Wentworth's Plane and Solid Geometry · Book IX, Props. XXIII–XXIV · Book 10: Book IX: Conic Sections
142. Area of an ellipse; the ellipse as a section of a cone
Goal: Your student can prove that the area of an ellipse is πab and explain why a slanting section of a cone is an ellipse.
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)
- A flashlight
What's in the book
Propositions XXIII–XXIV (§§ 942–944).
The lesson, step by step
Warm up
Ask: What is the area of a circle? How is the ellipse related to its auxiliary circle?
The new idea
The area of an ellipse is equal to πab.
The section of a right circular cone made by a plane that cuts all the elements of the surface of the cone is an ellipse.
Draw the figure
Shine a flashlight at a slant on the floor: the spot of light is an ellipse. Draw the side view for Prop. XXIV: the cone, the slanting plane, and two balls inside the cone touching the plane at F and F' (one above, one below).
Walk through the proof
XXIII: inside both the ellipse and the circle draw thin rectangles on the same bases; each ellipse rectangle is b/a of the circle rectangle. Take the limit: ellipse = (b/a) × πa² = πab.
XXIV: from any point P of the curve, PF equals the tangent PN to the upper sphere and PF' equals the tangent PN' to the lower sphere. NN' is the same length on every element, so PF + PF' is constant.
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.
Exercises
Compute the area of an ellipse with axes 10 and 6 cm. (π × 5 × 3 ≈ 47.1 sq cm.)
Wrap up
Ask: Why are these called "conic" sections? (They are all slices of a cone.)