Fractions of a turn: angles in degrees and radians

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A full turn measures 360° or 2π radians, so every fraction of a turn converts by multiplying: 1/3 is 120° and 1/6 is 60°. It is the same calculation as a fraction of a quantity, with the turn as the quantity. The table gathers the common angles, and the sections after it explain where the degree and radian columns come from. If you want to split data into sectors, see pie charts with fractions.

The first three columns describe the same rotation, so you can enter through the one you know and leave through another. To convert a fraction to its decimal form, use the fraction, decimal and percent table or the fraction to decimal calculator.

From degrees to a fraction of a turn

Divide the degrees by 360 and simplify with the greatest common factor (GCF, also called GCD or HCF). For 225°, the ratio is 225/360, and since 225 and 360 are both divisible by 45, it becomes 5/8 of a turn. The simplify fractions calculator does that reduction, and the factors calculator lists the divisors of 360 if you want to pick the greatest one by hand.

The same works for 90°, since 90/360 reduces to one quarter. For 80° you get 80/360 = 2/9, an angle that is not in the table but is solved the same way. To check, multiply back: 2/9 of 360 = 80. If you do not get the starting angle, the reduction went wrong.

From a fraction of a turn to radians

Multiply the fraction by 2π, because the turn measures 2π radians. A quarter turn is 2π/4 = π/2. The factor π is kept, so no approximation is needed. When an angle is already written as a fraction of π, divide that fraction by 2 to get the part of the turn. For example, 5π/6 has coefficient 5/6, and 5/6 ÷ 2 = 5/12 of a turn.

The minute hand and clock degrees

The minute hand covers 1/60 of a turn per minute, which is 360 ÷ 60 = 6° every minute. In 5 minutes it moves 30°, in 15 minutes it moves 90°, and in 30 minutes it reaches 180°. This is how clock minutes relate to fractions of a turn: 15 minutes is a quarter of an hour and a quarter of a turn at once.

At 3:00 the hour hand points to the 3 and the minute hand to the 12, with 90° between them. For trickier fractions of an hour, the fraction of an hour to minutes calculator converts the fraction to minutes, and from there to degrees by multiplying by 6.

Compass bearings

The eight main bearings divide the turn into eighths of 45°, counted from north clockwise.

Two opposite bearings are always 4/8 apart, which is half a turn: from northeast to southwest is 5/8 − 1/8 = 4/8, which is 1/2. To see the split drawn, the visual fraction models tool shows circles divided into equal parts.

When a fraction of a turn does not help

A fraction of a turn measures the size of a rotation. It does not relate the sides and angles of a triangle, so it does not replace trigonometry. For an angle like 17°, the fraction 17/360 is correct, but it does not simplify and adds nothing over the number of degrees. If a problem asks for the smaller angle between two directions, subtract from the turn, so 270° reads as 90°. In math, a counterclockwise turn is positive and a clockwise turn is negative; compass bearings, however, are counted clockwise.

To split a total into the sectors of a chart, the guide to pie charts with fractions applies this same multiplication to real data, and equivalent fractions explain why 2/8 and 1/4 are the same angle. The visual fractions hub collects the circle and bar models.

Sources

The English Wikipedia article Radian (accessed October 17, 2026) states that the circumference subtends an angle of 2π radians, so 2π radians equals 360 degrees. The tables and examples in this guide start from that equivalence between a turn, 2π radians and 360°.