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Areas: the rectangle, triangle, parallelogram and trapezoid

The plane and space, grade 10 (3 units) Lesson 8 of 15Learning

Areas: the rectangle, triangle, parallelogram and trapezoid

Area

The area of a shape is how many unit squares cover it.

  • A square metre (m²) is the area of a square of 1 m by 1 m; a square centimetre (cm²) – of a square of 1 cm by 1 cm.
  • To paint a wall, tile a floor or plant grass, you need the area.
  • A perimeter is measured in units of length, and an area in square units. Two shapes with the same perimeter can have different areas.
The formulas
  • A rectangle – a\displaystyle a, b\displaystyle b two sides that meet at a vertex:
A\displaystyle A =\displaystyle = a⋅b\displaystyle a\cdot b
  • A parallelogram – a side times the height to it, ha\displaystyle h_a:
A\displaystyle A =\displaystyle = a⋅ha\displaystyle a\cdot h_a
  • A triangle – half a parallelogram with the same side and the same height:
A\displaystyle A =\displaystyle = a⋅ha2\displaystyle \frac{a\cdot h_a}{2}
  • A trapezoid – a\displaystyle a, b\displaystyle b the bases, h\displaystyle h the height:
A\displaystyle A =\displaystyle = (a+b)⋅h2\displaystyle \frac{(a+b)\cdot h}{2}
  • A height is perpendicular to the side. In the drawing: the height h\displaystyle h of the trapezoid, not the slanting leg.
Example: painting a wall

A wall in a room is 4 m wide and 2.6 m high, and it has a door of 0.9 m by 2 m.
A can of paint covers 5 m².
The example below works out the area to paint, and how many cans to buy.

  • The wall: 4⋅2.6=10.4\displaystyle {4\cdot2.6}={10.4} m².
  • The door, which is not painted: 0.9⋅2=1.8\displaystyle {0.9\cdot2}={1.8} m².
  • The area to paint: 10.4−1.8=8.6\displaystyle {10.4-1.8}={8.6} m².
  • Cans: 8.6÷5=1.72\displaystyle {8.6\div5}={1.72}. One can is not enough, so you buy 2 cans – when buying, round up.
Exercise 1

Calculate the area, in m².

a
A sail shaped like a triangle: a side of 3 m, and the height to it 5 m

    / – 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

    A plot shaped like a parallelogram. The measurements in the drawing are in metres.

      / – 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

      A plot shaped like a trapezoid. The measurements in the drawing are in metres.

        / – 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 wall of a shed is a rectangle of 6 m by 3 m, with a triangle on top: the base 6 m and the height 2 m.
        A litre of paint covers 8 m².

        a
        The area of the wall, in m²

          / – 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
          How many litres of paint are needed?

            / – 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

            A lawn shaped like a trapezoid: the bases 12 m and 8 m, and the height 5 m.
            Turf costs 18 ₪ a square metre.

            a
            The area of the lawn, in m²

              / – 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
              The price of the turf, in shekels

                / – 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