📐 Solid Secrets Unlocked! Mastering Volume & Surface Area

Understanding how to calculate volume and surface area is **crucial** for succeeding in O-Level exams. From designing water tanks to estimating ice-cream scoops, these skills connect classroom maths to real-world situations.

Key Formulas and Concepts

For every solid, remember that **volume measures the space inside** while **surface area measures the external skin**. For example, a cone’s volume is

V=\frac{1}{3}\pi r^2 h

and its curved surface area is

A=\pi r l

, where

l

is the slant height.

Pyramid: $V=\dfrac{1}{3}(\text{base area})\times\text{height}$
Sphere: $V=\dfrac{4}{3}\pi r^3$ , $A=4\pi r^2$

Worked Example: Ice-Cream Cone with Hemispherical Scoop

Problem: A cone of radius

4\,\text{cm}

and height

10\,\text{cm}

is topped with a hemisphere of the same radius. Find the total volume of the composite solid, leaving

\pi

in your answer until the final step.

Step 1:
$V_{\text{cone}}=\frac{1}{3}\pi r^2 h=\frac{1}{3}\pi(4)^2(10)=\frac{160}{3}\pi\,\text{cm}^3$
Step 2:
$V_{\text{hemisphere}}=\frac{2}{3}\pi r^3=\frac{2}{3}\pi(4)^3=\frac{128}{3}\pi\,\text{cm}^3$
Step 3:
$V_{\text{total}}=V_{\text{cone}}+V_{\text{hemisphere}}=\frac{160}{3}\pi+\frac{128}{3}\pi=96\pi\,\text{cm}^3\approx301.6\,\text{cm}^3$

Practice combining these formulas with care, and you’ll handle any composite solid the exam throws at you. **Keep practising** to build confidence!