The Same Polygon in Degrees and Radians

Relabel the same geometry

Start with a regular hexagon in degrees. Each exterior angle is sixty degrees and each interior angle is one hundred twenty. The two markings describe adjacent parts of a straight angle.

Change the labels to radians while keeping the same hexagon. The exterior is pi divided by three and the interior is two pi divided by three. No corner changed shape. We changed the unit used to describe it.

A full turn is two pi radians, also written tau. The Circle lesson connects this with two pi radius-length arcs around a circumference. Degrees describe the same full turn as three hundred sixty equal parts.

Switch this gathered full-turn construction back to degrees. All six shares remain equal and unchanged in shape. Sixty degrees and pi divided by three are two names for the same one-sixth turn.

Read equal shares of a full turn

In radians, a square divides the full turn into four equal exterior angles. One fourth of two pi is pi divided by two. That is the same quarter-turn as ninety degrees.

A regular octagon gives eight equal shares. Each is pi divided by four radians, or forty-five degrees. Beginning with the fraction of a full turn makes the unit conversion direct.

A regular decagon gives ten equal shares. Two pi divided by ten simplifies to pi divided by five. In degrees, the same exterior turn is thirty-six.

Open the decagon. The pi-over-five angles return to its vertices without changing. The pi in an exact angle measure does not ask us to round it; it records a precise fraction of the turn.

Subtract one angle from a straight turn

Use a regular pentagon and look closely at a corner. Its exterior angle is two pi divided by five. The interior and exterior together make pi radians, the straight-angle total.

Write pi as five pi divided by five. Subtracting two pi divided by five leaves three pi divided by five inside. In degrees, those angles are seventy-two outside and one hundred eight inside.

Compare a hexagon corner. Subtract pi divided by three from pi to obtain two pi divided by three inside. The same supplement relationship works for both polygons.

View the full hexagon in degrees again. Its inside angle is one hundred twenty and the outside turn sixty. Keep one unit throughout an addition or subtraction; changing notation does not change the geometry being measured.

Add the interiors in radians

Keep the hexagon open in radians with Breakdown visible. Six vertices give six straight-angle pairs, totaling six pi. The exterior turns account for two pi, leaving four pi for the interior sum.

Each of its six equal interior angles is two pi divided by three. Multiplying by six gives four pi, the same total. The measure at one corner and the sum across all corners are different quantities.

Compare a pentagon. Five straight-angle pairs total five pi; removing the two pi outside leaves three pi inside. In general the interior sum is n minus two, multiplied by pi radians.

Switch the pentagon to degrees. Its interior sum is five hundred forty, which is three times one hundred eighty. Three pi radians and five hundred forty degrees express the same total in different units.

Fractions that do not simplify to pi over an integer

Choose nine sides and gather only the exterior turns. The full turn is two pi radians, so each of the nine equal wedges is two pi divided by nine. Do not drop the two in that numerator. Pi divided by nine would be half of the actual exterior angle and would fill only half a turn when repeated nine times.

Open the nonagon and restore both angle markings. Its interior is pi minus two pi divided by nine. Writing pi as nine pi divided by nine makes the subtraction visible in the arithmetic: seven pi divided by nine remains inside. The two fractions have a common denominator because they are parts of the same straight angle.

Look closely at one corner. In degrees, those measures would be one hundred forty inside and forty outside. In radians, seven pi over nine plus two pi over nine gives pi. The pair occupies the same geometric straight angle whichever unit we use; neither the denominator nor the pi symbol bends the sides.

Switch the same nonagon to degrees. Nine inside angles of one hundred forty sum to one thousand two hundred sixty. In radians the total was seven pi. Multiplying seven pi by one hundred eighty degrees per pi cancels pi and gives exactly that same total. A conversion changes the number and unit together.

Recover the count from an interior radian measure

Use a sixteen-sided regular polygon with radians shown. Its exterior is pi divided by eight and its interior is seven pi divided by eight. We will recover the sixteen from that interior angle. First compare seven eighths of pi with the complete straight angle, which is eight eighths of pi.

The missing eighth of pi is the exterior turn. Divide two pi, the full turn, by pi over eight. Multiplying by eight over pi cancels pi and gives sixteen. Dividing pi instead of two pi would give eight, but that would count only enough of these turns to cover a half-turn.

Gather the sixteen exterior wedges. Their angles add to two pi, confirming the recovered count. Sixteen times pi over eight is two pi because sixteen divided by eight is two. The gathering makes the full-turn choice explicit rather than leaving the factor of two as an isolated rule to remember.

Open the polygon again with its inside angles visible. Sixteen copies of seven pi over eight total fourteen pi. The interior-sum formula also gives sixteen minus two, times pi. Agreement between these calculations checks that the same side count was used for both the local angle and the whole polygon.

Gather the eight exterior turns of an octagon. Each is pi divided by four. The denominator four can tempt us to call that a quarter-turn, but the full turn is two pi. Divide pi over four by two pi and the result is one eighth. The reference whole must be included when interpreting a fraction.

Compare four gathered turns from a square. Each is pi over two, a genuine quarter of two pi. The square wedge is twice as wide as the octagon wedge even though its denominator is smaller. Counting the wedges around the center checks which fraction of the full turn the symbolic angle actually represents.

Choose the whole before doing the arithmetic

Return to a regular triangle in radians. Its exterior turns are two pi over three, and its interior angles are pi over three. Here the outside angle is larger than the inside angle. The terms inside and outside describe positions, not a rule that one of the two numbers must always be larger.

A square gives equal inside and outside angles, both pi over two. Four exterior turns total two pi, and four interior angles also total two pi. That equality of totals is special to four sides. It does not make the interior and exterior totals equal for every polygon.

Compare five sides. Each exterior is two pi over five, while each interior is three pi over five. Five exterior turns still total two pi, but five interior angles now total three pi. The number of straight-angle pairs increased, so subtracting the same outside total leaves a larger interior total.

Keep the pentagon and lengthen its triangle arms. The angle labels do not change. Pi over an integer is an angular measure here, not the length of an arm or the radius of a hidden circle. Choose the relevant whole first: two pi for a complete turn, pi for a straight-angle pair, or the sum across the specified vertices.

Identify a polygon from an exact radian angle

A regular polygon whose exterior angle is pi divided by six has twelve equal shares around the full turn. Two pi divided by pi over six is twelve. Here is that twelve-sided construction gathered at the center.

Open the construction and restore the interior markings. At each vertex, pi minus pi divided by six leaves five pi divided by six inside. The outside turn is small, while its interior supplement is close to a straight angle.

Show the same polygon in degrees. Its outside angle is thirty and its inside angle is one hundred fifty. Twelve inside angles total one thousand eight hundred degrees, equal to ten pi radians.

Return to radians for the final comparison. State the fraction of a full turn first, then the exterior measure and its interior supplement. You can choose another side count and check both units without changing the underlying angle construction.