Equating the Exterior Angle Measure

Oct 13, 2025 by pinkandtrash.com 91 views

Solving for the Value of x in a Regular Decagon

A regular decagon is a polygon with 10 equal sides and 10 equal angles. Each exterior angle of a regular decagon has a measure of ${$ (3x + 6)^{\circ}$}$. In this article, we will solve for the value of ${$ x$}$ using the properties of regular polygons.

A regular polygon is a polygon with equal sides and equal angles. The sum of the exterior angles of any polygon is always 360 degrees. Since a regular decagon has 10 equal exterior angles, each exterior angle measures ${$ \frac{360}{10} = 36^{\circ}$}$.

Equating the Exterior Angle Measure

We are given that each exterior angle of a regular decagon has a measure of ${$ (3x + 6)^{\circ}$}$. Since we know that each exterior angle measures 36 degrees, we can set up an equation:

${$ (3x + 6)^{\circ} = 36^{\circ}$]

To solve for x, we need to isolate x on one side of the equation. We can start by subtracting 6 from both sides of the equation:

[$3x + 6 - 6 = 36 - 6$]

This simplifies to:

[$3x = 30$]

Next, we can divide both sides of the equation by 3:

[$\frac{3x}{3} = \frac{30}{3}$]

This simplifies to:

[$x = 10$]

In this article, we solved for the value of x in a regular decagon. We used the properties of regular polygons and the fact that each exterior angle measures 36 degrees to set up an equation. By solving for x, we found that [ $x = 10\$}$ .

The correct answer is:

B. ${$ x = 10$}$

A regular decagon is a polygon with 10 equal sides and 10 equal angles.
The sum of the exterior angles of any polygon is always 360 degrees.
Each exterior angle of a regular decagon measures 36 degrees.
The value of x is 10.
Frequently Asked Questions (FAQs) about Regular Decagons and the Value of x

A: A regular decagon is a polygon with 10 equal sides and 10 equal angles. It is a type of regular polygon, which means that all its sides and angles are equal.

A: The sum of the exterior angles of any polygon is always 360 degrees. Since a regular decagon has 10 equal exterior angles, each exterior angle measures 36 degrees.

A: To find the measure of each exterior angle of a regular decagon, you can use the formula:

[$\frac{360}{n}$]

where n is the number of sides of the polygon. In this case, n = 10, so the measure of each exterior angle is:

[$\frac{360}{10} = 36^{\circ}$]

A: To solve for the value of x in a regular decagon, you can use the equation:

[$(3x + 6)^{\circ} = 36^{\circ}$]

By solving for x, you can find the value of x, which is 10.

A: The exterior angle of a polygon is related to the number of sides of the polygon. The sum of the exterior angles of any polygon is always 360 degrees. Since a regular decagon has 10 equal exterior angles, each exterior angle measures 36 degrees.

A: Yes, a regular decagon can be represented by a polygon with 10 equal sides and 10 equal angles. For example, a regular decagon can be drawn with each side measuring 10 units and each angle measuring 36 degrees.

A: Regular decagons have several real-world applications, including:

Architecture: Regular decagons are used in the design of buildings and bridges.
Art: Regular decagons are used in the creation of geometric patterns and designs.
Science: Regular decagons are used in the study of geometry and trigonometry.

A: Yes, here is a step-by-step solution to the problem of finding the value of x in a regular decagon:

Set up the equation: [$(3x + 6)^{\circ} = 36^{\circ}$]
Subtract 6 from both sides of the equation: [$3x + 6 - 6 = 36 - 6$]
Simplify the equation: [$3x = 30$]
Divide both sides of the equation by 3: [$\frac{3x}{3} = \frac{30}{3}$]
Simplify the equation: [$x = 10$]

In this article, we answered some frequently asked questions about regular decagons and the value of x. We covered topics such as the definition of a regular decagon, the sum of the exterior angles of a regular decagon, and the relationship between the exterior angle and the number of sides of a polygon. We also provided a step-by-step solution to the problem of finding the value of x in a regular decagon.