Angles

What it is

An angle is formed where two rays start at the same point, the vertex. Its size says how far one ray has to turn to land on the other. It is measured in degrees (a full turn is 360°) or in radians (a full turn is 2π, see radians).

The kinds of angle are named by size:

• Acute: less than 90°, for example 35°.
• Right: exactly 90°, the corner of a sheet of paper.
• Obtuse: between 90° and 180°, for example 120°.
• Straight: exactly 180°, a straight line.
• Reflex: between 180° and 360°, for example 270°.
• Full turn: 360°.

Examples:

• On a clock at three o’clock the hands make 90°. Between two hour marks the angle is 360° / 12 = 30°, so at four o’clock it is 4 · 30° = 120°.
• An equilateral triangle has three angles of 60° each, and 3 · 60° = 180°, see the angle sum of a triangle.
• Two angles of 40° and 140° are supplementary, because 40° + 140° = 180°.

What is not an angle in this sense:

• Two segments that do not meet have no vertex and form no angle until they are extended far enough to cross. Two parallel lines never do.
• The number 2 with no unit is not yet an angle: 2° and 2 rad are very different, and 2 rad ≈ 114.6°.

Why it matters

Angles run through nearly all of geometry. Sums of angles give rules: the angles of a triangle add to 180° and those of a polygon to (n − 2) · 180°, so a pentagon has 3 · 180° = 540°. Where a line crosses two parallel lines it creates angle pairs that are equal or add to 180°.

Trigonometry turns angles into lengths: sine, cosine and tangent link an angle in a right triangle to the ratio of its sides. The triangle solver finds missing angles and sides. With vectors there is the angle between two vectors, and to change units there is degrees to radians.

Common misconceptions

• “Longer rays make a bigger angle.” The angle depends on the turn, not on how long the rays are drawn. 30° is 30° on a ruler and on a large sheet.
• “A straight line makes no angle.” It is a straight angle, which is why the angles side by side along a line add up to 180°.
• “An angle that looks square is 90°.” Figures are not drawn to scale unless they say so. Only the small square in the corner, or the text, says that an angle is a right angle.
• “The angles of a quadrilateral are 90°.” That holds for a rectangle. The sum is 360° in every quadrilateral, but a rhombus can have angles of 60°, 120°, 60° and 120°.
• “The angle is the one smaller than 180°.” Two rays leave two turns between them, for example 100° and 260°, which add to a full turn. Which one is meant has to be stated. In a coordinate system the positive direction is counter-clockwise from the x-axis.

Common questions