Interior Angles Of Polygons - Math Is Fun

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Interior Angles of Polygons

An Interior Angle is an angle inside a shape:

interior exterior angles

Another example:

interior exterior angles

Triangles

The Interior Angles of a Triangle add up to 180°

Let's try a triangle: interior angles triangle 90 60 30 90° + 60° + 30° = 180°

It works for this triangle

Now tilt a line by 10°: interior angles triangle 80 70 30 80° + 70° + 30° = 180°

It still works! One angle went up by 10°, and the other went down by 10°

Quadrilaterals (Squares, and so on)

(A Quadrilateral has 4 straight sides)

Let's try a square: interior angles square 90 90 90 90 90° + 90° + 90° + 90° = 360°

A Square adds up to 360°

Now tilt a line by 10°: interior angles 100 90 90 80 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 ...

interior angles 90 (45,45) 90 (45,45)

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!

Pentagon

interior angles pentagon

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°

When the pentagon is regular (all angles equal), 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, then to pentagon, and so on), we add another 180° to the total:

If it is a Regular Polygon (all sides are equal, all angles are equal)
Shape Sides Sum of Interior Angles Shape Each Angle
Triangle 3 180° regular triangle 60°
Quadrilateral 4 360° regular quadrilateral 90°
Pentagon 5 540° pentagon regular 108°
Hexagon 6 720° hexagon regular 120°
Heptagon (also called Septagon) 7 900° heptagon refular 128.57...°
Octagon 8 1080° octagon regular 135°
Nonagon 9 1260° nonagon regular 140°
... ... .. ... ...
Any Polygon n (n−2) × 180° regular n gon (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

Here's an example to make it clearer:

Example: What about a Regular Decagon (10 sides) ?

regular decagon 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"

825, 826, 3314, 3315, 827, 828, 1779, 829, 1780, 3316 Interior Angles Exterior Angles Degrees (Angle) 2D Shapes Triangles Quadrilaterals Geometry Index

Tag » How To Find Interior Angles