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Chapter 2: Problem 14

Find the number of sides that a polygon has if the sum of the measures of its interior angles is: a) \(1980^{\circ}\) b) \(2340^{\circ}\)

Short Answer

Expert verified
(a) 13 sides; (b) 15 sides

Step by step solution

01

Use the formula for the sum of interior angles

The sum of the interior angles of a polygon with n sides is given by the formula \( (n-2) \times 180^{\circ} \). We will use this formula to find n for both (a) and (b).
02

Solve for n using equation for (a)

We know that the sum of the interior angles is \(1980^{\circ}\). Set up the equation: \((n-2) \times 180^{\circ} = 1980^{\circ}\).
03

Simplify and solve equation for (a)

Divide both sides by \(180\): \(n-2 = \frac{1980}{180}\). Simplify the right side to get \(n-2 = 11\). Add 2 to both sides to get \(n = 13\).
04

Solve for n using equation for (b)

The sum of the interior angles is given as \(2340^{\circ}\). Set up the equation: \((n-2) \times 180^{\circ} = 2340^{\circ}\).
05

Simplify and solve equation for (b)

Divide both sides by \(180\): \(n-2 = \frac{2340}{180}\). Simplify the right side to get \(n-2 = 13\). Add 2 to both sides to get \(n = 15\).

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Key Concepts

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

Interior Angles
Interior angles are a fundamental concept in geometry, particularly when it comes to understanding polygons. These angles are formed where two sides of a polygon meet. For any given polygon, the interior angle at any vertex is the angle formed inside the shape.
  • In a triangle, which is the simplest polygon, the interior angles add up to \(180^{\circ}\).
  • As the number of sides increases, the sum of the interior angles also increases.
Recognizing the pattern of how these angles accumulate in more complex polygons is key. This understanding helps solve problems related to polygonal shapes, like determining the number of sides based on the sum of its interior angles.
Sum of Interior Angles Formula
The formula for calculating the sum of the interior angles of a polygon is \((n-2) \times 180^{\circ}\), where \(n\) represents the number of sides of the polygon. This formula is derived from the concept that any polygon can be divided into triangles.
Here鈥檚 why this formula works:
  • Each triangle has a sum of angles that is \(180^{\circ}\).
  • By dividing a polygon into triangles, we know there are \((n-2)\) triangles.
  • As a result, the sum of the interior angles can be calculated by multiplying the number of triangles by \(180^{\circ}\).
This formula is extremely useful in geometry to find missing angle measurements or to determine the properties of a polygon. Understanding this can help in solving a wide array of geometric problems.
Geometry Problem Solving
Problem-solving in geometry often requires a logical and methodical approach, especially when dealing with shapes and angles. For instance, if you are asked to find the number of sides of a polygon based on the sum of its interior angles, you would follow these steps:
  • First, use the sum of interior angles formula \((n-2) \times 180^{\circ} = \text{Sum of angles}\).
  • Rearrange the formula to solve for \(n\): \(n = \left( \frac{\text{Sum of angles}}{180} \right) + 2\).
  • Substitute the given sum of interior angles into the formula and perform the calculation.
By using this approach, solving geometry problems becomes structured and clear. Not only does this help in obtaining the correct answer, but it also deepens understanding and improves problem-solving skills in mathematics.

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Most popular questions from this chapter

Find the number of sides in a regular polygon whose exterior angles each measure: a) \(24^{\circ}\) b) \(18^{\circ}\)

The given statement is true. Write an equivalent (but more compact) statement that must be true. If no two sides of a quadrilateral (figure with four sides) are parallel, then the quadrilateral is not a trapezoid.

Is it possible for a regular polygon to have the following measures for each interior angle? a) \(96^{\circ}\) b) \(140^{\circ}\)

Find the number of sides that a regular polygon has if the measure of each interior angle is: a) \(108^{\circ}\) b) \(144^{\circ}\)

Draw a conclusion where possible. 1\. If two triangles are congruent, then the triangles are similar. 2\. Triangles \(A B C\) and \(D E F\) are not similar. c. \(\therefore ?\)
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