Problem 30 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. The sum \(S\) of the measures of the interior angles of a polygon with \(n\) sides is \(S=180(n-2) .\) (a) Solve for \(n .\) (b) If \(S=3600^{\circ}\), how many sides does the polygon have?

Step-by-Step Solution

Answer

(a) \( n = \frac{S}{180} + 2 \); (b) 22 sides.

Step 1: Understand the formula

The sum of the interior angles of a polygon is given by the formula \( S = 180(n-2) \), where \( n \) is the number of sides of the polygon.

Step 2: Solve part (a): Isolate n

We start with the equation \( S = 180(n-2) \). To solve for \( n \), first divide both sides by 180: \[ \frac{S}{180} = n-2. \] Next, add 2 to both sides to isolate \( n \): \[ n = \frac{S}{180} + 2. \] This formula now expresses \( n \) in terms of \( S \).

Step 3: Solve part (b): Substitute S = 3600

Use the formula derived in part (a), \( n = \frac{S}{180} + 2 \). Substitute \( S = 3600 \) into the equation: \[ n = \frac{3600}{180} + 2. \] Simplify the division: \( \frac{3600}{180} = 20 \). Thus, \( n = 20 + 2 = 22 \).

Step 4: Conclusion

For a polygon with a sum of interior angles \( S = 3600^{\circ} \), it must have 22 sides, meaning it is a 22-sided polygon.

Key Concepts

Understanding Interior Angles in Polygons

Interior angles are the angles found inside a polygon, where two sides meet. These angles are crucial for understanding the shape and structure of the polygon. To calculate the sum of the interior angles of any polygon, there's a simple formula: \( S = 180(n-2) \).Here's a quick breakdown:

• The symbol \( S \) represents the sum of the interior angles.
• \( n \) denotes the number of sides in the polygon.
• By subtracting 2 from \( n \), you are essentially accounting for the number of triangles that can be formed within the polygon. Each triangle contributes 180 degrees to the sum.

For instance, in a triangle (a polygon with 3 sides), the formula gives:\( S = 180(3-2) = 180^{\circ} \).This confirms our knowledge that a triangle's interior angles sum up to 180 degrees. From triangles to decagons, this formula applies universally to any polygon configuration.

The Relationship Between Polygon Sides and Their Angles

The number of sides in a polygon, denoted by \( n \), directly influences the measure of its interior angles. Intuitively, as the number of sides increases, the magnitude of each interior angle increases, though the increase is not linear. To illustrate, a polygon with a low number of sides, such as a triangle, has relatively sharp angles. Conversely, in a dodecagon (a 12-sided figure), each angle is much larger as the shape becomes closer to a circle.Determining the number of sides using the sum of interior angles can be insightful:

• If the sum \( S \) is given, you can find \( n \) with the revised equation derived from the interior angle sum formula: \( n = \frac{S}{180} + 2 \).
• For a polygon where \( S = 3600^{\circ} \), plug this into the formula to find \( n \):\( n = \frac{3600}{180} + 2 = 22 \).

This tells us that if a polygon’s interior angles sum to 3600 degrees, it has 22 sides, evidencing how these geometric properties are interlinked.

Applying Mathematics Formulas in Geometry

Mathematical formulas offer us a bridge to explore, connect, and solve problems across different domains of geometry. For polygons, the formula \( S = 180(n-2) \) shows how powerful such tools can be; it enables us to explore intricate shapes, from understanding fundamental triangles to complex icosagons (20-sided figures).Using this formula:

• We can solve questions related to the sum of angles effortlessly by substituting known values.
• Rearranging it to \( n = \frac{S}{180} + 2 \) allows us to find out how many sides a shape has if the sum of its angles is known.

These kinds of formulas help streamline calculations and foster a deeper understanding of geometric principles. Being comfortable using them not only helps solve textbook problems but also lays a foundational skill set for tackling real-world architectural, engineering, and graphical tasks.