Types of angles: acute, right, obtuse and more

What an angle is

An angle is the amount of turn between two lines, called arms, that meet at a point called the vertex. Angles are measured in degrees: a full turn is 360°, a half turn is 180° and a quarter turn is 90°. The length of the arms makes no difference. A small angle drawn with long arms is still a small angle.

A clock face is a good picture to keep in mind. The minute hand turns 360° every hour, and the 12 numbers split that turn into 12 equal parts of 360 ÷ 12 = 30° each. From 12 to 3 the hand turns 3 × 30 = 90°, a right angle. From 12 to 6 it turns 180°, a straight line.

Types of angles

• Acute angle: more than 0° and less than 90°, such as 35° or 80°. It is sharper than the corner of a sheet of paper.
• Right angle: exactly 90°, like the corner of a square or a book. Diagrams mark it with a small square in the corner.
• Obtuse angle: more than 90° and less than 180°, such as 120°. It is wider than a corner but not yet a straight line.
• Straight angle: exactly 180°, a straight line or a half turn.
• Reflex angle: more than 180° and less than 360°, such as 270°.
• Full turn: exactly 360°, all the way round.

Acute, obtuse and right angles are the three named in the US grade 4 standards; reflex angles are named in England's Year 5 curriculum. Every angle has a partner on the outside of its arms that completes the full turn. A 70° angle leaves 360 − 70 = 290° on the outside, and that outside angle is reflex.

Measuring angles with a protractor

A protractor has two scales running in opposite directions, each from 0 to 180. Use the scale that reads 0 on the arm lying along the base line, and always estimate first. The two readings at the second arm always add up to 180, as 65 and 115 do in the example above, so an estimate is what tells you which one is right.

Most protractors only go to 180°, so a reflex angle needs one more step. Measure the smaller angle on the inside, then subtract it from 360. If the inside angle is 130°, the reflex angle is 360 − 130 = 230°. The practice set below does not ask you to read a protractor on screen; it practises naming angles and working out missing ones. Measuring is best practised with a real protractor and a pencil.

Finding missing angles without measuring

Angles that sit side by side add together. Three facts then solve most missing-angle questions: angles that make a right angle add up to 90°, angles on a straight line add up to 180°, and angles around a point add up to 360°. These examples get a little harder each time.

• Side by side: angles of 52° and 58° next to each other make 52 + 58 = 110°.
• In a right angle: a right angle is split into two parts and one is 37°. The other is 90 − 37 = 53°.
• On a straight line: one angle on a straight line is 133°. The other is 180 − 133 = 47°.
• Around a point: four angles are 86°, 70°, 128° and x. 86 + 70 + 128 = 284, so x = 360 − 284 = 76°.
• Fractions of a turn: 1/8 of a full turn is 360 ÷ 8 = 45°.

How it is taught in school

In the US, Common Core teaches angles in grade 4. Children learn that a one-degree angle turns through 1/360 of a circle (4.MD.C.5), measure and sketch angles with a protractor (4.MD.C.6), add angles and find unknown angles (4.MD.C.7), and draw and identify right, acute and obtuse angles (4.G.A.1). Complementary, supplementary and vertical angles follow in grade 7 (7.G.B.5).

In England, Year 3 pupils see angles as a property of shape or a description of a turn, identify right angles and decide whether an angle is greater or less than a right angle. Year 4 identify acute and obtuse angles and order angles up to two right angles. Year 5 measure angles in degrees, estimate and compare acute, obtuse and reflex angles, and use the facts that angles at a point make 360° and angles on a straight line make 180°. Year 6 find missing angles at a point, on a straight line and where lines cross.

How parents can help

• Go on an angle hunt at home: corners of doors and books are right angles, half-open scissors make an acute angle, and a laptop screen tipped back makes an obtuse one.
• Make a right-angle tester: fold any scrap of paper in half, then fold it again so the folded edge lies along itself. Use the corner to check whether angles are smaller, equal or bigger.
• Use clock times: at 3:00 the hands make a right angle, at 6:00 a straight angle, and at 1:00 an acute angle of 30°.
• Before any protractor reading, ask: more or less than a right angle?
• Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.