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All the polygons in these lessons are assumed to be convex polygons.
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The following diagrams give the formulas for the sum of the interior angles of a polygon and the sum ofexterior angles of a polygon. Scroll down the page if you need more examples and explanation.
Sum Of Interior Angles Of A Polygon
We first start with a triangle (which is a polygon with the fewest number of sides). We know that
The sum of interior angles in a triangle is 180°.
This is also called the Triangle Sum Theorem. Click here if you need aproof of the Triangle Sum Theorem.
Next, we can figure out the sum of interior angles of any polygon by dividingthe polygon into triangles. We can separate a polygon into triangles bydrawing all the diagonals that can be drawn from one single vertex.
In the quadrilateral shown below, we can draw only one diagonal from vertex A tovertex B. So, a quadrilateral can be separated into two triangles.
The sum of angles in a triangle is 180°. Since a quadrilateral is made upof two triangles the sum of its angles would be 180° × 2 = 360°
The sum of interior angles in a quadrilateral is 360º
Example:Find the sum of the interior angles of a heptagon (7-sided)
Solution:Step 1: Write down the formula (n - 2) × 180°
Step 2: Plug in the values to get (7 - 2) × 180° = 5 × 180° = 900°
Answer: The sum of the interior angles of a heptagon(7-sided) is 900°.
Example:Find the interior angle of a regular octagon.
Solution:Step 1: Write down the formula
Step 2: Plug in the values to get
Answer: Each interior angle of an octagon(8-sided) is 135°.
Worksheet using the Formula for the Sum of Interior Angles
How to find the sum of the interior angles of any polygon using triangles and then derive the generalized formula?
Problems Using The Sum Of Interior Angles
How to find a missing angle using the sum of interior angles of a polygon?
How to use the sum of interior angles to write an equation and solve for the unknown?Write an equation and solve for the unknown. Substitute your answer into each expression todetermine the measure of the angle. Give reasons for your answers.
Formula For The Sum Of Exterior Angles
The sum of exterior angles of any polygon is 360°.
The exterior angle of a regular n-sided polygon is 360°/n
Worksheet using the formula for the sum of exterior angles
Worksheet using the formula for the sum of interior and exterior angles
How to find the sum of the exterior angles and interior angles of a polygon?Every convex polygon has interior and exterior angles. The interior angles are inside the polygonformed by the sides. The exterior angles form a linear pair with the interior angles.
Example:Determine the measure of each exterior and interior angle of a regular polygon.
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Problems using the sum of exterior angles
The following video shows a problem involving the sum of exterior angles of a polygon.
Example:A regular polygon has an exterior angle that measures 40°. How many sides does the polygon have?
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.