7.3 Volume & Surface Area: Right Solids, Cavalieri's Principle & Nets

7.3 Volume & Surface Area: Right Solids, Cavalieri's Principle & Nets

Competency 3 skill 3 is worded unusually — it names right solids, Cavalieri's principle, and nets for non-right solids by name. Those three phrases tell you exactly what is tested: the standard formulas, the principle that extends them to oblique solids, and the unfolding technique that produces surface area.

Volume formulas

+---------------------------------------------------------------------------+
|  Prism (any base)        V = Bh          B = area of the base             |
|  Cylinder                V = pi*r^2*h                                     |
|  Pyramid (any base)      V = (1/3)Bh                                      |
|  Cone                    V = (1/3)*pi*r^2*h                               |
|  Sphere                  V = (4/3)*pi*r^3                                 |
+---------------------------------------------------------------------------+

The unifying idea: prisms and cylinders are "base area times height." Pyramids and cones are the pointed versions of the same base and height, and they hold exactly one third as much. Three cones fill a cylinder of the same base and height; three pyramids fill the matching prism.

A cylinder with r = 5 cm and h = 12 cm: V = π(25)(12) = 300π ≈ 942.5 cm³. A cone with the same base and height: V = (1/3)(300π) = 100π ≈ 314.2 cm³.

The one-third factor is the most-dropped element in the competency. If an answer choice is exactly three times another, one of them is the cone/pyramid trap.

Cavalieri's principle

If two solids have the same height and their cross sections at every level have equal areas, then the solids have equal volume.

The classic demonstration is a stack of coins: push the stack sideways into a leaning column and its volume does not change, because every horizontal cross section is still the same circle.

The consequence tested on the exam: an oblique prism has the same volume as a right prism with the same base and height, and the same holds for oblique cylinders, pyramids, and cones. So V = Bh applies whether or not the solid leans — provided h is the perpendicular height between the bases, not the slanted edge length.

An oblique cylinder with radius 4 and perpendicular height 10 has volume π(16)(10) = 160π, identical to the right cylinder with those dimensions.

This is precisely why the blueprint pairs "right solids" with "Cavalieri's principle": the formulas you memorize for right solids extend to oblique ones, and Cavalieri's principle is the justification.

Note that Cavalieri's principle governs volume only. Surface area is not preserved — an oblique prism has longer lateral faces and therefore greater lateral surface area than the right prism of the same height.

Nets and surface area

A net is the flat unfolding of a solid's surface. Rather than memorizing surface area formulas, read them off the net.

Rectangular prism. The net shows six rectangles in three congruent pairs:

SA = 2lw + 2lh + 2wh

Any prism. The net shows two congruent bases and one long rectangle whose length is the base perimeter and whose height is the prism height:

SA = 2B + Ph

For a triangular prism with a 3-4-5 right triangle base and length 10: B = ½(3)(4) = 6, P = 12, so SA = 2(6) + 12(10) = 132 square units.

Cylinder. The net shows two circles plus a rectangle that wraps around, whose width is the circumference:

SA = 2πr² + 2πrh

Pyramid. The net shows the base plus triangular faces. Each lateral triangle has base equal to a base edge and height equal to the slant height ℓ:

SA = B + ½Pℓ

Cone. The net shows a circle plus a sector:

SA = πr² + πrℓ

Sphere. A sphere has no net, since it cannot be flattened without distortion:

SA = 4πr²

Nets are also tested directly: given a net, identify the solid, or given a solid, choose the valid net. A common item shows several arrangements of six squares and asks which fold into a cube. Of the 35 hexomino arrangements, only 11 form a cube; the reliable test is that no two squares may overlap when folded, which rules out any arrangement with four squares around a single vertex.

Slant height versus vertical height

[!WARNING] Volume uses vertical height h. Lateral surface area of a pyramid or cone uses slant height ℓ. They are different numbers, related by the Pythagorean theorem.

For a cone: ℓ² = h² + r². For a square pyramid: ℓ² = h² + (s/2)², where s is the base edge, because the slant height runs to the midpoint of a base edge.

A cone has r = 6 and h = 8. Then ℓ = √(36 + 64) = √100 = 10. Volume = (1/3)π(36)(8) = 96π ≈ 301.6 Surface area = π(36) + π(6)(10) = 36π + 60π = 96π ≈ 301.6 square units

That the two happen to share the value 96π here is a coincidence of the chosen numbers — the units differ (cubic versus square), and items exploit the confusion by offering both.

Working backward and choosing the right measure

A cube has surface area 150 in². Find its volume. 6s² = 150 → s² = 25 → s = 5, so V = 125 in³.

Deciding which measure a context wants is itself assessed:

The "open top" qualifier changes the formula: an open cylindrical tank has SA = πr² + 2πrh, with one circle rather than two.