Wentworth's Plane and Solid Geometry · Book IX, Props. IX–X · Book 10: Book IX: Conic Sections

136. The area of a parabolic segment; the parabola as a section of a cone

Goal: Your student can explain why a parabolic segment is two thirds of the triangle of its tangents, and why a parabola is a section of a cone.

⏱ About 40 minutes · 6 steps

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You'll need

  • The book, open to these pages
  • A ruler (straightedge) and a pair of compasses
  • Sharp pencil, eraser and plain paper (or a geometry notebook)
  • Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)

What's in the book

Propositions IX–X (§§ 893–895).

The lesson, step by step

  1. Warm up

    Ask: State Prop. VIII.

  2. The new idea

    The area of a parabolic segment made by a chord is two thirds the area of the triangle formed by the chord and the tangents drawn through the ends of the chord.

    The section of a right circular cone made by a plane parallel to one, and only one, element of the surface is a parabola.

  3. Draw the figure

    Shine a flashlight on a wall at a slant until the edge of the beam just becomes open: that edge is a parabola, a section of the cone of light.

    For Prop. X, draw the side view: the cone, the slanting plane parallel to one element, and a ball resting inside the cone touching the plane at F.

  4. Walk through the proof

    IX: each inner triangle is twice the outer triangle made by the tangents (§ 405); repeat forever, and the segment is ⅔ of the whole triangle.

    X: a sphere inside the cone touches the cone in a circle and the plane at F. From any point P of the curve, PF and PK (along the cone to the circle) are tangents to the sphere, so they are equal; and PK equals the distance from P to a fixed line. So P is as far from F as from that line.

  5. Your student proves it back

    Close the book. Your student draws the curve and the figure from memory, letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason. Point to each line as it is named.

    If a step is missing, ask "How do we know that?" rather than supplying it.

    Prop. X is hard. It is enough that your student explains why the two tangent lengths PF and PK are equal.

  6. Wrap up

    Ask: What kind of section do you think you get if the plane cuts every element of the cone? (An ellipse, coming next.)

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