How to Name Polygons and Find Their Interior Angle Sums? (+FREE Worksheet!)

This lesson explains how to name polygons, calculate interior angle sums, and find each interior angle in regular polygons, with step-by-step examples and practice questions.

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

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

1. Count the sides to identify the polygon name.
2. Use \(\color{blue}{(n – 2) \times 180^{\circ}}\) to find the total interior angle sum.
3. For a regular polygon, divide the sum by n to find each angle.
4. 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

1. What is the sum of interior angles of a quadrilateral?
2. Each angle of a regular polygon is 108°. Name the polygon.
3. A hexagon has five angles of 110°, 125°, 130°, 115°, and 100°. Find the sixth angle.
4. What is each interior angle of a regular decagon (10 sides)?
5. How many sides does a polygon have if the sum of its interior angles is 1,080°?
6. Is a stop sign (octagon) regular? What is each of its interior angles?

Answers

1. \(\color{blue}{(4-2)\times 180}\) = 360°
2. \(\color{blue}{108 = \frac{((n-2)\times 180)}{n}}\) → \(\color{blue}{n = 5}\); regular pentagon
3. \(\color{blue}{\text{ Sum } = 720}\)°; sixth \(\color{blue}{\text{ angle } = 720 – (110+125+130+115+100) = 720 – 580}\) = 140°
4. \(\color{blue}{\frac{((10-2)\times 180)}{10} = \frac{1440}{10}}\) = 144°
5. \(\color{blue}{(n-2)\times 180 = 1080}\) → \(\color{blue}{n-2 = 6}\) → n = 8 sides (octagon)
6. Yes, a regular octagon; each \(\color{blue}{\text{ angle } = \frac{1080}{8}}\) = 135°

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Frequently Asked Questions

What is a polygon in geometry?

A polygon is a flat, closed figure made of three or more straight line segments that meet only at their endpoints. The segments are the sides, and the corner points are the vertices. Triangles, rectangles, pentagons and hexagons are all polygons. A circle is not one, because it has no straight sides, and an open shape like the letter V is not one because it does not close.

What are some examples of polygons?

Polygons are named by how many sides they have: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6, an octagon 8 and a decagon 10. A stop sign is a regular octagon, and the cells of a honeycomb are hexagons. Each name also tells you the interior angle sum; for a pentagon it is (5 − 2) × 180° = 540°.

Are triangles polygons?

Yes. A triangle is the simplest polygon, with three straight sides, three vertices and three interior angles that always add up to 180°. No polygon can have fewer than three sides. The triangle also explains the angle sum formula: any polygon with n sides can be split into n − 2 triangles from one vertex, so a hexagon splits into 4 triangles and its angles total 4 × 180° = 720°.

Are all quadrilaterals polygons?

Yes. Every quadrilateral is a polygon with four straight sides and four vertices, so squares, rectangles, rhombuses, trapezoids and kites all count. The interior angles of any quadrilateral add up to (4 − 2) × 180° = 360°. That fact solves many test questions: if three angles of a quadrilateral are 80°, 95° and 110°, the fourth is 360° − 285° = 75°.

Can polygons have curved sides?

No. Every side of a polygon must be a straight line segment, so any shape with a curve, such as a circle, an oval or a semicircle, is not a polygon. The figure must also be closed, and its sides can meet only at their endpoints. If a shape has even one curved edge, the interior angle formula (n − 2) × 180° does not apply to it.

Can polygons be 3D?

No, polygons are always flat, two-dimensional figures. A three-dimensional solid whose faces are all polygons is called a polyhedron instead. A cube, for example, is a polyhedron with six square faces, and a pyramid with a square base has one square face and four triangular faces. So polygons are the building blocks that make up the surfaces of many 3D shapes.

Related Topics

• Triangles
• Area of a Triangle
• Area of a Parallelogram
• Area of a Trapezoid
• Finding Area of Compound Figures