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Chapter 5: Problem 82

Solve for the specified variable or expression. $$ S n=(n-2) 180 \text { for } n $$

Short Answer

Expert verified
The solution for \( n \) is \( n = \frac{S_n}{180} + 2 \).

Step by step solution

01

Understand the equation

The equation is given as \( S_n = (n-2) \times 180 \). This is a formula used to find the sum of the interior angles of a polygon with \( n \) sides.
02

Isolate the variable \( n \)

To solve for \( n \), we need to isolate it on one side of the equation. Start by dividing both sides of the equation by 180:\[ \frac{S_n}{180} = n - 2 \]
03

Solve for \( n \)

Add 2 to both sides of the equation to solve for \( n \):\[ n = \frac{S_n}{180} + 2 \]
04

Conclusion

The variable \( n \) is isolated, and the solution is \( n = \frac{S_n}{180} + 2 \).

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Key Concepts

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

Polygons
A polygon is a two-dimensional shape with straight sides. These sides must be connected, forming a closed figure. Simple polygons, like triangles and rectangles, are encountered most often in daily life. However, polygons can vary in complexity, including forms such as pentagons and hexagons.

Every polygon can be categorized based on the number of its sides, known as edges. For example:
  • A triangle has 3 sides.
  • A quadrilateral has 4 sides.
  • A pentagon has 5 sides.
  • A hexagon has 6 sides.
The number of sides directly impacts the polygon's properties, including the total measure of its interior angles. This relationship is essential to understanding and solving geometry problems.
Interior Angles
Interior angles are the angles formed inside a polygon by its sides. The sum of the interior angles of a polygon depends on the number of sides it possesses. This sum can be calculated using the formula \( S_n = (n-2) \times 180 \), where \( n \) represents the number of sides.

Let's break this down:
  • This formula accounts for the fact that a polygon can be divided into triangles from a single vertex, each contributing 180 degrees to the overall angle sum.
  • For a triangle (\( n=3 \)), the sum is \( (3-2) \times 180 = 180掳 \).
  • For a quadrilateral (\( n=4 \)), the sum is \( (4-2) \times 180 = 360掳 \).
Understanding this concept allows you to determine the sum of interior angles for any polygon efficiently. This knowledge is crucial when solving for unknowns in geometric problems.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations to solve for an unknown variable. In the context of polygons, this might mean solving for the number of sides \( n \) when given the sum of the interior angles \( S_n \).

Here is how you can manipulate the equation \( S_n = (n-2) \times 180 \):
  • First, isolate the term containing \( n \) by dividing both sides by 180: \( \frac{S_n}{180} = n - 2 \).
  • Next, solve for \( n \) by adding 2 to each side to get \( n = \frac{S_n}{180} + 2 \).
These steps illustrate basic algebraic manipulation, a fundamental skill in not only geometry but all mathematical problem-solving. Practicing these skills regularly will help you become more proficient in solving various types of equations.

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Most popular questions from this chapter

Solve each inequality. Graph the solution set and write it using interval notation. $$ |x|>7 $$

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