All the polygons in this lesson are assumed to be convex polygons.Related Topics:
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 180º.
This is also called the Triangle Sum Theorem. Click here if you need a proof of the Triangle Sum Theorem.
Next, we can figure out the sum of interior angles of any polygon by dividing the polygon into triangles. We can separate a polygon into triangles by drawing 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 to vertex B. So, a quadrilateral can be separated into two triangles.
The sum of angles in a triangle is 180º. Since a quadrilateral is made up of two triangles the sum of its angles would be 180º × 2 = 360º
The sum of interior angles in a quadrilateral is 360º
A pentagon (five-sided polygon) can be divided into three triangles. The sum of its angles will be 180º × 3 = 560º
The sum of interior angles in a pentagon is 560º.
A hexagon (six-sided polygon) can be divided into four triangles. The sum of its angles will be 180º × 4 = 720º
The sum of interior angles in a hexagon is 720º.
We can see from the above examples that the number of triangles in a polygon is always two less than the number of sides of the polygon. We can then generalize the results for a n-sided polygon to get a formula to find the sum of the interior angles of any polygon.
If n is the number of sides of the polygon then,
Sum of interior angles = (n - 2) × 180°
For a regular polygon,
Each interior angle (of a regular polygon) =
Find the sum of the interior angles of a heptagon (7-sided)
Step 1: Write down the formula (n - 2) × 180°
Step 2: Plug in the values (7 - 2) × 180° = 5 × 180° = 900°
Answer: The sum of the interior angles of a heptagon (7-sided) is 900°.
Find the interior angle of a regular octagon.
Step 1: Write down the formula
Step 2: Plug in the values
Answer: Each interior angle of an otagon (8-sided) is 135°.
Worksheet using the Formula for the Sum of Interior AnglesHow to find the sum of the interior angles of any polygon using triangles and then derive the generalized formula?
The sum of exterior angles of any polygon is 360º.
The exterior angle of a regular polygon is
Worksheet using the formula for the sum of exterior angles
Worksheet using the formula for the sum of interior and exterior anglesHow to find the sum of the exterior angles and interior angles of a polygon?
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