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Geometry

Polygon Angles

Lesson

The angle sums of a polygon follow simple formulas you can apply to any number of sides.

Sum of interior angles

(n−2)⋅180∘(n - 2) \cdot 180^\circ

Where nn is the number of sides. A quadrilateral (4 sides) sums to 360°, pentagon 540°, hexagon 720°, and so on.

Sum of exterior angles

360∘360^\circ

Always 360° — regardless of the number of sides.

Regular polygons

A polygon is regular if all sides and angles are equal. Each interior angle of a regular n-gon:

(n−2)⋅180∘n\frac{(n - 2) \cdot 180^\circ}{n}

Each exterior angle: 360∘n\frac{360^\circ}{n}.

Worked example

Regular hexagon (n=6n = 6):

interior sum=4⋅180=720∘\text{interior sum} = 4 \cdot 180 = 720^\circ
each interior=720/6=120∘\text{each interior} = 720/6 = 120^\circ
each exterior=360/6=60∘\text{each exterior} = 360/6 = 60^\circ

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Interior sum: triangle\text{Interior sum: triangle}

Problem 2

Interior sum: quadrilateral\text{Interior sum: quadrilateral}

Problem 3

Interior sum: pentagon\text{Interior sum: pentagon}

Problem 4

Sum of exterior angles of any polygon\text{Sum of exterior angles of any polygon}

Practice

Standard problems matching the lesson.

Problem 5

Hexagon interior sum\text{Hexagon interior sum}

Problem 6

Heptagon (7-gon) interior sum\text{Heptagon (7-gon) interior sum}

Problem 7

Octagon interior sum\text{Octagon interior sum}

Problem 8

Decagon interior sum\text{Decagon interior sum}

Problem 9

Each interior angle of regular pentagon\text{Each interior angle of regular pentagon}

Problem 10

Each interior angle of regular hexagon\text{Each interior angle of regular hexagon}

Problem 11

Each interior angle of regular octagon\text{Each interior angle of regular octagon}

Problem 12

Each exterior angle of regular pentagon\text{Each exterior angle of regular pentagon}

Problem 13

Each exterior angle of regular hexagon\text{Each exterior angle of regular hexagon}

Problem 14

Each exterior angle of regular octagon\text{Each exterior angle of regular octagon}

Problem 15

Each exterior angle of equilateral triangle\text{Each exterior angle of equilateral triangle}

Problem 16

Interior sum 1620∘. Sides?\text{Interior sum } 1620^\circ. \text{ Sides?}

Problem 17

Regular polygon interior 144∘. Sides?\text{Regular polygon interior } 144^\circ. \text{ Sides?}

Problem 18

Sum of interior angles of 12-gon\text{Sum of interior angles of 12-gon}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Each interior angle of regular dodecagon\text{Each interior angle of regular dodecagon}

Problem 20

Regular polygon exterior 20∘. Sides?\text{Regular polygon exterior } 20^\circ. \text{ Sides?}

Problem 21

Pentagon 4 angles: 100∘,110∘,120∘,130∘. Fifth?\text{Pentagon 4 angles: } 100^\circ, 110^\circ, 120^\circ, 130^\circ. \text{ Fifth?}

Problem 22

Hexagon 5 angles each 130∘. Sixth?\text{Hexagon 5 angles each } 130^\circ. \text{ Sixth?}

Problem 23

Interior sum of 15-gon\text{Interior sum of 15-gon}

Problem 24

Each interior of regular 20-gon\text{Each interior of regular 20-gon}

Problem 25

Regular polygon interior 162∘. Sides?\text{Regular polygon interior } 162^\circ. \text{ Sides?}

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