How Many Degrees In Pentagon

When dealing with polygons in geometry, it sometimes helps to know the total measure of all interior angles. Nosotros tin also find the exact measure of each angle in a regular polygon.

And then, how many degrees are in a polygon? If nosotros sum the interior angles of a polygon with N sides (an N-gon), in that location are 180(N-ii) degrees. If the polygon is regular, then every interior bending has the same mensurate: 180(Due north-2)/n. The outside angles of an N-sided polygon always sum to 360 degrees, regardless of the value of N.

Of course, for a polygon that is not regular, we would need to practise a little more work to find the measure of an private interior bending.

In this article, we'll talk almost how many degrees are in a polygon. We'll also look at some specific examples for polygons from N = 3 to N = 10 sides.

Permit's get started.

How Many Degrees In A Polygon?

There are 180(N – two) degrees in a polygon if we add upwards the measures of every interior angle:

Sum of Interior Angles of an N-gon = 180(Northward – 2) degrees.

For example, a polygon with N = 22 sides has 180(22 – ii) = 180(20) = 3600 degrees. That is, the sum of all interior angles in a 22-sided polygon is 3600 degrees.

If we add up the measure of every exterior angle, we get a total of 360 degrees:

Sum of Exterior Angles of an N-gon = 360 degrees.

The image below illustrates interior and exterior angles (in a hexagon).

interior and exterior angles hexagon — This picture illustrates an interior and exterior bending of a hexagon. The two must sum to 180 degrees.

Remember that for a regular polygon:

every side has the same length
every angle has the aforementioned mensurate

Given this second fact, nosotros can hands calculate the measures of each interior and exterior angle for a regular polygon with Due north sides.

Since there are N angles of the aforementioned measure, nosotros only need to split up the sum of the angles by Northward. Therefore:

Measure of 1 Interior Bending of a Regular North-gon = 180(Due north – 2) / N degrees.
Mensurate of One Exterior Bending of a Regular N-gon = 360 / Due north degrees.

Note what happens when we take the sum of an interior angle and an outside angle for a regular N-gon:

[Interior Bending of Regular N-gon] + [Exterior Bending of Regular N-gon]
=[180(North – 2) / Due north] + [360 / N]
=[(180N – 360) / North] + [360 / N]
=(180N – 360 + 360) / N
=180N / North
=180

As expected, we become a sum of 180 degrees for the sum of an interior angle in a polygon and its corresponding exterior angle.

How Many Degrees In A Triangle?

There are 180 degrees in a triangle.

interior and exterior angles triangle — This figure illustrates the interior and exterior angles of a triangle.

A triangle has three angles (or if you lot prefer, a trigon has 3 sides – this is where trigonometry comes from!). We tin can apply the formula for the sum of interior angles to verify this:

180(N – 2)
=180(3 – 2)
=180(1)
=180 degrees

For a regular iii-gon (that is, an equilateral triangle), the measure of each interior bending is:

180(N – 2) / N
=180(3 – 2) / iii
=60 degrees

Every bit e'er, the sum of the outside angles is 360 degrees. The measure of an exterior angle for a regular 3-gon (equilateral triangle) is:

360 / Due north
=360 / 3
=120 degrees

How Many Degrees In A Quadrilateral, Rectangle, Or Square?

At that place are 360 degrees in a quadrilateral, rectangle, or square (that is, a 4-gon).

We can utilize the formula for the sum of interior angles to verify this:

180(North – 2)
=180(4 – ii)
=180(two)
=360 degrees

For a regular 4-gon (that is, a square), the measure out of each interior angle is:

180(Due north – 2) / N
=180(4 – 2) / iv
=180(ii) / iv
=xc degrees

Every bit always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 4-gon (square) is:

360 / N
=360 / four
=ninety degrees

Notation: this is the only case where the interior and exterior angles for a regular N-gon have the aforementioned measure. Here is the proof:

Measure of Interior Angle of Regular N-gon = Mensurate of Exterior Angle of Regular N-gon
180(N – 2) / N = 360 / N
180(North – 2) = 360
180N – 360 = 360
180N = 720
Northward = iv

How Many Degrees In A Pentagon?

In that location are 540 degrees in a pentagon (that is, a 5-gon).

Nosotros tin can utilise the formula for the sum of interior angles to verify this:

180(North – 2)
=180(v – 2)
=180(3)
=540 degrees

For a regular five-gon (that is, a regular pentagon), the mensurate of each interior angle is:

180(N – 2) / N
=180(5 – two) / v
=180(3) / 5
=108 degrees

As always, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 5-gon (regular pentagon) is:

360 / Northward
=360 / five
=72 degrees

How Many Degrees In A Hexagon?

In that location are 720 degrees in a hexagon (that is, a 6-gon).

We can use the formula for the sum of interior angles to verify this:

180(N – 2)
=180(6 – 2)
=180(4)
=720 degrees

For a regular 6-gon (that is, a regular hexagon), the measure of each interior angle is:

180(Northward – 2) / N
=180(6 – ii) / half dozen
=180(4) / 6
=120 degrees

Every bit ever, the sum of the exterior angles is 360 degrees. The measure of an outside angle for a regular six-gon (regular hexagon) is:

360 / North
=360 / 6
=60 degrees

How Many Degrees In A Heptagon?

There are 900 degrees in a heptagon (that is, a 7-gon).

Nosotros tin employ the formula for the sum of interior angles to verify this:

180(N – 2)
=180(7 – 2)
=180(5)
=900 degrees

For a regular seven-gon (that is, a regular heptagon), the measure of each interior bending is:

180(N – 2) / N
=180(7 – 2) / 7
=180(five) / vii
=900 / 7
~128.57 degrees

As always, the sum of the exterior angles is 360 degrees. The measure of an outside bending for a regular 7-gon (regular heptagon) is:

360 / Due north
=360 / 7
~51.43 degrees

How Many Degrees In An Octagon?

At that place are 1080 degrees in an octagon (that is, an viii-gon).

Nosotros can use the formula for the sum of interior angles to verify this:

180(North – two)
=180(8 – 2)
=180(six)
=1080 degrees

For a regular viii-gon (that is, a regular octagon), the measure of each interior angle is:

180(N – 2) / Due north
=180(8 – 2) / 8
=180(half dozen) / eight
=1080 / eight
=135 degrees

As always, the sum of the exterior angles is 360 degrees. The measure of an exterior bending for a regular 8-gon (regular octagon) is:

360 / N
=360 / 8
=45 degrees

How Many Degrees In A Nonagon?

At that place are 1260 degrees in a nonagon (that is, a 9-gon).

A nonagon has 9 sides and 9 angles. The interior angles sum to 1260.

We can employ the formula for the sum of interior angles to verify this:

180(N – two)
=180(9 – 2)
=180(7)
=1260 degrees

For a regular 9-gon (that is, a regular nonagon), the measure of each interior angle is:

180(N – 2) / N
=180(ix – two) / 9
=180(vii) / 9
=1260 / 9
=140 degrees

As always, the sum of the exterior angles is 360 degrees. The measure out of an outside bending for a regular 9-gon (regular nonagon) is:

360 / N
=360 / 9
=forty degrees

How Many Degrees In A Decagon?

At that place are 1440 degrees in a decagon (that is, a 10-gon).

Nosotros can use the formula for the sum of interior angles to verify this:

180(N – 2)
=180(10 – 2)
=180(eight)
=1440 degrees

For a regular 10-gon (that is, a regular decagon), the measure of each interior angle is:

180(N – ii) / N
=180(10 – ii) / 10
=180(8) / 10
=1440 / ten
=144 degrees

Every bit e'er, the sum of the exterior angles is 360 degrees. The measure of an exterior angle for a regular 10-gon (regular decagon) is:

360 / N
=360 / 10
=36 degrees

The tabular array below summarizes the interior angles and exterior angles for polygons with North sides, for North = three to ten.

N	Interior Angle Sum	Interior Angle Measure (Regular N-gon)	Exterior Angle Measure (Regular Due north-gon)
3	180	sixty	120
4	360	xc	90
5	540	108	72
half dozen	720	120	sixty
7	900	~128.57	~51.43
eight	1080	135	45
9	1260	140	40
10	1440	144	36

The table below summarizes the interior
angles and exterior angles for polygons
with Northward sides, for N = 3 to 10.

Conclusion

Now you know how many degrees are in a polygon (that is, the sum of the interior angles), depending on the number of sides. Y'all also know how to find the measures of interior and exterior angles for regular polygons.

I hope y'all found this commodity helpful. If so, please share it with someone who can use the data.

Don't forget to subscribe to my YouTube channel & go updates on new math videos!

~Jonathon