From One Turn to the Interior Angle Sum

Two adjacent angles at one vertex

Start with the opened pentagon. Both the inside angle and the outside turn are marked. Follow the two sides of the polygon into one corner. The interior angle occupies the region inside; the exterior turn describes the change of direction outside.

Together these adjacent angles make a straight angle: one hundred eighty degrees. The exterior turn is seventy-two, so the interior is one hundred eight. One hundred eight plus seventy-two is one hundred eighty.

Move to the next corner and compare its pair. Regularity gives the same two values there. The straight-angle relationship is local: it concerns one vertex, before we add anything across the whole polygon.

Now view all five corners. Each contributes one interior-exterior pair totaling one hundred eighty degrees. Five such pairs would total nine hundred degrees. We will separate that total into all the outside turns and all the inside angles.

Separate one complete outside turn

Keep the pentagon open with Breakdown visible. Across its five vertices, the paired angles total nine hundred degrees. The exterior angles account for one complete turn, or three hundred sixty of that total.

Gather the exterior-angle triangles. Their five seventy-two-degree turns still fill a complete turn. The interior angle markings recede as the central polygon closes; we are collecting the outside contributions here, not claiming that the inside angles fit around this one point.

Return the turns to the pentagon and restore the inside markings. Subtracting the exterior total from the five straight-angle pairs leaves five hundred forty degrees for the interior sum.

Five equal interior angles share that five hundred forty, giving one hundred eight each. This agrees with subtracting seventy-two from one hundred eighty at a single vertex. The local and total calculations describe the same construction.

Compare several polygons

A triangle has three interior angles of sixty degrees. Their total is one hundred eighty. Its three exterior turns are one hundred twenty each, still totaling three hundred sixty. A large exterior turn leaves a smaller interior supplement.

A square has four interior angles of ninety degrees. Their sum is three hundred sixty. At every corner, the ninety-degree inside angle and the ninety-degree outside turn make one hundred eighty.

A regular hexagon has inside angles of one hundred twenty degrees and outside turns of sixty. Six inside angles total seven hundred twenty. Six straight-angle pairs total one thousand eighty; subtracting the full outside turn leaves seven hundred twenty.

A regular octagon has inside angles of one hundred thirty-five degrees and outside turns of forty-five. Its interior total is one thousand eighty. Adding corners makes each equal outside turn smaller and each inside supplement larger.

Connect the displayed pattern to a general expression

Return to the pentagon. For n vertices, there are n straight-angle pairs, totaling n times one hundred eighty degrees. The exterior turns always use three hundred sixty. Subtract that common total to obtain the interior sum.

Three hundred sixty is two times one hundred eighty. Factoring out one hundred eighty leaves n minus two, multiplied by one hundred eighty. For five vertices, three times one hundred eighty is five hundred forty.

Use ten vertices. Eight times one hundred eighty gives one thousand four hundred forty degrees in the interior total. Dividing by ten gives one hundred forty-four at each regular vertex. The exterior turn is thirty-six.

Look closely at one of those ten corners. The inside angle one hundred forty-four and outside turn thirty-six sum to one hundred eighty. Keep the total over all vertices separate from the individual angle displayed at this one vertex.

A larger total does not mean a larger outside turn

Start with six sides and the breakdown visible. The interior total is seven hundred twenty degrees. At a single corner, the interior is one hundred twenty and the exterior is sixty. Keep these three numbers separate: one total, one inside angle and one outside turn. Each answers a different question about the same polygon.

Change to seven sides. The interior total rises to nine hundred degrees, an increase of one hundred eighty. One extra vertex adds one extra straight-angle pair to the accounting, while the combined exterior contribution remains three hundred sixty. We subtract the same full turn from a larger collection of straight angles.

Look at one of the seven corners. Its exterior is three hundred sixty divided by seven, a little over fifty-one degrees. Its interior is nine hundred divided by seven, a little under one hundred twenty-nine. The interior total increased, each interior increased, and each exterior decreased. Their directions of change are different.

Use eight sides next. The total becomes one thousand eighty, another increase of one hundred eighty. Each inside angle is one hundred thirty-five and each outside turn forty-five. Equal steps in the total do not imply equal steps in each individual angle, because the number of angles sharing that total also changes.

Inside angles approach a straight angle

A twelve-sided regular polygon has an interior angle of one hundred fifty degrees. The outside turn is thirty. Enlarge the marked corner with the camera and trace the two adjacent angles. The inside is already fairly close to a straight angle, but the remaining thirty-degree outside turn is essential to following the boundary.

Compare twenty-four sides at a marked corner. The exterior turn has halved to fifteen degrees, leaving one hundred sixty-five inside. Doubling the number of sides did not double the interior angle. It halved the amount missing from a straight angle, which is the exterior supplement.

Return to the full twenty-four-sided construction. Its interior total is twenty-two times one hundred eighty, or three thousand nine hundred sixty degrees. A total can exceed a complete turn many times because it adds angles at many different vertices. It is not one angle that has been drawn around the center.

Show thirty-six sides. Each outside turn is ten degrees and each inside angle one hundred seventy. Every finite side count still leaves a positive outside turn. If each interior were exactly one hundred eighty, its exterior would be zero and following successive edges would never change direction enough to close this convex polygon.

A regular fifteen-sided polygon has an interior angle of one hundred fifty-six degrees. If that single angle were the information given, subtract it from one hundred eighty to get twenty-four outside. Three hundred sixty divided by twenty-four is fifteen. We have recovered the number of corners without using an interior total.

Return to the whole polygon. Its fifteen interior angles total two thousand three hundred forty degrees. That total is thirteen times one hundred eighty. The number one hundred fifty-six and the number two thousand three hundred forty both describe interiors, but only the first describes one corner. Labeling the quantity prevents choosing the wrong inverse calculation.

Recover a polygon from its interior total

Suppose the interior total is two thousand eight hundred eighty degrees. Divide by one hundred eighty to obtain sixteen. That is the number of straight-angle units remaining after the outside turn was removed. It is not yet the side count. Add the two removed units back to obtain eighteen vertices, shown here.

At a corner of this eighteen-sided polygon, the interior is one hundred sixty degrees. Eighteen copies give the required two thousand eight hundred eighty. The exterior supplement is twenty, and eighteen copies of twenty give three hundred sixty. The total, the individual angle and the outside check all agree.

Return to nine sides. Its interior total is one thousand two hundred sixty. Dividing by one hundred eighty gives seven, then adding two gives nine. This calculation uses a total, while dividing three hundred sixty by an exterior angle uses a single corner. Recognizing which kind of information you were given determines the operation.

Gather just the nine exterior turns, hiding the inside markings while the polygon closes. Their sum remains three hundred sixty. We have changed neither the count nor the forty-degree turns. This collected outside contribution is the fixed amount subtracted in every one of the interior-total calculations, regardless of how large that interior total becomes.

Let an interior angle identify the regular polygon

This regular nonagon has nine sides. Its marked interior angle is one hundred forty degrees. Subtracting that from one hundred eighty gives an exterior turn of forty degrees.

Nine forty-degree turns make the full three hundred sixty. That explains how the angle determines the side count: first find its exterior supplement, then divide the complete turn by that supplement.

The interior total is nine times one hundred forty, or one thousand two hundred sixty. The general expression gives seven times one hundred eighty, also one thousand two hundred sixty. Two connected calculations agree on the same polygon.

As a contrast, an interior angle of one hundred forty-five would leave a thirty-five-degree outside turn. That does not divide three hundred sixty into a whole number of equal parts, so it cannot be every interior angle of a convex regular polygon. The displayed nonagon remains our valid example.