Surface Area and Volume (Cylinder, Cone, Sphere)

Chapter in a nutshell: Find the curved/total surface area and volume of the cylinder, cone, sphere and hemisphere, and of combined or melted/recast solids. When a solid is melted and recast, its volume is conserved.

1. Key Ideas

• CSA = curved surface area; TSA = total surface area (CSA + flat faces); Volume = space occupied.
• Recasting/melting: volume stays the same → equate volumes to find unknowns.
• Combined solids: add the appropriate CSAs (not the hidden faces) for surface area; add volumes for volume.

2. Formulas (current chapter — all of them)

• CSA $=2\pi r h$ · TSA $=2\pi r(r+h)$ · Volume $=\pi r^2 h$
• Hollow cylinder (R outer, r inner): Volume $=\pi(R^2-r^2)h$; TSA $=2\pi(R+r)(h+R-r)$.

Cone (radius r, height h, slant $l$):

• Slant height $l=\sqrt{r^2+h^2}$
• CSA $=\pi r l$ · TSA $=\pi r(l+r)$ · Volume $=\dfrac{1}{3}\pi r^2 h$

Sphere (radius r):

• Surface area $=4\pi r^2$ · Volume $=\dfrac{4}{3}\pi r^3$

Hemisphere (radius r):

• CSA $=2\pi r^2$ · TSA $=3\pi r^2$ · Volume $=\dfrac{2}{3}\pi r^3$

3. Prerequisite Formulas (earlier classes — in full)

• Square: area $=a^2$, perimeter $=4a$.
• Rectangle: area $=l\times b$, perimeter $=2(l+b)$.
• Triangle: area $=\tfrac12\times\text{base}\times\text{height}$; equilateral $=\tfrac{\sqrt3}{4}a^2$.
• Circle: area $=\pi r^2$, circumference $=2\pi r$.
• Parallelogram: $=\text{base}\times\text{height}$; trapezium $=\tfrac12(a+b)h$.

3-D (cube & cuboid):

• Cube: volume $=a^3$, TSA $=6a^2$, diagonal $=a\sqrt3$.
• Cuboid: volume $=l b h$, TSA $=2(lb+bh+hl)$, diagonal $=\sqrt{l^2+b^2+h^2}$.
• Use $\pi=\dfrac{22}{7}$ or 3.14 as told.

4. Worked Example (one, for the method)

5. Common Mistakes to Avoid

• Using height instead of slant height in a cone's CSA (CSA uses $l$).
• Forgetting the $\tfrac13$ in a cone's volume or $\tfrac43$ in a sphere's volume.
• Adding the hidden/joined faces when finding the surface area of a combined solid.
• Unit slips — surface area in cm², volume in cm³; convert litres (1 L = 1000 cm³).
• Using $2\pi r^2$ (hemisphere CSA) when TSA $3\pi r^2$ is needed.

6. Likely Exam Questions (with crisp answers)

1. CSA and volume of a cylinder? → $2\pi rh$ and $\pi r^2h$.
2. Slant height of a cone? → $\sqrt{r^2+h^2}$.
3. CSA and volume of a cone? → $\pi rl$ and $\tfrac13\pi r^2h$.
4. Surface area and volume of a sphere? → $4\pi r^2$ and $\tfrac43\pi r^3$.
5. TSA and volume of a hemisphere? → $3\pi r^2$ and $\tfrac23\pi r^3$.
6. What is conserved when a solid is melted and recast? → Its volume.
7. Volume of a cuboid and a cube? → $lbh$ and $a^3$.
8. TSA of a cylinder? → $2\pi r(r+h)$.
9. Volume of a hollow cylinder? → $\pi(R^2-r^2)h$.
10. 1 litre equals how many cm³? → 1000 cm³.
11. TSA of a cube? → $6a^2$.
12. Area of an equilateral triangle of side a? → $\frac{\sqrt3}{4}a^2$.

Extended & Prerequisite Formulas (full)

• Cylinder CSA 2πrh, TSA 2πr(r+h), V πr²h; hollow V π(R²−r²)h.
• Cone l=√(r²+h²), CSA πrl, TSA πr(l+r), V ⅓πr²h.
• Sphere SA 4πr², V (4/3)πr³; Hemisphere CSA 2πr², TSA 3πr², V (2/3)πr³.
• Prereq: cube V a³ TSA 6a²; cuboid V lbh TSA 2(lb+bh+hl) diag √(l²+b²+h²); circle πr²,2πr; triangle ½bh; equilateral (√3/4)a²; 1 L = 1000 cm³.