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Chapter 3: Problem 4

For each polygon, find (a) the interior angle sum and (b) the exterior angle sum. Octagon

Short Answer

Expert verified
The interior angle sum of the octagon is 1080 degrees and the exterior angle sum is 360 degrees.

Step by step solution

01

Find the sum of the interior angles

To find the sum of the interior angles of the octagon, substitute \(n = 8\) in the interior angle sum formula: \((n-2)*180\). Calculate \((8-2)*180 = 6*180 = 1080\) degrees.
02

Find the sum of the exterior angles

The formula for the sum of the exterior angles of a polygon is constant and equals 360 degrees, irrespective of the number of sides of the polygon. Hence, the sum of the exterior angles of the octagon is 360 degrees.

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Key Concepts

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

Interior Angle Sum
Understanding the interior angle sum of a polygon is an essential concept in geometry. This sum is a key characteristic that helps us understand more about polygonal shapes.
To grasp this idea, remember this simple formula:
  • The interior angle sum is given by \((n - 2) \cdot 180\), where \(n\) represents the number of sides in the polygon.
For an octagon, which has 8 sides, you would calculate the interior angle sum by substituting \(n = 8\) into the formula:
  • i.e., \((8 - 2) \cdot 180 = 6 \cdot 180 = 1080\) degrees.
This tells us that the sum of all interior angles in an octagon is 1080 degrees.

This process becomes very handy, allowing you to quickly find the interior angle sum for any polygon, just by knowing the number of its sides.
Exterior Angle Sum
The sum of the exterior angles of a polygon is a marvelously simple yet powerful concept. Regardless of the number of sides, the sum of the exterior angles in any polygon always equals 360 degrees.
  • This concept is powerful because it is universal for all polygons.
  • No matter if it's a triangle or a dodecagon, this rule is the same.
For an octagon, or any polygon, you can remember that: The exterior angles add up to 360 degrees.
If you were to "walk" around the octagon, turning at each vertex to face the next side, those turns or "exterior angles" would ultimately create a full circle, which is why they sum to 360 degrees.
This consistent property makes exterior angles a reliable tool when solving problems related to polygons of any size.
Octagon Geometry
The geometry of an octagon is fascinating due to its versatility and symmetric properties. An octagon is an eight-sided polygon often seen in architecture and various designs.
  • Each interior angle in a regular octagon, which has all sides and angles equal, measures \(135\) degrees, calculated as \(\frac{1080}{8}\).
  • Understanding the angles is crucial, especially in real-world applications like stop signs in traffic, which are typically shaped as regular octagons.
In addition to angles, octagons are known for their symmetry.
  • They have rotational symmetry, which means that they look the same after a rotation of \(45\) degrees about their center.
  • They also exhibit reflectional symmetry, visible when the octagon is folded or reflected along its principal axes.
Learning about octagon geometry helps in recognizing symmetry and designing efficient structures, since octagons can pack tightly without gaps, such as in tiling patterns.

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Most popular questions from this chapter

Tell whether the reasoning process is deductive or inductive. By using the definitions of equilateral triangle (a triangle with three congruent sides) and of perimeter (the sum of the lengths of the sides of a figure), Katie concludes that the perimeter of every equilateral triangle is three times the length of a side.

Complete the table for convex polygons. $$\begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Number of sides } & 3 & 4 & 5 & 6 & 7 & 8 & n \\ \hline \text { Number of diagonals } & 0 & 2 & ? & ? & ? & ? & ? \\ \hline \end{array}$$

In pentagon \(P Q R S T, m \angle P=60\) and \(m \angle Q=130 . \angle S\) and \(\angle T\) are each three times as large as \(\angle R.\) a. Find the measures of \(\angle R, \angle S,\) and \(\angle T\) b. Which pair of sides of \(P Q R S T\) must be parallel?

Complete each statement with the word always, sometimes, or never. Two lines parallel to a third line are \(\underline{?}\) parallel to each other.

Accept the two statements as given information. State a conclusion based on deductive reasoning. If no conclusion can be reached, write none. Polygon G has more than 6 sides. Polygon G has fewer than 8 sides.
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