

Cylinder with base radius $r$, height $h$:

• Volume $= \pi r^2 h$
• Curved surface area $= 2\pi r h$
• Total surface area $= 2\pi r(r + h)$

Cone with base radius $r$, height $h$, slant height $l$:

• Slant height $l = \sqrt{r^2 + h^2}$ (from Pythagoras)
• Volume $= \dfrac13 \pi r^2 h$
• Curved surface area $= \pi r l$
• Total surface area $= \pi r(l + r)$

Key Point: A cone's volume is exactly one-third of the cylinder on the same base and height. Always find $l$ from $r$ and $h$ before using $\pi r l$.

Sphere and Hemisphere, and a Ready Table

Sphere of radius $r$: Surface area $= 4\pi r^2$; Volume $= \dfrac43\pi r^3$.

Hemisphere of radius $r$ (half a sphere): CSA $= 2\pi r^2$; TSA $= 3\pi r^2$ (curved $+$ the flat circular face $\pi r^2$); Volume $= \dfrac23\pi r^3$.

Solid	CSA / Curved	TSA	Volume
Cuboid	$2h(l+b)$	$2(lb+bh+hl)$	$lbh$
Cube	$4a^2$	$6a^2$	$a^3$
Cylinder	$2\pi rh$	$2\pi r(r+h)$	$\pi r^2 h$
Cone	$\pi rl$	$\pi r(l+r)$	$\tfrac13\pi r^2 h$
Sphere	$4\pi r^2$	$4\pi r^2$	$\tfrac43\pi r^3$
Hemisphere	$2\pi r^2$	$3\pi r^2$	$\tfrac23\pi r^3$

[Board Important] Memorise this table cold — most of the chapter is choosing the right rows and combining them.

Solved Examples

Example 1: Cylinder

Find the volume and total surface area of a cylinder of radius 7 cm and height 10 cm. $\left(\pi=\dfrac{22}{7}\right)$

Solution:

1. Volume $= \pi r^2 h = \dfrac{22}{7}\times 49\times 10 = 1540$ cm$^3$.
2. TSA $= 2\pi r(r+h) = 2\times\dfrac{22}{7}\times 7\times(7+10) = 44\times 17 = 748$ cm$^2$.

Final Answer: Volume $= 1540$ cm$^3$, TSA $= 748$ cm$^2$.

Example 2: Cone slant height

A cone has radius 6 cm and height 8 cm. Find its slant height, CSA and volume. $\left(\pi=3.14\right)$

Solution:

1. $l = \sqrt{r^2+h^2} = \sqrt{36+64} = \sqrt{100} = 10$ cm.
2. CSA $= \pi r l = 3.14\times 6\times 10 = 188.4$ cm$^2$.
3. Volume $= \dfrac13\pi r^2 h = \dfrac13\times 3.14\times 36\times 8 = 301.44$ cm$^3$.

Final Answer: $l=10$ cm, CSA $=188.4$ cm$^2$, Volume $=301.44$ cm$^3$.

Takeaway: $(6,8,10)$ is a Pythagorean triple — find $l$ first.

Example 3: Sphere and hemisphere

Find the surface area of a sphere of radius 7 cm, and the TSA of a hemisphere of the same radius. $\left(\pi=\dfrac{22}{7}\right)$

Solution:

1. Sphere: $4\pi r^2 = 4\times\dfrac{22}{7}\times 49 = 616$ cm$^2$.
2. Hemisphere TSA $= 3\pi r^2 = 3\times\dfrac{22}{7}\times 49 = 462$ cm$^2$.

Final Answer: Sphere $616$ cm$^2$; hemisphere TSA $462$ cm$^2$.

Takeaway: Hemisphere TSA is $3\pi r^2$ (not $2\pi r^2$) — it includes the flat circle.

Example 4: Cube

A cube has edge 5 cm. Find its volume, total surface area and diagonal.

Solution:

1. Volume $= a^3 = 125$ cm$^3$.
2. TSA $= 6a^2 = 6\times 25 = 150$ cm$^2$.
3. Diagonal $= a\sqrt3 = 5\sqrt3 \approx 8.66$ cm.

Final Answer: $125$ cm$^3$, $150$ cm$^2$, $5\sqrt3\approx 8.66$ cm.

Takeaway: Cube diagonal $= a\sqrt3$; face diagonal $= a\sqrt2$.