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Chapter 1: Problem 22

Find the sum of the angles of an octagon (8-sided figure).

Short Answer

Expert verified
The sum of the angles of an octagon is 1080 degrees.

Step by step solution

01

Identify the Formula

To find the sum of the interior angles of a polygon, we use the formula: \( (n - 2) \times 180^\circ \), where \( n \) represents the number of sides in the polygon.
02

Substitute Values into the Formula

In this exercise, we have an octagon, which has 8 sides. Substitute \( n = 8 \) into the formula: \( (8 - 2) \times 180^\circ \).
03

Simplify the Expression

Perform the subtraction inside the parentheses: \( 8 - 2 = 6 \). Now, multiply the result by 180: \( 6 \times 180^\circ \).
04

Calculate the Sum of Angles

Multiply 6 by 180 to find the sum of the angles: \( 6 \times 180 = 1080^\circ \). Thus, the sum of the interior angles of an octagon is \( 1080^\circ \).

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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 polygon with eight sides. This simple definition lays the groundwork for understanding more complex geometric concepts. Octagons can be regular or irregular. If all sides and angles are equal, it's a regular octagon, like a stop sign. If not, it's irregular.
Understanding the octagon's structure helps us calculate its properties, such as angles and perimeter.
  • Regular Octagon: Where each interior angle is the same.
  • Irregular Octagon: The angles and side lengths differ.
Knowing whether an octagon is regular or irregular can simplify problems involving angle calculations by using symmetry in regular shapes.
Sum of Angles
The sum of the interior angles of a polygon is essential in geometry. For any polygon, the sum can be found using a basic formula. This relates to how we can break polygons down into simpler shapes.
The formula to determine the sum of interior angles is:
  • \[(n - 2) \times 180^\circ\]
Here, \(n\) is the number of sides. By subtracting 2, we calculate how many triangles can fit within the polygon. Each triangle has angles summing up to \(180^\circ\). Thus, for an octagon (\(n = 8\):
  • The sum is \((8 - 2) \times 180^\circ = 1080^\circ\).
Angle Calculation
Angle calculation involves using established formulas to determine the measures of angles within polygons. For octagons, knowing the sum of their angles allows for various computations.
In a regular octagon, all interior angles are equal, so you divide the total by the number of sides:
  • Interior Angle of a Regular Octagon: \[\frac{1080^\circ}{8} = 135^\circ\]
To verify or find angles in irregular octagons, one may need additional information about specific angles or side lengths. However, starting with the known sum (\(1080^\circ\) for octagons) is crucial, providing a reference for further geometry work.

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Most popular questions from this chapter

Draw and label each angle: \(\angle \mathrm{XIZ}=90^{\circ}\).

Draw and label each angle: \(\angle R S T=72^{\circ}\).

Draw and label each angle: \(\angle \mathbb{E H G}=60^{\circ}\).

Find \(x\) and \(A C\) if \(B\) is the midpoint of \(A C\) and \(A B=3(x-5)\) and \(B C=x+3\).

Find the supplement of an angle of (a) \(30^{\circ}\) (b) \(37^{\circ}\) (c) \(90^{\circ}\) (d) \(120^{\circ}\)
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