Another example:
Triangles
The Interior Angles of a Triangle add up to 180°
Let's try a triangle:
90° + 60° + 30° = 180°
It works for this triangle
Now tilt a line by 10°:
80° + 70° + 30° = 180°
It still works!
One angle went up by 10°,
and the other went down by 10°
Quadrilaterals (Squares, etc)
(A Quadrilateral has 4 straight sides)
Let's try a square:
90° + 90° + 90° + 90° = 360°
A Square adds up to 360°
Now tilt a line by 10°:
80° + 100° + 90° + 90° = 360°
It still adds up to 360°
The Interior Angles of a Quadrilateral add up to 360°
Because there are 2 triangles in a square ...
The interior angles in a triangle add up to 180° ...
... and for the square they add up to 360° ...
... because the square can be made from two triangles!
A pentagon has 5 sides, and can be made from three triangles, so you know what ...
... its interior angles add up to 3 × 180° = 540°
And when it is regular (all angles the same), then each angle is 540° / 5 = 108°
(Exercise: make sure each triangle here adds up to 180°, and check that the pentagon's interior angles add up to 540°)
The Interior Angles of a Pentagon add up to 540°
The General Rule
Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total:
| If it is a Regular Polygon (all sides are equal, all angles are equal) | ||||
| Triangle | 3 | 180° | 60° | |
| Quadrilateral | 4 | 360° | 90° | |
| Pentagon | 5 | 540° | 108° | |
| Hexagon | 6 | 720° | 120° | |
| Heptagon (or Septagon) | 7 | 900° | 128.57...° | |
| Octagon | 8 | 1080° | 135° | |
| Nonagon | 9 | 1260° | 140° | |
| ... | ... | .. | ... | ... |
| Any Polygon | n | (n−2) × 180° | (n−2) × 180° / n |
So the general rule is:
Sum of Interior Angles = (n−2) × 180°
Each Angle (of a Regular Polygon) = (n−2) × 180° / n
Perhaps an example will help:
Sum of Interior Angles = (n−2) × 180°
= (10−2) × 180°
= 8 × 180°
= 1440°
And for a Regular Decagon:
Each interior angle = 1440°/10 = 144°
Note: Interior Angles are sometimes called "Internal Angles"
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Interior angle formulas are used to find interior angles associated with a polygon and their sum. Interior angles are the angles that lie inside a shape, generally a polygon. Also, the angles lying in the area enclosed between two parallel lines that are intersected by a transversal are also called interior angles. Let us understand the interior angle formula in detail in the following section.
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What is Interior Angle Formula?
The interior angle formula is used to:
- find the sum of all interior angles of a polygon.
- find an unknown interior angle of a polygon.
- find each interior angle of a regular polygon.
Let us consider a polygon of n sides. Then by interior angle formula to find the sum of interior angles of a polygon is given as,
The sum of interior angles = 180(n-2)º
The interior angles of a polygon always lie inside the polygon and the formula to calculate it can be obtained in three ways.
Formula 1: For “n” is the number of sides of a polygon, formula is as,
Interior angles of a Regular Polygon = [180°(n) – 360°] / n
Formula 2: The formula to find the interior angle, if the exterior angle of a polygon is given,
Interior Angle of a polygon = 180° – Exterior angle of a polygon
Formula 3: If the sum of all the interior angles of a regular polygon, the measure of interior angle can be calculates using the formula,
Interior Angle = Sum of the interior angles of a polygon / n
where,
“n” is the number of polygon sides
Let us understand interior angle formulas better using solved examples.
Solved Examples Using Interior Angle Formula
-
Solution: To find: The sum of all interior angles of a heptagon.
We know that the number of sides of a heptagon is, n = 7.
By interior angle formula
The sum of interior angles = 180(n-2)º
= 180 (7-2)º
= 180 (5)º= 900º
Answer: The sum of all interior angles of a heptagon = 900°.
-
Solution:
To find: The measure of each interior angle of a regular polygon of 23 sides.
The number of sides of the given polygon is n = 23.
By interior angle formula,
The sum of interior angles = 180(n-2)º= 180 (23-2)º
= 180 (21) º
= 3780º
The measure of each interior angle is obtained by dividing the above sum by 23.
Each interior angle = 3780 / 23 = 164.35°
Answer: Each interior angle of a polygon of 23 sides = 164.35°.
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