An angle is a turn
Two straight lines that meet at a point make an angle. The point where they meet is the vertex, and the two lines are the arms of the angle.
Start with both arms on top of each other, then turn one arm around the vertex. The size of the angle is how far that arm has turned away from the other one. Angles are measured in degrees, and degrees are written with a small circle: 90° is read "ninety degrees".
A quarter turn is a right angle
Turn the arm a quarter of the way round, and the angle is a right angle, 90°. The corner of a page and the corner of a door are right angles. A right angle is marked with a small square in the corner instead of a curve.
an angle is the turn between two arms; scaling both arms by any factor leaves θ unchanged
Make a right angle
Turn the arm until the small square appears in the corner. That is a quarter turn, a right angle of 90°.
Less than a right angle: acute
An angle that has turned less than a quarter turn is smaller than a right angle. An angle smaller than 90° is called acute. An angle of 40° is acute, because 40° is less than 90°.
The arm has turned 40°, less than a right angle, so the angle is acute.
More than a right angle: obtuse
Keep turning past the quarter turn and the angle grows bigger than a right angle. An angle between 90° and a straight line is called obtuse. An angle of 130° is obtuse, because 130° is more than 90°.
Two right angles make a straight line, which is 180°. So an obtuse angle is more than 90° and less than 180°.
The arm has turned 130°, past a right angle, so the angle is obtuse.
Compare with a right angle
To name an angle, compare it with a right angle. Smaller than 90° is acute, exactly 90° is a right angle, and bigger than 90° is obtuse. An angle of exactly 90° is not acute and not obtuse.
The length of the arms does not change the size of an angle. Making both arms longer does not make the turn bigger. A small angle drawn with long arms is still a small angle, so judge an angle by its turn, not by how long its sides are.
Worked example: The Angles of a Shape Sorted Against a Right Angle
Question A kite maker cuts a shape with five corners, A, B, C, D and E. She measures the angle at each corner. Angle A is 90°, angle B is 120°, angle C is 140°, angle D is 60° and angle E is 130°. (a) How many of the angles are greater than a right angle? (b) How many of the angles are smaller than a right angle?
1.A right angle is 90°. Compare each of the five angles with 90°.
A right angle is 90°. Compare each angle with 90°. 2.120°, 140° and 130° are all more than 90°. So angles B, C and E are greater than a right angle.
120°, 140° and 130° are more than 90°: angles B, C and E. 3.(a) 3 of the angles are greater than a right angle.
(a) 3 of the angles are greater than a right angle. 4.60° is less than 90°, so angle D is smaller than a right angle. Angle A is exactly 90°, so it is a right angle and it belongs to neither group.
60° is less than 90°, so angle D is smaller. Angle A is exactly a right angle. 5.(b) 1 angle is smaller than a right angle. Check: 3 + 1 + 1 = 5, so every corner has been sorted.
(b) 1 angle is smaller than a right angle. Check: 3 + 1 + 1 = 5.
Answer: (a) 3 angles; (b) 1 angle
Common mistakes
- Counting angle A as greater than a right angle. Angle A is exactly 90°. It is equal to a right angle, not greater than one.
- Judging an angle by the length of its sides. Angle D has the longest sides, but it is the smallest angle. The size of an angle is how far one side has turned from the other.
More thousands, angles and time problems, worked step by step →