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Volume

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Volume

Study Guide

Key Definition

Volume is the amount of space a 3-dimensional object occupies, measured in cubic units like $cm^3$ or $m^3$.

Important Notes

Ensure all measurements are in the same unit before calculating volume.
Prisms: $V = B \cdot h$
Cubes: $V = s^3$
Cylinders: $V = \pi r^2 h$
Spheres: $V = \frac{4}{3} \pi r^3$

Mathematical Notation

$\pi$ is approximately 3.14159

$V$ represents volume

$r$ represents radius

$h$ represents height

$B$ represents the area of the base

Remember to use proper notation when solving problems

Why It Works

Volume formulas are derived by slicing shapes into layers: a cylinder consists of circular cross-sections of area $\pi r^2$ stacked along its height $h$, giving $V = \pi r^2 h$; prisms multiply base area $B$ by height $h$ for $V = B \cdot h$; and spheres sum infinitesimal disks via integration to yield $V = \frac{4}{3} \pi r^3$.

Remember

Always check units and use the appropriate $key formulas$.

Quick Reference

Prism:$V = B \cdot h$

Cube:$V = s^3$

Cylinder:$V = \pi r^2 h$

Sphere:$V = \frac{4}{3} \pi r^3$

Understanding Volume

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Volume is the space a shape occupies, calculated with $V = B \cdot h$ for prisms.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the volume formula for a cylinder?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Calculate the volume of a cylindrical water bottle with radius $5 \, cm$ and height $20 \, cm$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a cube has a side length of $4 \, cm$, what is its volume?

Challenge Quiz

Single Choice Quiz

Advanced

What is the volume of a sphere with radius $r$?

Recap

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Review key concepts and takeaways