The document discusses formulas for calculating angles in regular polygons. It explains that the sum of interior angles in an n-sided polygon is (n-2)×180 degrees. The central angle of a regular polygon with n sides is calculated as 360 degrees divided by n. Similarly, the exterior angle is 360 degrees divided by n. The interior angle is calculated as 180(n-2) degrees divided by n. Examples are given applying these formulas to triangles, squares, pentagons, hexagons, and octagons.

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Equilateral
Triangle
Square
Pentagon Hexagon
Heptagon Octagon Enneagon
Decagon Hendecagon Dodecagon

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20 sides
Eicosagon

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Interior Angle
What is the sum of the interior angles of this enneagon?
7 x 180° = 1260°
A 9–sided polygon is split into 7 triangles

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What is the sum of the interior angles of this enneagon?
What is the sum of the interior angles of a polygon
with n sides?
A n - sided polygon can be split into triangles
n – 2
( 2)
n - 180
´ ° 180( 2)
n
= -
=
sum

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sum of the interior angles of various polygons
triangle
180°
quadrilateral
180° x 2 = 360°
pentagon
180° x 3 = 540°
hexagon
180° x 4 = 720°
heptagon
180° x 5 = 900°
octagon
180° x 6 = 1080°

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Consider the following polygon
What do the exterior angles of a polygon add up to?
Exterior Angle

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360°
Consider the following polygon
What do the exterior angles of a polygon add up to?

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Exterior Angle
Consider the following polygon
What do the exterior angles of a polygon add up to?

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Consider the following polygon
What do the exterior angles of a polygon add up to?

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Consider the following polygon
What do the exterior angles of a polygon add up to?

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Consider the following polygon
What do the exterior angles of a polygon add up to?

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Consider the following polygon
What do the exterior angles of a polygon add up to?

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Consider the following polygon
What do the exterior angles of a polygon add up to?

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Consider the following polygon
What do the exterior angles of a polygon add up to?

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Consider the following polygon
360°
What do the exterior angles of a polygon add up to?

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Central Angle
Central Angle
The central angle of a regular polygon
How do we find the central
angle of a regular polygon
with n sides?
Central angle =
360°
n

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The central angle of a regular polygon
How do we find the central
angle of a regular polygon
with n sides?
Central angle of a pentagon =
360°
5
= 72°

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The central angle of a regular polygon
How do we find the central
angle of a regular polygon
with n sides?
Central angle of an octagon =
360°
8
= 45°

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The exterior angle of a regular pentagon
exterior angle =
360°
5
The exterior angles of any polygon add up to 360°
= 72°

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The exterior angle of a regular octagon
exterior angle = 360°
8
= 45°
The exterior angles of any polygon add up to 360°

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The interior angle of a regular polygon
A n - sided polygon can be split into triangles
n – 2
( 2)
n - 180
´ ° 180( 2)
n
= -
=
sum
Interior angle =
180(n – 2)
n

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the interior angles of various regular polygons
equilateral
triangle
180°
square
180° x 2 = 360°
pentagon
180° x 3 = 540°
hexagon
180° x 4 = 720°
heptagon
180° x 5 = 900°
octagon
180° x 6 = 1080°
÷ 3 = 60° 360° ÷ 4 = 90° 540° ÷ 5 = 72°
720° ÷ 6 = 120° 900° ÷ 7 ≈ 128.6° 1080° ÷ 8 = 135°

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central angle =
360°
n
exterior angle =
360°
n
For every regular polygon
interior angle =
180(n – 2)
n
These formulae are very easy to derive

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