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

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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 $SA = 2(4·3) + 2(4·2) + 2(3·2) = 24 + 16 + 12 = 52\,cm^2$.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

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

A$2\pi rh$B$\pi r^2$C$2\pi rh + 2\pi r^2$D$\pi rh$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 cereal box has length 10 cm, width 5 cm, and height 8 cm.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

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

Challenge Quiz

Single Choice QuizAdvanced

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

A$\pi r l + \pi r^2$B$\pi r^2$C$\pi r l$D$2\pi r l$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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