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The volume of a prism, a pyramid and a sphere

Measures and solids, grade 6 Lesson 14 of 16Learning

The volume of a prism, a pyramid and a sphere

The volume of a prism
  • V\displaystyle V – the volume
  • S\displaystyle S – the base’s area
  • h\displaystyle h – the height

h

The volume of a prism, like that of a box and a cylinder, is the base’s area times the height:

V\displaystyle V =\displaystyle = S×h\displaystyle S\times h
  • The base can be a triangle, a pentagon or any other polygon.
  • The height is the distance between the two bases.
  • A box is a prism whose base is a rectangle.
Finding a prism’s volume

A prism’s base is a triangle: one of its sides is 6 cm, and the height to that side is 4 cm.
The prism is 10 cm high.
What is the prism’s volume?

  • The base’s area: 6×4:2=12\displaystyle {6\times4:2}={12} cm².
  • The volume: 12×10=120\displaystyle {12\times10}={120} cm³.
Exercise 1

Find the prism’s volume, in cm³.

a
the base 15 cm², the height 8 cm

    / – 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 base a triangle: a side of 10 cm, the height to it 6 cm; the prism 20 cm high

      / – 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
      the base a hexagon of 26 cm², the height 5 cm

        / – 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

        The volume of a pyramid

        h

        A pyramid and a prism with the same base and height are like the cone and the cylinder: three pyramids of sand fill the prism. So the volume of a pyramid is a third of the prism’s:

        V\displaystyle V =\displaystyle = S×h3\displaystyle \frac{S\times h}{3}
        • The pyramid’s height is the distance from its top vertex to its base (the dashed line in the drawing).
        • For example, a pyramid whose base is a 6 cm by 6 cm square and whose height is 5 cm: 6×6×5:3=60\displaystyle {6\times6\times5:3}={60} cm³.
        Exercise 2

        Find the pyramid’s volume.

        a
        a 3 cm by 3 cm square base, 4 cm high; in cm³

          / – 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 10 cm by 6 cm rectangle base, 9 cm high; in cm³

            / – 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
            the Great Pyramid of Giza: a 230 m by 230 m square base, about 147 m high; in m³

              / – 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

              The volume of a sphere

              A sphere is a solid that is round all over, like a basketball: all the points on its surface are the same distance from its centre – the radius.

              • Take a cylinder that wraps a sphere exactly: its radius is the sphere’s radius, and its height is the diameter, 2r\displaystyle 2r.
              • Fill the cylinder with water and push the sphere in: two thirds of the water spill out, and a third stays.

              r2r

              So the volume of a sphere is 23\displaystyle \frac{2}{3} of the cylinder around it:

              V\displaystyle V =\displaystyle = 23×πr2×2r\displaystyle \frac{2}{3}\times\pi r^2\times2r

              For example, a sphere whose radius is 3 cm:

              • The cylinder around it: 3.14×3×3×6=169.56\displaystyle {3.14\times3\times3\times6}={169.56} cm³.
              • The sphere: 169.56:3×2=113.04\displaystyle {169.56:3\times2}={113.04} cm³.
              Exercise 3

              Find the sphere’s volume, in cm³ (π≈3.14\displaystyle \pi\approx3.14).

              a
              the radius is 6 cm

                / – 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 basketball: the radius is 12 cm

                  / – 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 4

                  Design two different solids – a prism, a pyramid, a cylinder or a cone – each with a volume of 60 cm³.
                  Write their measurements, and show by calculating that the volume comes out 60 cm³.

                  Exercise 5

                  A pyramid and a prism have the same base – a 4 cm by 4 cm square.
                  Each has a volume of 48 cm³.
                  How high is the prism? And the pyramid?

                  The prism’s height

                    / – 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

                    The pyramid’s height

                      / – 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