\text{Perimeter} = n \times b

\text{Sum of interior angles} = (n-2) \times 180^\circ

\text{Area} = \frac{1}{2} \times n \times b \times \sin\left(\frac{360}{n}\right)

\text{Interior angle} = \left(180 – \frac{360}{n} \right)^\circ

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\text{Exterior angle} = \frac{360}{n}^\circ

` ````
```Where:
n: number of polygon sides,
b: each length of polygon.

A regular polygon is a polygon in which all sides are of equal length and all angles are of equal measure. The number of sides an n-sided regular polygon has will determine its name. For example, a regular polygon with 3 sides is called a triangle, a regular polygon with 4 sides is called a square, and so on.

- Side length: the length of each side of the polygon

- Apothem: the distance from the center of the polygon to the midpoint of one of its sides

- Perimeter: the total distance around the outside of the polygon, calculated by multiplying the length of one side by the number of sides

- Interior angle: the angle formed inside the polygon at each vertex, calculated by 180*(n-2)/n degrees

- Exterior angle: the angle formed outside the polygon at each vertex, calculated by 360/n degrees

These parameters can be calculated and used to determine various properties and measurements of regular polygons.

Let's consider a regular polygon with n sides. Each side has a length of b units. The total perimeter of the polygon is the sum of all the side lengths.

The perimeter of a regular polygon with n sides can be calculated using the formula:

\text{Perimeter} = n \times b

The interior angles of a regular polygon with n sides can be calculated using the formula:

\text{Interior angle} = \left(180 - \dfrac{360}{n}\right)

The sum of the interior angles of a regular polygon with n sides can be calculated using the formula:

\text{Sum of interior angles} = (n-2) \times 180

The exterior angles of a regular polygon with n sides can be calculated using the formula:

\text{Exterior angle} = \dfrac{360}{n}

The area of a regular polygon with n sides can be calculated using the formula:

\text{Area} = \dfrac{n \times b^2}{4 \times \tan\left(\dfrac{\pi}{n}\right)}

These formulas can be used to calculate the properties of any regular polygon with n sides each of length b. Make sure to substitute the values of n and b into the respective formulas to find the desired property of the regular polygon.

Now, you have a better understanding of the properties of a regular polygon with n sides each of length b. Remember to use the formulas provided to calculate the perimeter, interior and exterior angles, and area of any regular polygon.

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Regular Polygon

Find the interior angle of a regular polygon with 6 sides.

180 - \frac{360}{n}

180 - \frac{360}{6}

180 - 60

120

Determine the exterior angle of a regular polygon with 8 sides.

\frac{360}{n}

\frac{360}{8}

45

Calculate the sum of the interior angles of a regular polygon with 5 sides.

(n-2) \times 180

(5-2) \times 180

3 \times 180

540

Find the measure of each exterior angle of a regular polygon with 12 sides.

\frac{360}{n}

\frac{360}{12}

30

Determine the number of sides of a regular polygon if the measure of each interior angle is 150 degrees.

180 - \frac{360}{n} = 150

\frac{360}{n} = 30

n = 12

More Worked Examples on Regular Polygon
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1. Find the interior angle of a regular hexagon.

2. Calculate the sum of the interior angles of a regular pentagon.

3. Determine the exterior angle of a regular octagon.

4. Find the measure of each interior angle of a regular heptagon.

5. Calculate the sum of the exterior angles of a regular nonagon.

6. Determine the interior angle of a regular decagon.

7. Find the measure of each exterior angle of a regular dodecagon.

8. Calculate the sum of the interior angles of a regular 20-gon.

9. Determine the exterior angle of a regular 15-gon.

10. Find the measure of each interior angle of a regular 7-gon.

11. Calculate the sum of the exterior angles of a regular 18-gon.

12. Determine the interior angle of a regular 30-gon.

13. Find the measure of each exterior angle of a regular 12-gon.

14. Calculate the sum of the interior angles of a regular 25-gon.

15. Determine the exterior angle of a regular 9-gon.

1. 120^\circ

2. 540^\circ

3. 45^\circ

4. 128.57^\circ

5. 360^\circ

6. 144^\circ

7. 30^\circ

8. 3240^\circ

9. 24^\circ

10. 128.57^\circ

11. 360^\circ

12. 156^\circ

13. 30^\circ

14. 4500^\circ

15. 40^\circ

Quick Answers for Regular Polygon
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