Find the SUM of the interior angles of an OCTAGON. Must Know Geometry Formula!

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[williamgeorgefraser]

(2n - 4) right angles, where n is the number of sides. If you draw lines from each side to the centre you will create as many triangles as there are sides. Each triangle = 2 right angles (180°) while the angles around the centre = 360° (4 right angles). Therefore, multiply each side by 2 and subtract the 4 right angles in the centre.

[dazartingstall]

The sum of the interior angles of a polygon, where n is the number of sides, is:
(n −2) × 180
For an octagon, that's:
(8 − 2) × 180
6 × 180
1, 080

[rickhunter]

Surveyors would use this formula back in the day before computers to determine the sum total of the angles of the parcel of land (i.e. - giant polygon) being surveyed. Many times the angle was measured down to degrees, minutes, seconds and this would aid in determining amount of error in their survey.

[nixxonnor]

A triangle has a sum of internal angles of 180°. For every extra corner you add to the polygon you add another 180° to the sum. So the octagon has 3 + 5 extra corners = 8 corners. 180° + 5 x 180° = 6 x 180° = 1080°. The octagon has a sum of internal angles of 1080°. Each angle must be 1080° / 8 corners = 135°

[dazartingstall]

This is for @624400bmb and anyone else who wants a proof.

Imagine you're stood on the mid-point of the base-line of the regular polygon of your choice. I'm going to use an octagon as in the video, as an example. We're going to walk the perimeter of the figure.

The angle that we have to turn through at each vertex is known as the exterior angle. Note that, since (obviously) when we return to the point we started from, we'll be facing in the same direction, we'll have turned through 360°.

More formally, then, the sum of the exterior angles is 360°.

Now note that each exterior angle and its adjacent interior angle lies on a straight line, and thus each such pair must add up to 180°.

Since there are 8 vertices, the sum of all interior and exterior angles is:
8 × 180 = 1440°

Thus we can find the sum of the interior angles by subtracting the sum of the exterior angles (360) from 1440, for a result of 1080°

The general formula for the sum of the interior angles of any polygon, where n is the number of vertices, or the number of sides, whichever phrasing you prefer, is thus:
n × 180 – 360
Which can be, and often is, phrased as:
(n – 2) × 180

[GetMeThere]

How does the rule work if the polygon bends inward? Take, for example, a square, where two of the sides head inward, to touch around the center?

[russelllomando]

got it 1080 formula # of sides -2 x 180 thanks for the fun

[mikeo]

I solved this in about 10 seconds in my head. Just draw vertical and horizontal lines between the vertices, you make 4 pairs of right angles and 4 pairs of 45° triangles or 4 more 90°, so that is 12*90=1080

[tomtke]

The secret?? EVERY TRIANGLE contains exactly 180°. So how many triangles, distinct triangles, can be drawn inside the given figure? Pick a corner and then draw a line to other corners thus filling the interior with triangles. 5 such lines results in 6 triangles. 6 × 180° = 1080°

Alternately.. the figure appears to be a regular octagon having 8 sides. That each new side is 45° off from original side. As a straight line = 180° and that 180-45 = 135°. Then 8 such angles = 8×135 = 1080°

[chrisdissanayake]

Inside an octagon, there can be 8 triangles constructed with the mid point.

The sum of the angels in one triangle = 180 degrees

The sum of the angels of the 8 triangles = 180x8 = 1440 degrees

The number of degrees around the midpoint = 360 degrees

Therefore, the sum of the rest of the angels in the octagon = 1440-360=1080 degrees

1080 degrees

[Vipre-]

Does this mean that as the figure approaches a perfect circle the sum of the interior angles approaches infinity?

[stevenjohnson]

Looking at an Octagon from the midpoint 8 triangles can be made 8-2 =6×180 =1080

[bradlewis]

(N-2)180; 8 - 2 = 6. 6 x 180 = 1, 080 total degrees of interior octagon.

[johnmarchington]

I broke the octagon down into 8 isosceles triangles and, knowing that the angle of each of the triangles at the very centre of the octagon is 360°/8 or 45°, the other two angles of each triangle are each equal to 67.5°. Two of those angles are required to make up each of the octagon's internal angles so the grand total of those internal angles is 8 x 135° - i.e. 2 x 67.5° - or 1080°

[johnplong]

I remember having to prove this in Geometry

[Ayelmar]

Solved at the thumbnail, the sum is 1080 degrees.
Each angle is going to be 180 + 45 deg = 135 deg., and 8 * 135 deg = 1080 degrees.
The way I visualized it is that each pair of angles would be created by taking a right angle, 90 deg, and "clipping" the corner to add another 45 deg to each of the two resulting corner, making 135 for each.

[PaxAlotin-jr]

*I find it interesting how the sum of the interior angles* equals 1080 degrees - yet ---
--- completes a circle that can never be more than 360 degrees & all without any internal lines ever crossing to create new angles..
Is this another example of the weird properties of a Quasi-Topological shape ? along the lines of a Necker Cube - Klein Bottle & Mobius strip ? 🤨🤔

[argonwheatbelly]

1080 in about 10 seconds in my head. Took way too long. My bad.

[Kleermaker]

And what about a line? It has 180 degrees, but no angles.

[bmb]

Nonsense. Each angle is 45⁰. 8x45 = 360⁰. No polygon can have less than or greater than 360⁰ in total angles, as every closed polygon is in essence circular, or fully enclosed within a circle - which by definition is 360⁰. Any other result is shear garbage.