Wentworth's Plane and Solid Geometry · Book VII, §§ 630–644 (Prop. XIV) · Book 8: Book VII: Polyhedrons, Cylinders, and Cones
99. Pyramids and their lateral area
Goal: Your student can define a pyramid and its parts, a regular pyramid, slant height and frustum, and prove that the lateral area of a regular pyramid is half the slant height times the perimeter.
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
Definitions of pyramids (§§ 630–642), Proposition XIV (§§ 643–644), and Exercises 657–659.
The lesson, step by step
Warm up
Ask: What is the area of a triangle? (Half the base times the altitude.)
The new idea
A pyramid is a polyhedron of which one face, called the base, is a polygon of any number of sides and the other faces are triangles having a common vertex.
The lateral area of a regular pyramid is equal to half the product of its slant height by the perimeter of its base.
Draw the figure
Draw a square pyramid with its altitude (dashed, inside) and its slant height on one face. Mark a frustum by cutting it parallel to the base.
Walk through the proof
XIV: the lateral faces are equal isosceles triangles (§ 637), each with altitude L. Each has area ½ L times its base; adding, S = ½ L × P.
§ 644: each face of the frustum is a trapezoid of altitude L, so the frustum's lateral area is ½ (P + p) × L.
Your student proves it back
Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Ex. 657. Find the lateral area of a regular pyramid if the slant height is 16 feet, and the base is a regular hexagon with side 12 feet.
Ex. 659. Find the total surface of a regular pyramid if the slant height is 6 feet, and the base is a square with side 4 feet.
Wrap up
Ask: What is the difference between the altitude and the slant height of a pyramid? Which is longer? (The slant height.)
Answers to the book's problems
Ex. 657: ½ × 16 × 72 = 576 sq ft. Ex. 658: ½ × 8 × 25 = 100 sq ft. Ex. 659: 48 + 16 = 64 sq ft.
Worked out for this site from the scan. If a number in your copy differs, trust the book.