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Chapter 2: Problem 19

What is the measure of each interior angle of a stop sign? (IMAGES CANNOT COPY).

Short Answer

Expert verified
Each interior angle of a stop sign is 135 degrees.

Step by step solution

01

Identify Shape and Number of Sides

A stop sign is shaped like a regular octagon. An octagon has 8 sides.
02

Use Formula for the Sum of Interior Angles

The formula for the sum of the interior angles of a polygon with \( n \) sides is \( (n - 2) \times 180 \). For an octagon, \( n = 8 \), so the sum is \( (8 - 2) \times 180 = 6 \times 180 = 1080 \) degrees.
03

Calculate One Interior Angle of the Regular Polygon

In a regular polygon, all interior angles are equal. Therefore, one interior angle of a regular octagon is the sum of the interior angles divided by the number of sides: \( \frac{1080}{8} = 135 \) degrees.

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Key Concepts

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

Octagon
An octagon is a type of polygon that consists of eight sides. In the world around us, a classic example of an octagon is the shape of a stop sign.
This geometric figure is common in many designs due to its symmetrical nature.
An octagon can either be regular or irregular. A regular octagon means all its sides and interior angles are equal. This uniformity gives it a pleasing aesthetic which is often utilized in symmetric patterns and architectural designs.
  • Regular octagon: Equal sides and angles.
  • Irregular octagon: Unequal sides or angles.
The stop sign octagon is a regular polygon, making it easier to calculate its geometric properties like the interior angles.
Sum of Interior Angles
The sum of the interior angles of any polygon can be determined by a specific formula.
This formula is \[ (n - 2) \times 180 \]where \( n \) is the number of sides of the polygon.
This rule applies to polygons with three or more sides.When it comes to an octagon, which has eight sides, the formula gives us:\[ (8 - 2) \times 180 = 1080 \text{ degrees} \]
This means that the total of all interior angles in an octagon adds up to 1080 degrees.
  • This calculation is critical when you need to determine if an octagon is regular, by checking if dividing this sum by the number of sides gives equal individual angles.
Regular Polygon
A regular polygon is fascinating because of its symmetry. Each side and angle is identical, making calculations straightforward.
In a regular polygon like our octagon (the stop sign), every interior angle can be figured out simply by dividing the total sum of the interior angles by the number of sides.
In our case, we take the sum 1080 degrees, and divide by the eight equal sides, to determine:\[ \frac{1080}{8} = 135 \text{ degrees} \]
So, each interior angle of a regular octagon measures 135 degrees.
  • Regular polygons are easier to manage in geometry because of their repeating patterns.
  • Their predictability makes them a popular choice in design and architecture.
This characteristic ensures that regular polygons are not only mathematically interesting but also practically useful.

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Most popular questions from this chapter

,j \| \mathrm{k}\( and \)\triangle \mathrm{ABC}\(.$$ siven: \)\quad \triangle A B C\( with \)m \angle B=m \angle C=\frac{x}{2}\( and$$\mathrm{m} \angle B A C=x$$. Given: \)\quad \triangle A B C\( with \)\mathrm{m} \angle B=\mathrm{m} \angle C=\frac{x}{2}\( and \)\mathrm{m} \angle B A C=x\( Find: \)\quad x, \mathrm{m} \angle B A C,\( and \)\mathrm{m} \angle B$

Give the indirect proof for each problem or statement. If \(x^{2} \neq 25,\) then \(x \neq 5\)

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Find the total number of diagonals for a polygon of \(n\) sides if: a) \(n=5\) b) \(n=10\)

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