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Chapter 1: Problem 22

The perimeter of an equilateral octagon is \(42 \mathrm{cm}\) Find the length of each side.

Short Answer

Expert verified
Each side is 5.25 cm.

Step by step solution

01

Understand the Problem

An equilateral octagon means all eight sides are of equal length. The perimeter of an octagon is the sum of the lengths of all its sides.
02

Use the Formula for Perimeter

The formula for the perimeter of an equilateral octagon is given by: \[ P = 8s \]where \(P\) is the perimeter and \(s\) is the length of each side. In this case, \(P = 42 \) cm.
03

Solve for the Side Length

To find the length of each side, rearrange the formula to solve for \(s\):\[ s = \frac{P}{8} \]Substitute \(P = 42 \mathrm{cm}\) into the equation:\[ s = \frac{42}{8} \]
04

Calculate the Side Length

Divide 42 by 8 to find the length of each side:\[ s = 5.25 \mathrm{cm} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

equilateral octagon
An equilateral octagon is an eight-sided polygon where all sides are of equal length. This unique characteristic greatly simplifies the calculation of geometric properties like the perimeter. Instead of needing to measure and sum different side lengths, you only need to consider the length of one side. Equilateral shapes, like this octagon, are often found in tiling and decorative patterns because their uniform sides make for smooth tessellation. Visualize an octagon like a stop sign, but with the added feature that all edges are exactly the same length.
perimeter
The perimeter is the total distance around a two-dimensional shape. For an equilateral octagon, this means summing the lengths of all eight sides. Fortunately, with all sides being equal, you can use a simple formula to find the perimeter:
  • The formula is \( P = 8s \), where \( P \) represents the perimeter and \( s \) represents the side length.
  • Given the perimeter, this formula can also be rearranged to find the side length if you know the total perimeter.
Understanding perimeter is crucial not just in classroom exercises but also in real-world applications like fencing a garden or framing a picture. Knowing how to calculate it quickly can save time and ensure accuracy in these tasks.
side length calculation
To solve for the side length of an equilateral octagon, you rearrange the perimeter formula to isolate the variable representing the side length. Here are the steps broken down:
  • Start with the perimeter formula for an equilateral octagon: \( P = 8s \)
  • To find \( s \), divide both sides of the equation by 8: \( s = \frac{P}{8} \)
In the given problem, the perimeter \( P \) is 42 centimeters. Substituting this into the formula gives:
  • \( s = \frac{42}{8} \)
  • When you do the division, you get: \( s = 5.25 \) cm
This step-by-step approach ensures you understand why the formula works and helps you apply it to other problems involving equilateral polygons.

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