Volume Of A Cone - Varsity Tutors

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Volume of a Cone

Study Guide

Key Definition

The volume of a cone is calculated as $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height.

Important Notes

  • Ensure all measurements are in the same unit before calculating volume.
  • The formula $V = \frac{1}{3} \pi r^2 h$ applies to both right and oblique cones.
  • The volume of a cylinder with the same height is three times the volume of a cone.
  • Volume is measured in cubic units like $cm^3$ or $m^3$.
  • A cone has one circular base and a curved surface.

Mathematical Notation

$\pi$ is approximately 3.14159$r$ represents the radius$h$ represents the height$V$ represents volume$\frac{1}{3}$ signifies one-thirdRemember to use proper notation when solving problems

Why It Works

The formula for the volume of a cone is derived from the proportional relationship between the volumes of a cone and a cylinder with the same base and height.

Remember

The volume formula $V = \frac{1}{3} \pi r^2 h$ is key for solving cone volume problems.

Quick Reference

Volume Formula:$V = \frac{1}{3} \pi r^2 h$

Understanding Volume of a Cone

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

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Beginner Explanation

The volume of a cone is $V = \frac{1}{3} \pi r^2 h$, a simple way to calculate the space inside a cone.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

What is the volume of a cone with $r = 3 \text{ cm}$ and $h = 9 \text{ cm}$?

A$\frac{1}{3} \pi (3)^2 (9)$B$\pi (3)^2 (9)$C$2 \pi (3)^2 (9)$D$\frac{1}{2} \pi (3)^2 (9)$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

A cone-shaped cup has a radius of $5 \text{ cm}$ and a height of $12 \text{ cm}$. How much liquid can it hold?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

If a cone has a volume of $100 \text{ cm}^3$ and a radius of $4 \text{ cm}$, what is its height?

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

A right circular cone has a height three times its radius. If the volume is $150 \pi \text{ cm}^3$, what is the radius?

A$3 \text{ cm}$B$4 \text{ cm}$C$5 \text{ cm}$D$6 \text{ cm}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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