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The prism: its parts, lateral area and surface area

Space geometry and vectors, grade 12 (4 units) Lesson 4 of 20Learning

The prism: its parts, lateral area and surface area

The prism

hABCA′B′C′H

A prism is a solid whose two bases are congruent polygons in parallel planes, each vertex of one base joined by an edge to the matching vertex of the other.

  • The bases – ABC\displaystyle ABC and A′B′C′\displaystyle A'B'C'.
  • The lateral faces – parallelograms, such as ABB′A′\displaystyle ABB'A'.
  • The edges – the base edges (AB\displaystyle AB, B′C′\displaystyle B'C' …) and the side edges (AA′\displaystyle AA', BB′\displaystyle BB', CC′\displaystyle CC'). The side edges are equal and parallel to each other.
  • A diagonal – a segment joining two vertices that are not on one face. A triangular prism has no diagonals; a box has four.
  • The height – the distance between the bases’ planes: a perpendicular segment from a vertex of one base to the other base’s plane, such as A′H\displaystyle A'H. Its foot H\displaystyle H may also be outside the base.
The parallelepiped, the right prism, the box and the cube

ABCDA′B′C′D′

  • A parallelepiped – a prism whose bases are parallelograms. All six of its faces are parallelograms.
  • A right prism – the side edges are perpendicular to the base plane. The lateral faces are rectangles, and the height equals the side edge.
  • A box – a right prism whose base is a rectangle.
  • A cube – a box whose edges are all equal.
The lateral area and the surface area

The lateral area M\displaystyle M is the sum of the areas of the lateral faces. The surface area S\displaystyle S is the lateral area and the two bases (B\displaystyle B – the area of one base):

S\displaystyle S =\displaystyle = M+2B\displaystyle M+2B

In a right prism the lateral faces are rectangles, all of height h\displaystyle h, and their bases add up to the perimeter P\displaystyle P of the base:

M\displaystyle M =\displaystyle = P⋅h\displaystyle P\cdot h

In a slanted prism each lateral face is worked out on its own. The area of a parallelogram with sides a\displaystyle a, b\displaystyle b and the angle α\displaystyle \alpha between them is a⋅b⋅sin⁡α\displaystyle a\cdot b\cdot\sin\alpha (grade 11).

Example: a right triangular prism

34ABCA′B′C′

The bases of a right prism are equilateral triangles of side 3. The side edge is 4 long.

  • M=P⋅h=9⋅4=36\displaystyle {M}={P\cdot h}={9\cdot4}={36}.
  • The base’s area: B=12⋅3⋅3⋅sin⁡60∘=934\displaystyle {B}={\frac{1}{2}\cdot3\cdot3\cdot\sin60^{\circ}}={\frac{9\sqrt{3}}{4}}.
  • S=36+2⋅934=36+932≈43.79\displaystyle {S}={36+2\cdot\frac{9\sqrt{3}}{4}}={36+\frac{9\sqrt{3}}{2}\approx43.79}.
Exercise 1

8610ABCA′B′C′

The base of a right prism is the triangle ABC\displaystyle ABC, right-angled at B\displaystyle B: AB=8\displaystyle {AB}={8}, BC=6\displaystyle {BC}={6}.
The side edge is 10 long.

a
AC\displaystyle AC

    / – 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 lateral area

      / – 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 surface area

        / – 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

        ABCDA′B′C′D′

        The base of a right prism is a rhombus with diagonals 6 and 8 long.
        The prism is 5 high.

        a
        The rhombus’s side

          / – 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 lateral area

            / – 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 surface area

              / – 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

              60°ABCDA′B′C′D′

              The base of the parallelepiped ABCDA′B′C′D′\displaystyle ABCDA'B'C'D' is a square of side 4.
              The side edge is AA′=5\displaystyle {AA'}={5}, and ∠A′AB=∠A′AD=60∘\displaystyle {\angle A'AB}={\angle A'AD}={60^{\circ}}.
              Find exactly:

              a
              The area of the face ABB′A′\displaystyle ABB'A'

                / – 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 lateral area

                  / – 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 surface area

                    / – 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

                    ABCA′B′C′

                    In the prism ABCA′B′C′\displaystyle ABCA'B'C':

                    a

                    Which faces hold the edge BB′\displaystyle BB'?

                    b

                    Which three vertices determine the plane of the face ACC′A′\displaystyle ACC'A'?

                    c

                    A prism whose base is a pentagon: how many vertices, edges and faces?

                    Vertices

                      / – 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

                      Edges

                        / – 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

                        Faces

                          / – 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