Wentworth's Plane and Solid Geometry · Book VIII, Exs. 768–790 · Book 9: Book VIII: The Sphere
129. Numerical exercises: surfaces
Goal: Your student computes surfaces of spheres, zones, lunes and spherical triangles and polygons.
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 402–403: Numerical Exercises, Ex. 768–790.
The lesson, step by step
Warm up
Ask: Surface of a sphere? of a zone? of a lune? of a spherical triangle?
Spheres and domes
Ex. 768. Find the surface of a sphere, if the diameter is (i) 10 inches; (ii) 1 foot 9 inches; (iii) 2 feet 4 inches; (iv) 7 feet; (v) 10.5 feet.
Ex. 770. The circumference of a dome in the shape of a hemisphere is 66 feet. How many square feet of lead are required to cover it?
Triangles and polygons
Ex. 775. The angles of a spherical triangle are 60°, 70°, and 80°. The radius of the sphere is 14 feet. Find the area of the triangle.
Ex. 777. Find the area of a spherical polygon on a sphere whose radius is 10½ feet, if its angles are 100°, 120°, 140°, and 160°.
Zones
Ex. 780. The diameter of a sphere is 21 feet. Find the curved surface of a segment whose height is 5 feet.
Ex. 783. The altitude of the torrid zone is 3200 miles. Find its area, assuming the earth to be a sphere with a radius of 4000 miles.
A thinking problem
Ex. 788. How much of the earth's surface would a man see if he were raised to the height of the radius above it?
Draw the figure: the tangent lines from his eye touch the earth along a circle. Find the height of the zone he sees.
Wrap up
Check the answers together.
Tip
The book seems to expect π = 3.1416 or 22/7. With 22/7, many answers come out to whole numbers (Ex. 769, 770).
Answers to the book's problems
Ex. 768: (i) 100π ≈ 314.2 sq in; (ii) 441π ≈ 1385 sq in; (iii) 784π ≈ 2463 sq in; (iv) 49π ≈ 153.9 sq ft; (v) 110.25π ≈ 346.4 sq ft. Ex. 769: (i) about 14 in; (ii) about 3.5 ft; (iii) about 56 ft. Ex. 770: C²/(2π) ≈ 693 sq ft. Ex. 771: about 16,286 sq in, about $1140. Ex. 772: ½. Ex. 773: ⅓ sq ft. Ex. 774: the angles total 240°, so E = 60 and the triangle is 1/12 of the surface. Ex. 775: E = 30; about 102.6 sq ft. Ex. 776: the polar triangle has angles 100°, 106°, 52°; E = 78; about 266.8 sq ft. Ex. 777: E = 160; about 307.9 sq ft. Ex. 779: E = 60; πR²/3 = 4π gives R = √12 ≈ 3.46 ft. Ex. 780: 2π × 10.5 × 5 = 105π ≈ 329.9 sq ft. Ex. 781: R/2. Ex. 782: π(r² + h²), a circle whose radius is the chord of the generating arc (§ 828); on the same sphere, doubling the height doubles the area. Ex. 783: 2π × 4000 × 3200 ≈ 80,425,000 sq mi. Ex. 788: the visible zone has height R/2, one quarter of the surface. Ex. 789: the visible zone must have height R/3; the height above the earth is R/2. Ex. 790: the diagonal of the cube is the diameter, and is √3 times the edge.
Worked out for this site from the scan. If a number in your copy differs, trust the book.