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Measuring angles

Geometry, grade 7 Lesson 2 of 18Learning

Measuring angles

Degrees

The size of an angle is measured in degrees. Degrees are written with a small circle: 90∘\displaystyle 90^{\circ} (read: 90 degrees).

  • A full turn around a point is 360∘\displaystyle 360^{\circ}: one degree is one of 360 equal parts of a full turn.
  • A straight angle is half a turn: 180∘\displaystyle 180^{\circ}.
  • A right angle is a quarter of a turn: 90∘\displaystyle 90^{\circ}.
  • An acute angle is smaller than 90∘\displaystyle 90^{\circ}.
  • An obtuse angle is bigger than 90∘\displaystyle 90^{\circ} and smaller than 180∘\displaystyle 180^{\circ}.
The protractor

An angle is measured with a protractor. In the drawing, a protractor lies on an angle:

18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180

  • The centre of the protractor is on the vertex of the angle.
  • The baseline – the straight line through the centre – lies along one of the arms.
  • The arc has two scales of numbers, from 0 to 180: the outer scale starts at 0 on the left, and the inner scale starts at 0 on the right.
  • You read the scale whose 0 is on the arm along the baseline. Here that arm points right, so you read the inner scale: the other arm passes 50. The angle is 50∘\displaystyle 50^{\circ}.
  • The 130 on the outer scale belongs to another angle: the one between the other arm and the left part of the baseline.
Estimating before measuring

An estimate is a guess at about how big the angle is, without measuring. An estimate before measuring shows which scale to read:

  • Compare the angle with a right angle, 90∘\displaystyle 90^{\circ}, and with half of it, 45∘\displaystyle 45^{\circ}.
  • The angle on the protractor above is acute, so it is smaller than 90∘\displaystyle 90^{\circ}: it is 50∘\displaystyle 50^{\circ}, not 130∘\displaystyle 130^{\circ}.
  • A measurement far from the estimate is a sign of reading the wrong scale.
Drawing an angle of a given size

The steps below draw an angle of 70∘\displaystyle 70^{\circ} with a protractor:

  1. We draw a ray – one arm of the angle. Its start is the vertex.
  2. We put the centre of the protractor on the vertex, and the baseline along the ray.
  3. We find 70 on the scale whose 0 is on the ray, and mark a point there.
  4. We lift the protractor and draw a ray from the vertex through the point.
  5. We mark the angle with an arc and write 70∘\displaystyle 70^{\circ} beside it.
Exercise 1

How many degrees is the angle on each protractor?

a

18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

    18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

      18001701016020150301404013050120601107010080909080100701106012050130401403015020160101700180

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        Exercise 2

        The sizes of the three angles are 25∘\displaystyle 25^{\circ}, 85∘\displaystyle 85^{\circ} and 140∘\displaystyle 140^{\circ}, not in this order.
        Write the size of each angle, without measuring.
        Explain.

        αβγ

        α\displaystyle \alpha

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          β\displaystyle \beta

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            γ\displaystyle \gamma

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              Exercise 3

              Draw an angle of the given size with a protractor.
              Mark the angle and write its size beside it.

              a

              35∘\displaystyle 35^{\circ}

              b

              120∘\displaystyle 120^{\circ}

              Exercise 4

              A clock has 12 numbers, equally far apart.
              The hands turn around the centre of the clock.
              Calculate the angle between the two hands at each time.

              a
              At one o’clock (1:00)

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                b
                At four o’clock (4:00)

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  c
                  At five o’clock (5:00)

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power