Finding the amount of inner & Exterior Angles

Polygons are prefer the little houses the two-dimensional geometry world. They create insides, referred to as the interior, and outsides, referred to as the exterior. You deserve to measure inner angles and also exterior angles. You deserve to also add up the sums that all inner angles, and the sums of all exterior angles, of consistent polygons. Our formula functions on triangles, squares, pentagons, hexagons, quadrilaterals, octagons and also more.

You are watching: Sum of angles in a square

What Is A constant Polygon?

For a polygon to be a regular polygon, that must fulfill these four requirements:

Be two-dimensionalEnclose a space, creating an interior and exteriorUse just line segments because that sidesHave every sides same in length to one another, and all inner angles equal in measure up to one another


Sum of internal Angles of a Polygon

Regular polygons exist without limit (theoretically), however as you get an ext and much more sides, the polygon looks an ext and much more like a circle. The continuous polygon through the fewest political parties -- 3 -- is the it is provided triangle. The constant polygon with the most sides generally used in geometry classes is more than likely the dodecagon, or 12-gon, through 12 sides and also 12 interior angles:


Pretty fancy, isn"t it? however just because it has actually all those sides and interior angles, do not think girlfriend cannot figure out a lot about our dodecagon. Suppose, for instance, you desire to know what all those inner angles include up to, in degrees?

Sum of interior Angles

Triangles room easy. Their interior angles include to 180°. Likewise, a square (a continuous quadrilateral) adds to 360° because a square have the right to be separated into two triangles.

The native "polygon" way "many angles," despite most civilization seem to notice the sides an ext than they an alert the angles, therefore they developed words choose "quadrilateral," which method "four sides."

Regular polygons have as many interior angles as they have actually sides, for this reason the triangle has three sides and three interior angles. Square? four of each. Pentagon? Five, and also so on. Ours dodecagon has actually 12 sides and 12 inner angles.

Sum of inner Angles Formula

The formula for the amount of the polygon"s internal angles is refreshingly simple. Let n same the number of sides that whatever consistent polygon you are studying. Right here is the formula:


Sum of angles in a triangle

You have the right to do this. Shot it first with ours equilateral triangle:


(n - 2) × 180°

(3 - 2) × 180°

Sum of inner angles = 180°

Sum of angles of a square

And again, try it for the square:


(n - 2) × 180°

(4 - 2) × 180°

2 × 180°

Sum of interior angles = 360°

How To uncover One inner Angle

To discover the measure up of a single interior angle, then, you merely take that full for all the angles and also divide the by n, the variety of sides or angle in the continuous polygon.

The brand-new formula looks really much favor the old formula:


Again, check it for the equilateral triangle:

(3 - 2) × 180°3


One inner angle = 60°

And for the square:

(4 - 2) × 180°4

2 × 180°4


One internal angle = 90°

Hey! the works! and also it functions every time. Let"s tackle that dodecagon now.

Interior angle Examples

Remember what the 12-sided dodecagon watch like? Let"s find the sum of the interior angles, and one inner angle:


Find the sum of interior angles that a dodecagon

(n - 2) × 180°

(12 - 2) × 180°

10 × 180°

Sum of interior angles = 1,800°

Now, let"s discover one interior angle

(n - 2) × 180°n

(12 - 2) × 180°12

10 × 180°12


One internal angle = 150°


Sum that Exterior Angles

Every constant polygon has exterior angles. These are not the reflex angle (greater than 180°) created by rotating native the exterior the one side to the next. That is a typical misunderstanding. For instance, in an it is provided triangle, the exterior edge is not 360° - 60° = 300°, as if us were rotating indigenous one side every the method around the vertex come the various other side.


Exterior angles Examples

What do we have left in our arsenal of continual polygons? that dodecagon! we know any kind of interior edge is 150°, so the exterior angle is:


180° - 150°

Exterior angle = 30°

Checking her Work

Look closely at the three exterior angle we provided in ours examples:

Triangle = 120°

Square = 90°

Dodecagon = 30°

Prepare to it is in amazed. Multiply each of those dimensions times the number of sides the the regular polygon:

Triangle = 120° × 3 = 360°Square = 90° × 4 = 360°Dodecagon = 30° × 12 = 360°

It looks favor magic, yet the geometric factor for this is actually simple: to move around these shapes, you room making one complete rotation, or turn, of 360°.

Still, this is an easy idea to remember: no matter how fussy and also multi-sided the consistent polygon gets, the sum of that exterior angle is constantly 360°.

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Lesson Summary

After working with all that, currently you are able to define a constant polygon, measure one internal angle of any type of polygon, and identify and apply the formula offered to find the sum of inner angles of a regular polygon. You additionally can describe to who else just how to uncover the measure up of the exterior angles of a consistent polygon, and you understand the amount of exterior angle of every constant polygon.

Next Lesson:

Ratios and also Proportions

What you learned:

After working your way through this lesson and also the video, girlfriend learned to:

Define a consistent polygonIdentify and apply the formula supplied to uncover the sum of interior angles of a continuous polygonMeasure one internal angle the a polygon utilizing that same formulaExplain exactly how you find the measure of any type of exterior edge of a constant polygonKnow the sum of the exterior angles of every continual polygon