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The pyramid and the cone: volume and surface area

The plane and space, grade 12 (3 units) Lesson 6 of 15Learning

The pyramid and the cone: volume and surface area

The volume of a pyramid and a cone

Fill a pyramid with sand and pour it into a prism with the same base and the same height. After three full pyramids the prism is full. So it is with a cone and a cylinder:

  • The volume of a pyramid is a third of the volume of the prism with the same base and height.
  • The volume of a cone is a third of the volume of the cylinder with the same base and height.
V\displaystyle V =\displaystyle = B⋅h3\displaystyle \frac{B\cdot h}{3}
V\displaystyle V =\displaystyle = πr2h3\displaystyle \frac{\pi r^2h}{3}

B\displaystyle B – the area of the base, h\displaystyle h – the height: the distance from the apex to the base.

Heights in a square pyramid

hkOMa

In a square pyramid the apex is exactly above the centre of the base, O\displaystyle O. It has two heights:

  • h\displaystyle h – the pyramid’s height: from the apex to O\displaystyle O. It gives the volume.
  • k\displaystyle k – the height of a side face: from the apex to M\displaystyle M, the midpoint of a side of the base. It gives the area of the triangles.

The segment OM\displaystyle OM is half a side of the base. h\displaystyle h is perpendicular to the base, so h\displaystyle h, OM\displaystyle OM and k\displaystyle k are the sides of a right-angled triangle, and k\displaystyle k is the hypotenuse:

k2\displaystyle k^2 =\displaystyle = h2+(a2)2\displaystyle h^2+\left(\frac{a}{2}\right)^2

The surface area: the base a2\displaystyle a^2, plus four triangles, a⋅k2\displaystyle \frac{a\cdot k}{2} each:

A\displaystyle A =\displaystyle = a2+2ak\displaystyle a^2+2ak
The slant height and surface area of a cone

hrl

The slant height of a cone, l\displaystyle l, is the segment from the apex to the edge of the base. The height h\displaystyle h, the radius r\displaystyle r and the slant height are the sides of a right-angled triangle, and l\displaystyle l is the hypotenuse:

l2\displaystyle l^2 =\displaystyle = r2+h2\displaystyle r^2+h^2

A cone’s curved surface unrolls into a sector of a disc, and its area is πrl\displaystyle \pi rl. The surface area is the base plus the curved surface:

L\displaystyle L =\displaystyle = πrl\displaystyle \pi rl
A\displaystyle A =\displaystyle = πr2+πrl\displaystyle \pi r^2+\pi rl
Example: the glass pyramid

At the entrance to the Louvre museum in Paris stands a square pyramid of glass.
The side of its base is 35 m, and its height 21.6 m.
What is its volume, and how many m² of glass are in its four side faces?

  • The volume: V=352⋅21.63=8820\displaystyle {V}={\frac{35^2\cdot21.6}{3}}={8820} m³.
  • The height of a face: OM=35÷2=17.5\displaystyle {OM}={35\div2}={17.5}, so k=21.62+17.52≈27.80\displaystyle {k}={\sqrt{21.6^2+17.5^2}\approx27.80} m.
  • The glass: four triangles, 4⋅35⋅27.802≈1946\displaystyle 4\cdot\frac{35\cdot27.80}{2}\approx1946 m².
Exercise 1

A pile of sand on a building site is a cone: the radius is 2 m, and the height 1.5 m.
A lorry carries up to 8 m³ of sand.
Is one lorry enough to clear the pile?

    / – 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

    A tepee is a cone: the radius of its floor is 1.5 m, and its height 2 m.
    The cloth covers the curved surface.

    a
    The slant height, 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

      b
      How many m² of cloth does the tent have? Round to two decimal places.

        / – 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 pavilion’s roof is a square pyramid: the side of the base is 8 m, and the height 3 m.
        The four faces are covered with roof tiles. A pack of tiles covers 1.5 m².

        a
        The height of a face, 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

          b
          The area of the four faces, 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

            c
            How many packs of tiles must be bought?
            Exercise 4

            An ice-cream cone is a cone: the radius is 2.5 cm, and the height 10 cm.
            How many ml of ice cream fit in the cone, up to the rim? Round to two decimal places.

              / – 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