5.2 Volume & Surface Area of Pyramids, Cones, Spheres & Composite Solids

Why This Topic Matters on the GED

Indicators Q.5.d (pyramids and cones), Q.5.e (spheres), and Q.5.f (composite 3-D figures) round out the three-dimensional geometry assessment targets, contributing another 2–3 of the roughly 46 scored items. These questions have a reputation for being some of the trickier measurement items because they introduce a new variable — slant height — that is easy to confuse with the plain height used in volume formulas, and because composite-solid questions require you to recognize which basic shapes are combined and whether to add or subtract their volumes or surface areas. The payoff for mastering this section is real: once you can see a grain silo as "a cylinder plus a cone" or a scoop of ice cream as "a cone plus a hemisphere," the arithmetic is no harder than what you already practiced with prisms and cylinders in the previous section.

Core Vocabulary

• Apex — the single point opposite the base of a pyramid or cone.
• Height (h) — the perpendicular distance straight up from the base to the apex; used in every volume formula.
• Slant height (s) — the distance from the apex down to the edge of the base, measured along the outer surface (not straight down); used only in surface-area formulas for pyramids and cones, never in volume formulas.
• Composite solid — a figure built by joining two or more basic solids together, or by removing one solid from another; tested directly under Q.5.f.
• Hemisphere — exactly half of a sphere, common inside composite figures such as domes, tanks, and scoops.

How the GED Frames These Questions

Expect everyday objects built from these shapes: ice cream cones, party hats, tents, balls, globes, domed roofs, and grain silos. Some items give you every dimension and simply ask for volume or surface area. Others — matching the Q.5.d and Q.5.e wording "solve for side lengths, height, radius, or diameter when given volume or surface area" — give you the volume or surface area plus all but one dimension, and expect you to substitute and solve algebraically, exactly as you practiced with prisms and cylinders. Composite-figure items (Q.5.f) typically describe the object in words ("a cylindrical can topped with a cone-shaped lid") rather than showing a labeled diagram, so the first step is always to mentally split the description into the basic solids you already know.

The Formula Sheet: Pyramids, Cones & Spheres

(B = area of the base, p = perimeter of the base, s = slant height, h = height, r = radius; π ≈ 3.14)

The "One-Third Rule." A pyramid's volume is exactly one-third the volume of a right prism sharing its base and height. A cone's volume is exactly one-third the volume of a cylinder sharing its radius and height. If you ever blank on whether the fraction is ⅓ or ½, remember: a shape that tapers to a single point holds far less than the "full" solid it sits inside — one-third, not one-half.

Worked Examples

Example 1 — Pyramid volume. A square-based paperweight has a base side of 6 cm and a height of 10 cm. Base area B = 6 × 6 = 36 cm². V = (1/3)(36)(10) = 120 cm³.

Example 2 — Cone, solving for height. A conical measuring cup has a radius of 3 cm and holds 113.04 cm³ (π ≈ 3.14). Substituting: 113.04 = (1/3)(3.14)(9)(h) = 9.42h, so h = 12 cm.

Example 3 — Sphere surface area. A ball has a radius of 4.5 in. SA = 4πr² = 4(3.14)(20.25) = 254.34 in².

Example 4 — Composite solid, adding volumes. A grain silo is a cylinder (radius 6 ft, height 20 ft) topped with a cone of the same 6 ft radius and a height of 8 ft. Cylinder volume = πr²h = 3.14(36)(20) = 2,260.8 ft³. Cone volume = (1/3)πr²h = (1/3)(3.14)(36)(8) = 301.44 ft³. Total volume = 2,260.8 + 301.44 = 2,562.24 ft³.

Example 5 — Composite solid, subtracting volumes. A 10 cm cube of ice has a cylindrical hole (radius 2 cm) drilled straight through it, top to bottom. Cube volume = 10³ = 1,000 cm³. Cylinder removed = πr²h = 3.14(4)(10) = 125.6 cm³. Remaining volume = 1,000 − 125.6 = 874.4 cm³.

Common Traps

• Height vs. slant height. Using the vertical height h inside a cone or pyramid's surface-area formula (instead of the slant height s) is the single most common Q.5.d error — surface area needs s, while volume needs h.
• Forgetting the one-third factor. Treating a cone like a cylinder or a pyramid like a prism (dropping the ⅓) triples the correct volume.
• Hemisphere shortcuts. A hemisphere's volume is half of (4/3)πr³. If its flat circular face is exposed (a mixing bowl, for instance), total surface area needs the curved half (half of 4πr², which is 2πr²) plus the flat circle (πr²) — dropping either piece is a common miss.
• Double-counting hidden surfaces. In a composite solid like the silo above, the circle where the cylinder meets the cone is internal and covered — it should never be added into a total surface-area calculation, only the outward-facing surfaces count.

Key Takeaways

Keep height and slant height straight (volume uses h, surface area of cones/pyramids uses s), never drop the ⅓ factor on pyramids and cones, and when a solid is composite, decide first whether the question wants you to add two volumes together or subtract one solid's volume from another before you touch a formula.