Wentworth's Plane and Solid Geometry · Book VIII, Exs. 811–835 · Book 9: Book VIII: The Sphere
131. Miscellaneous exercises (1)
Goal: Your student proves theorems and solves problems drawing on all of solid geometry.
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 calculator (optional)
What's in the book
Pages 405–406: Miscellaneous Exercises, Ex. 811–835.
The lesson, step by step
Warm up
Ask your student to name one big result from each of Books VI, VII and VIII.
Computations
Ex. 819. Find the area of the surface of the sphere inscribed in a regular tetrahedron whose edge is 6 inches.
Ex. 821. Find the difference between the volume of a frustum of a pyramid and the volume of a prism each 24 feet high, if the bases of the frustum are squares with sides 20 feet and 16 feet, respectively, and the base of the prism is the section of the frustum parallel to the bases and midway between them.
Ex. 822. If the earth is assumed to be a sphere of 4000 miles radius, how far at sea can a lighthouse 100 feet high be seen?
Theorems
Your student picks two:
Ex. 812. The portion of a tetrahedron cut off by a plane parallel to any face is a tetrahedron similar to the given tetrahedron.
Ex. 830. The straight lines joining the middle points of the opposite edges of a tetrahedron all pass through the centre of gravity of the tetrahedron, and are bisected by the centre of gravity.
or Ex. 813, 829 or 832.
A puzzle
Ex. 834. To cut a cube by a plane so that the section shall be a regular hexagon.
Try it with a cube of clay or a potato.
Wrap up
Your student reads one proof aloud.
Answers to the book's problems
Ex. 818: (4/3)πR³ : (2R/√3)³ = π : (2/3)√3. Ex. 819: r = √6/2, so the surface is 6π ≈ 18.85 sq in. Ex. 820: if h is the height of the zone, (2πRh)² = 4πR² × (4πR² − 2πRh), so h² + 2Rh − 4R² = 0 and h = R(√5 − 1) ≈ 1.236R; the base is R(√5 − 2) ≈ 0.236R from the centre, the zone being larger than a hemisphere. Ex. 821: 7808 − 7776 = 32 cu ft. Ex. 822: about √(2Rh) ≈ 12.3 miles. Ex. 823: (4/3)π(4050³ − 4000³) ≈ 10,180,000,000 cubic miles. Ex. 829: compare Book III, medians of a triangle. Ex. 832: join the point to the four vertices; the four pyramids have the faces as bases and add up to the whole. Ex. 834: pass the plane through the middle points of six edges, perpendicular to a diagonal at its middle.
Worked out for this site from the scan. If a number in your copy differs, trust the book.