Interior Angles of Polygons

# 2D Shapes

Regular Polygons

A polygon is a plane (2D) shape with straight sides.
To be a regular polygon all the sides and angles must be the
same:

 Triangle – 3 Sides Square – 4 Sides Pentagon – 5 Sides Hexagon – 6 sides Heptagon – 7 Sides Octagon – 8 Sides Nonagon – 9 Sides Decagon – 10 Sides More …

Other Common Polygons

 Quadrilateral Any 4 sided 2D shape Rectangle – 4 SidesAll right angles And many more!

Curved Shapes

These 2D shapes have curves, so are not polygons:

 Circle Ellipse And many more!

Polygons (more detail) Geometry Index

ClipArt ETC

# 7-sided Polygon

## Description

Polygon consisting of 7 sides

## Keywords

7 sides , 7-sided , closed figure , faces , Geometric , geometry , polygon , septagon , seven sides

Heptagons

## Source:

Florida Center for Instructional Technology Clipart ETC (Tampa, FL: University of South Florida, 2009)

# Interior Angles of Polygons

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°

(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°

### 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!

## 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°

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

Shape Sides Sum of Interior Angles Shape Each Angle 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:

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

Sum of Interior Angles = (n-2) × 180°
= (10-2)×180° = 8×180° = 1440°

And it is a Regular Decagon so:

Each interior angle = 1440°/10 = 144°

Note: Interior Angles are sometimes called "Internal Angles"

Interior Angles Exterior Angles Degrees (Angle) 2D Shapes Triangles Quadrilaterals Geometry Index