Question

A regular polygon has equal angle measures and equal side lengths. For a regular \(n\)-sided polygon ( \(n \geq 3\) ), the measure \(a_n\) of an interior angle is given by \(a_n=\frac{180(n-2)}{n}\). a. Write the first five terms of the sequence. b. Write a rule for the sequence giving the sum \(T_n\) of the measures of the interior angles in each regular \(n\)-sided polygon. c. Use your rule in part (b) to find the sum of the interior angle measures in the Guggenheim Museum skylight, which is a regular dodecagon.

Short answer

a. The first five terms of the sequence are: 60, 90, 108, 120, 128.57. b. The rule for the sequence is \(T_n=180(n-2)\). c. The sum of the interior angle measures in a regular dodecagon is 1800 degrees.

Step-by-step solution

Formulating the First Five Terms

By using the given formula, the measures of the interior angles for the first five regular polygons (triangles, squares, and so on) can be found. The formula is applied as: \(a_n=\frac{180(n-2)}{n}\)for \(n=3, 4, 5, 6, 7\). This results in the sequence: \(60, 90, 108, 120, 128.57\).

Formulating the Rule for the Sequence

The sum \(T_n\) of the measures of the angles in a regular \(n\)-sided polygon is given by multiplying the measure of one angle by the number of sides, 'n'. Thus, \(T_n = n * a_n\)which simplifies to \(T_n = n * \frac{180(n-2)}{n}\).After simplifying, we obtain the rule as \(T_n = 180(n-2)\).

Application of the Rule to a Dodecagon

The rule from Step 2 is used to find the sum of the interior angle measures for a regular dodecagon. Substituting \(n=12\) into \(T_n=180(n-2)\), we find that the sum is 1800 degrees.

Key concepts

Interior Angle Measure Formula

Understanding the interior angle measure formula is essential for analyzing regular polygons. A regular polygon is a shape with equal side lengths and equal internal angles. For instance, imagine looking at a stop sign; it's an octagon — a type of regular polygon with all sides and angles being equal.

To determine the measure of an interior angle, we use the formula: \[\begin{equation}a_n=\frac{180(n-2)}{n}\end{equation}\]where \(a_n\) represents the interior angle measure, and \(n\) is the number of sides the polygon has. If we're dealing with a triangle, which has three sides (\(n=3\)), plugging into the formula gives us an angle measure of 60 degrees for each angle. Similarly, a square, with four sides (\(n=4\)), would have interior angles measuring 90 degrees each.

• For a triangle (\(n=3\)): \(60\) degrees.
• For a square (\(n=4\)): \(90\) degrees.
• For a pentagon (\(n=5\)): \(108\) degrees.
• And so on...

Understanding this formula helps students solve various geometry problems involving regular polygons and lays the foundation for more complex geometrical concepts.

Polygon Angle Sum Rule

The polygon angle sum rule provides a quick way to calculate the sum of internal angles in any polygon, whether it’s a simple triangle or a complex dodecagon. The rule states that this sum, which we denote as \(T_n\) for a polygon with \(n\) sides, can be found using the formula:\[\begin{equation}T_n = 180(n-2)\end{equation}\]This formula derives from understanding that any polygon can be divided into triangles. Since each triangle's angles sum up to 180 degrees, a polygon can be 'triangulated' to demonstrate how the sum of the interior angles is based on the number of triangles that fit within it.

The Triangulation Concept:

• A quadrilateral can be divided into 2 triangles, thus the angle sum is \(180(4-2) = 360\) degrees.
• A pentagon breaks down into 3 triangles, leading to \(180(5-2) = 540\) degrees.

Applying the polygon angle sum rule is not only efficient but also reinforces understanding of how polygons relate to triangles, one of the simplest geometric shapes.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers following a specific pattern. In the context of geometry and regular polygons, the sequence related to the interior angle measures is a fascinating example. Once we know the angle measure formula for a regular polygon, we can generate a sequence for the interior angles' measures for different-sized polygons.

As we’ve seen in our original problem's step-by-step solution, applying the interior angle measure formula for the first five regular polygons (\(n=3\) to \(n=7\)) yields a sequence: \(60, 90, 108, 120, 128.57\). This sequence isn’t arithmetic or geometric, but it reflects the pattern that as the number of sides increases, so do the measures of the interior angles, approaching but never reaching 180 degrees.

Patterns in Polygon Sequences:

Recognizing patterns in sequences allows us to predict values without direct calculation, which is precisely what the sum rule for interior angles offers for polygons. The exploration of sequences in polygon angles illustrates how patterns emerge in mathematics, providing insights into more profound mathematical structures and relationships.