Ray's New Higher Arithmetic · Art. 410 · Unit 28: Mensuration

122. Solids: surfaces and volumes

Goal: Your student names prisms, cylinders, pyramids, cones, frustums and spheres, and finds their surfaces and volumes.

⏱ About 45 minutes · 7 steps

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You'll need

  • The book
  • A slate, small whiteboard or scrap paper
  • A notebook and pencil
  • A calculator, for checking only (optional)
  • A box, a can, and a ball if handy

What's in the book

Solids: definitions (solid, faces and edges, prism, parallelepiped, cube, cylinder, pyramid, cone, frustum, sphere, slant height, altitude, volume, similar solids), eight general rules, notes, and examples (answers printed).

The lesson, step by step

  1. Warm-up

    Hold up a box, a can and a ball. Ask your student to describe each without naming it.

  2. The idea

    Read:

    A Prism is a solid having two bases which are parallel polygons, and faces which are parallelograms.

    A Cylinder is a solid having two bases which are equal parallel circles

    A Pyramid is a solid with only one base, and whose faces are triangles with a common vertex.

    A Sphere is a solid bounded by a curved surface, every point of which is at the same distance from a point within, called the center.

  3. Volume

    Read:

    The Volume of a solid is the number of solid units it contains; the assumed unit of measure is a cube.

  4. The rules

    Read:

    Multiply the perimeter of the base by the altitude.

    Multiply the base by the altitude.

    Multiply the base by one third of the altitude.

    (These are the side surface of a prism or cylinder, its volume, and the volume of a pyramid or cone.) Then for a sphere:

    Multiply the circumference by the diameter.

    Multiply the cube of the diameter by .5235987

  5. Guided practice

    Find the whole surface of a right triangular prism, the sides of the base 60, 80, and 100 ft.; altitude 90 ft.

    The base is a right triangle (60, 80, 100), so each end is ½ × 60 × 80.

  6. Independent practice

    Your student works more examples, such as:

    Whose altitude is 15.24 in., and whose base is a triangle having each side 1 ft.

    (a pyramid). Check the printed answers.

  7. Wrap-up

    Ask: A cone and a cylinder have the same base and height. How many cones of water fill the cylinder? (3.)

Afterwards

Two sittings: definitions and surfaces, then volumes.

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