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Chapter 10: Problem 13

Find the sum of the measures of the interior angles of each polygon. octagon

Short Answer

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

Step by step solution

01

Understanding the Problem

We need to find the sum of the interior angles of an octagon. An octagon is a polygon with 8 sides.
02

Identifying the Formula

The sum of the interior angles of a polygon with \( n \) sides can be found using the formula \((n-2) \times 180^\circ\), where \( n \) is the number of sides of the polygon.
03

Applying the Formula

Substitute 8 for \( n \) in the formula: \[(8-2) \times 180^\circ = 6 \times 180^\circ\]
04

Calculating the Sum

Calculate \(6 \times 180\) to find the sum of the interior angles. \[6 \times 180^\circ = 1080^\circ\]

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Key Concepts

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

Understanding Polygons
A polygon is a two-dimensional geometric figure made up of straight line segments that form a closed shape. These line segments are called sides or edges. Polygons are classified based on the number of sides they have. For example:
  • A triangle has 3 sides.
  • A quadrilateral has 4 sides.
  • A pentagon has 5 sides.
  • And so on...
These figures can have any number of sides, and each kind of polygon has its own unique properties. Understanding the basic concept of polygons is important as it provides a foundation for exploring more complex properties, like interior angles.
Exploring the Octagon
An octagon is a specific type of polygon that has eight sides. The most well-known octagon is the shape of a stop sign you might see at an intersection. The shape is symmetrical and has equal-length sides in a regular octagon, meaning every side and angle are the same size. It's helpful to visualize or draw an octagon while solving problems about them, as this aids in understanding its properties, such as symmetrical balance and uniformity.
Sum of Interior Angles
The sum of the interior angles of any polygon depends on the number of sides it has. This is given by the formula: \[(n-2) \times 180^\circ\] where \(n\) is the number of sides in the polygon. This formula derives from the fact that any polygon can be divided into triangles, and each triangle has an interior angle sum of \(180^\circ\). For any polygon, you subtract 2 from the number of sides to determine how many triangles can be formed. This formula is instrumental in determining the total degree sum of a polygon鈥檚 interior angles.
Performing Angle Calculation
To calculate the sum of the interior angles of an octagon, which is an 8-sided polygon, the formula is applied as follows:
  • First, substitute 8 for \(n\) into the formula \((n-2) \times 180^\circ\).
  • This equation becomes \((8-2) \times 180^\circ\).
  • Calculate \(6 \times 180^\circ\) which results in a total of \(1080^\circ\).
This means that the sum of all interior angles in an octagon is \(1080^\circ\). Such calculations are useful in various mathematical problems, as well as in geometric constructions and designs.

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