How to Find the Perimeter of Polygons? (+FREE Worksheet!)
Polygons are flat, closed figures made of straight line segments. They include triangles, squares, pentagons, and many other familiar shapes. On the GED, you need to know how to name polygons, calculate their interior angle sums, and work with regular polygons. This lesson covers it all.
What Is a Polygon?
A polygon is a closed, two-dimensional figure made of three or more straight sides. Key vocabulary:
- Vertices — the corner points where two sides meet
- Sides — the line segments forming the boundary
- Interior angles — angles formed inside the polygon at each vertex
- Regular polygon — all sides equal AND all interior angles equal
- Irregular polygon — sides or angles differ in size
Common Polygon Names
By Number of Sides
| Sides | Name | Interior Angle Sum |
|---|---|---|
| 3 | Triangle | 180° |
| 4 | Quadrilateral | 360° |
| 5 | Pentagon | 540° |
| 6 | Hexagon | 720° |
| 8 | Octagon | 1,080° |
| 10 | Decagon | 1,440° |
Interior Angle Sum Formula
For any polygon with n sides:
Sum of interior \(\color{blue}{\text{ angles } = (n – 2) \times 180}\)°
For a regular polygon, each interior angle equals:
Each \(\color{blue}{\text{ angle } = ((n – 2) \times 180^{\circ}) \div n}\)
- Square (\(\color{blue}{n=4}\)): each \(\color{blue}{\text{ angle } = ((4-2)\times 180) \div 4 = 360 \div 4}\) = 90°
- Regular pentagon (\(\color{blue}{n=5}\)): each \(\color{blue}{\text{ angle } = ((5-2)\times 180) \div 5 = 540 \div 5}\) = 108°
- Regular hexagon (\(\color{blue}{n=6}\)): each \(\color{blue}{\text{ angle } = ((6-2)\times 180) \div 6 = 720 \div 6}\) = 120°
Step-by-Step Summary
- Count the sides to identify the polygon name.
- Use \(\color{blue}{(n – 2) \times 180^{\circ}}\) to find the total interior angle sum.
- For a regular polygon, divide the sum by n to find each angle.
- For a missing angle in an irregular polygon, subtract the known angles from the total sum.
Watch: Math Antics — Polygons (Video Lesson)
Math Antics explains polygon names, properties, and the interior angle sum formula clearly:
Worked Examples
Example 1: What is the sum of the interior angles of a hexagon?
\(\color{blue}{(6 – 2) \times 180 = 4 \times 180 = 720^{\circ}}\)
Example 2: Each interior angle of a regular polygon is 120°. How many sides does it have?
Set up: \(\color{blue}{[(n-2)\times 180] / n = 120 \rightarrow (n-2)\times 180 = 120n \rightarrow 180n – 360 = 120n \rightarrow 60n = 360 \rightarrow n = 6}\). It is a regular hexagon.
Example 3: A pentagon has four angles of 100°, 110°, 120°, and 95°. Find the fifth angle.
Sum of pentagon \(\color{blue}{\text{ angles } = (5-2)\times 180 = 540}\)°.
Fifth \(\color{blue}{\text{ angle } = 540 – (100 + 110 + 120 + 95) = 540 – 425}\) = 115°.
Example 4: What is each interior angle of a regular octagon?
\(\color{blue}{[(8-2)\times 180] / 8 = [6\times 180] / 8 = \frac{1080}{8} = 135^{\circ}}\)
More Practice: Polygons (Video)
This Math Antics video dives deeper into polygon properties and real-world examples:
Exercises
- What is the sum of interior angles of a quadrilateral?
- Each angle of a regular polygon is 108°. Name the polygon.
- A hexagon has five angles of 110°, 125°, 130°, 115°, and 100°. Find the sixth angle.
- What is each interior angle of a regular decagon (10 sides)?
- How many sides does a polygon have if the sum of its interior angles is 1,080°?
- Is a stop sign (octagon) regular? What is each of its interior angles?
Answers
- \(\color{blue}{(4-2)\times 180}\) = 360°
- \(\color{blue}{108 = \frac{((n-2)\times 180)}{n}}\) → \(\color{blue}{n = 5}\); regular pentagon
- \(\color{blue}{\text{ Sum } = 720}\)°; sixth \(\color{blue}{\text{ angle } = 720 – (110+125+130+115+100) = 720 – 580}\) = 140°
- \(\color{blue}{\frac{((10-2)\times 180)}{10} = \frac{1440}{10}}\) = 144°
- \(\color{blue}{(n-2)\times 180 = 1080}\) → \(\color{blue}{n-2 = 6}\) → n = 8 sides (octagon)
- Yes, a regular octagon; each \(\color{blue}{\text{ angle } = \frac{1080}{8}}\) = 135°
Frequently Asked Questions
What is the difference between a regular and irregular polygon?
A regular polygon has all sides equal in length and all interior angles equal in measure. An irregular polygon has sides or angles that differ. A square is regular; a rectangle (unless it is a square) is irregular.
Are all quadrilaterals polygons?
Yes. All quadrilaterals — squares, rectangles, parallelograms, trapezoids, rhombuses — are polygons with exactly four sides.
What is the exterior angle sum of any polygon?
The sum of all exterior angles (one at each vertex) of any polygon is always 360°, regardless of the number of sides.
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