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Surface Area of a Sphere

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Surface Area of a Sphere

Study Guide

Key Definition

The surface area of a sphere is calculated using the formula $4\pi r^2$, where <a href="radius">$r$</a> is the radius of the sphere.

Important Notes

A sphere is a 3D object with all points on its surface equidistant from its center.
The surface area formula $4\pi r^2$ is derived from the lateral surface area of a cylinder.
Archimedes discovered the relationship between a sphere and a cylinder.
The diameter of a sphere is twice its radius: $d = 2r$.
The surface area helps in determining the amount of material needed to cover the sphere.

Mathematical Notation

$\pi$ represents the mathematical constant Pi, approximately 3.14159

<a href="radius">$r$</a> represents the radius of the sphere

$4\pi r^2$ is the formula for surface area of a sphere

Remember to use proper notation when solving problems

Why It Works

The formula for the surface area of a sphere is derived by comparing it to the lateral surface area of a cylinder with the same radius and height equal to the diameter of the sphere.

Remember

Always use $4\pi r^2$ to calculate the surface area of a sphere.

Quick Reference

Surface Area:$4\pi r^2$

Understanding Surface Area of a Sphere

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The surface area of a sphere is calculated using $4\pi r^2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the surface area of a sphere with <a href="radius">$r = 3$</a>?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are painting a spherical ball with a <a href="radius">$5$</a> cm.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If the surface area of a sphere is $314\pi$ cm², what is its radius?

Challenge Quiz

Single Choice Quiz

Advanced

A sphere and a cylinder have the same radius, and the cylinder's height is equal to the sphere's diameter. What is the ratio of their surface areas?

Recap

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Review key concepts and takeaways