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Chapter 11: Problem 23

What is the sum of the measures of the interior angles of a dodecagon (12 sides)?

Short Answer

Expert verified
The sum of the interior angles of a dodecagon is 1800 degrees.

Step by step solution

01

Understand the Formula

To find the sum of the interior angles of any polygon, we use the formula: \(S = (n - 2) \times 180\) degrees, where \(n\) is the number of sides of the polygon.
02

Identify the Number of Sides

A dodecagon is a polygon with 12 sides. Thus, \(n = 12\).
03

Plug Values into the Formula

Substitute the value of \(n\) into the formula: \(S = (12 - 2) \times 180\).
04

Perform the Calculation

Calculate inside the parentheses: \(12 - 2 = 10\). Then multiply the result by 180: \(10 \times 180 = 1800\).
05

Conclusion

The sum of the interior angles of a dodecagon is 1800 degrees.

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Key Concepts

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

Sum of Interior Angles Formula
Calculating the sum of the interior angles of a polygon is a fundamental concept in geometry. The formula used is \( S = (n - 2) \times 180 \) degrees, where \( n \) represents the number of sides of the polygon. Here's how it works:
This formula is derived from dividing the polygon into triangles. Each triangle has an angle sum of 180 degrees. Since any polygon can be divided into \( n-2 \) triangles, multiplying this by 180 gives us the desired sum of interior angles.
For example, in a dodecagon, which has 12 sides, substituting \( n = 12 \) into the formula gives \( S = (12 - 2) \times 180 = 1800 \) degrees.
Polygon Angles
A polygon is a flat shape with straight sides that are fully closed. Angles in polygons can be classified into interior and exterior angles. Focusing on interior angles:
  • The interior angles are the angles inside the polygon at each vertex where two sides meet.
  • The sum of these interior angles depends on the number of sides the polygon has.
As the number of sides increases, so does the sum of the interior angles. For example, in a triangle (3 sides), the angles sum up to 180 degrees. In a quadrilateral, the sum is 360 degrees, and as seen, a dodecagon totals 1800 degrees.
Geometry Calculation
Performing geometry calculations involves understanding and correctly using formulas. Here鈥檚 a concise guide on how you approach calculations involving polygons:
  • First, identify the problem and what you need to find, in this case, the sum of interior angles.
  • Second, determine the relevant formula. For any polygon, it's \( S = (n - 2) \times 180 \) for interior angles.
  • Next, substitute known values into your formula. In a dodecagon, substitute 12 for \( n \).
  • Then, solve the equation step by step, performing any arithmetic needed, like subtraction and multiplication in this formula.
  • Finally, interpret the result. In this exercise, you find that the sum of the interior angles of a dodecagon is 1800 degrees.
Taking a step-by-step approach reduces errors and ensures a deep understanding of each calculation.

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