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Surface Area

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Surface Area

Study Guide

Key Definition

The total surface area of a three-dimensional solid figure is the sum of the areas of all of the faces or surfaces that enclose the figure.

Important Notes

Surface area is measured in $units^2$
Lateral surface area excludes the bases
For a cylinder, lateral surface area can be visualized as a rectangle
Use the formula $LSA = \frac{1}{2}p l$ for pyramids
The total surface area includes all faces

Mathematical Notation

$\times$ represents multiplication

$\div$ represents division

$\pi$ represents pi (≈3.14)

$r^2$ means r squared

$units^2$ indicates square units

$SA$ denotes surface area

$LSA$ denotes lateral surface area

Remember to use proper notation when solving problems

Why It Works

Surface area formulas work by unfolding the faces of a solid into flat regions. Each face’s area can be calculated using plane geometry and then summed. For example, the lateral area of a cylinder becomes a rectangle of width $2\pi r$ and height $h$.

Remember

Surface area formulas are essential for calculating the areas of three-dimensional figures.

Quick Reference

Cylinder (Total):$2\pi r^2 + 2\pi r h$

Cone (Total):$\pi r^2 + \pi r l$

Sphere (Total):$4\pi r^2$

Prism (Total):$2B + pL$

Pyramid (Lateral):$\frac{1}{2}pl$

Pyramid (Total):$\frac{1}{2}pl + B$

Understanding Surface Area

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The surface area of a 3D shape is the sum of its faces. For a rectangular prism, $SA = 2lw + 2lh + 2wh$. For example, if $l=4\,cm$, $w=3\,cm$, and $h=2\,cm$, then $SA = 2(4·3) + 2(4·2) + 2(3·2) = 24 + 16 + 12 = 52\,cm^2$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the lateral surface area of a cylinder with radius $r$ and height $h$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A cereal box has length 10 cm, width 5 cm, and height 8 cm.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine the surface area of a square pyramid with base side length 6 cm and slant height 5 cm.

Challenge Quiz

Single Choice Quiz

Advanced

Find the total surface area of a cone with radius $r$ and slant height $l$.

Recap

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Review key concepts and takeaways