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Volume & Surface Area

Overview

Wrapping vs. Filling.

What is it?

Volume is the space inside a 3D shape. Surface Area is the total area of all outer faces.

History

Archimedes was so proud of proving the volume of a sphere is 2/3 of its cylinder that he requested the diagram be carved on his tombstone.

Key Idea

Prisms: V = BaseArea * Height. Pyramids/Cones: V = (BaseArea * Height) / 3.

Practice This Topic

Concept Guide

Plain English: Imagine a shape is a hollow container. Volume is how much water fits inside. Surface area is how much paint you need to cover the outside. Volume is 3D (cubic units), Surface Area is 2D (square units).

Real-world example: HVAC Engineering. Calculating the volume of a room to determine how powerful the Air Conditioner needs to be.

How to do it

  1. Identify the 'Base' shape (circle, square, etc).
  2. Calculate the Area of the Base (B).
  3. Identify the Vertical Height (h).
  4. Multiply B * h.
  5. If it comes to a point (cone/pyramid), divide by 3.
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Common Pitfall

Confusing Slant Height (side of a cone) with Vertical Height (straight up the middle). Volume always uses Vertical Height.

Word Problem
"A cylindrical swimming pool has a diameter of 10m and is 2m deep. How many cubic meters of water does it hold?"
Reasoning: Radius is 5m. Base Area (Circle) = πr² = 25π. Volume = Base * Height = 25π * 2 = 50π. Approx 157 cubic meters.

Practice Examples

Cylinder
______ / \ | | h \______/ r
V = πr²h