Finding the Regular Polygon with a 135° Interior Angle

The question asks us to identify a regular polygon based on the measure of its interior angle, which is given as 135°.

A regular polygon is a polygon that is both equilateral (all sides are equal) and equiangular (all interior angles are equal). The measure of each interior angle in a regular polygon with n sides can be calculated using the following formula:

$\text{Interior Angle} = \frac{(n-2) \times 180^\circ}{n}$

We are given that the interior angle is 135°. We can set up an equation using the formula and solve for n, the number of sides:

$135^\circ = \frac{(n-2) \times 180^\circ}{n}$

Now, let's solve this equation for n step-by-step:

1. Multiply both sides of the equation by n to remove the denominator:

   $135n = (n-2) \times 180$

2. Distribute the 180 on the right side:

   $135n = 180n - 360$

3. Subtract $135n$ from both sides:

   $0 = 180n - 135n - 360$

   $0 = 45n - 360$

4. Add 360 to both sides:

   $360 = 45n$

5. Divide both sides by 45 to find the value of n:

   $n = \frac{360}{45}$

   $n = 8$

Since n represents the number of sides, and we found $n=8$, the polygon has 8 sides. A polygon with 8 sides is known as an octagon.

Therefore, a regular polygon with an interior angle of 135° is an octagon.

Checking Other Polygon Options

We can quickly calculate the interior angles for the polygons listed in the options to verify:

• Hexagon (6 sides): Interior Angle = $\frac{(6-2) \times 180}{6} = \frac{4 \times 180}{6} = \frac{720}{6} = 120^\circ$
• Square (4 sides): Interior Angle = $\frac{(4-2) \times 180}{4} = \frac{2 \times 180}{4} = \frac{360}{4} = 90^\circ$
• Pentagon (5 sides): Interior Angle = $\frac{(5-2) \times 180}{5} = \frac{3 \times 180}{5} = \frac{540}{5} = 108^\circ$
• Octagon (8 sides): Interior Angle = $\frac{(8-2) \times 180}{8} = \frac{6 \times 180}{8} = \frac{1080}{8} = 135^\circ$

Our calculation confirms that the octagon is the correct polygon.

Revision Table: Regular Polygon Properties

Additional Information on Regular Polygon Angles

Besides the interior angle, other angles associated with a regular polygon include the exterior angle and the sum of interior angles.

• Exterior Angle: The exterior angle of a regular polygon is the angle formed by one side and the extension of an adjacent side. The sum of the exterior angles of any convex polygon (including regular ones) is always 360°. For a regular polygon with n sides, each exterior angle measures $\frac{360^\circ}{n}$. Note that the interior angle and the corresponding exterior angle at a vertex are supplementary (they add up to 180°). Using this, we could also find n: Exterior angle = $180^\circ - 135^\circ = 45^\circ$. Then $n = \frac{360^\circ}{45^\circ} = 8$.
• Sum of Interior Angles: The sum of all interior angles in any polygon with n sides (regular or irregular) is given by the formula $(n-2) \times 180^\circ$. For a regular polygon, since all interior angles are equal, each angle is simply this sum divided by the number of sides, n, which brings us back to our original formula.

Understanding these formulas helps in solving various problems related to regular polygons and their angles.