ACT® · Math: Geometry — Geometric Measurement and Modeling
Composite Solids
A composite solid is two or more standard shapes stuck together. You find its volume or surface area by splitting it into the familiar pieces, computing each separately, then combining them correctly.
No single formula covers a cone with a scoop of ice cream on top, because that shape isn't one of the standard solids you've memorized. It's a cone glued to a sphere, and nobody hands you a one-step formula for 'cone-plus-scoop.' What you do have, cold, are the formulas for cones, cylinders, spheres, prisms, and pyramids. So the move is always the same: break the weird shape into the boring shapes you already know, solve each piece, then combine.
For volume, combining almost always means adding. A silo made of a cylinder with a hemisphere on top has a total volume equal to the cylinder's volume plus the hemisphere's volume, found separately then summed. Half a sphere means half of four thirds π r cubed, which simplifies to two thirds π r cubed. Nothing gets subtracted here because both pieces are solid and fully present. For surface area, combining means adding the outside surfaces but subtracting the hidden ones. Wherever two solids are fused, the circle or rectangle where they touch is buried inside the shape. It still has area, but it isn't part of the boundary you can touch from outside, so it never gets counted.
This topic is the payoff for knowing your basic volume and surface area formulas cold, since composite solids are built entirely from pieces you've already handled separately. It also connects forward to geometric probability problems, where you sometimes need the volume of an oddly shaped region as a denominator or numerator in a ratio.
Key terms
- Composite solid
- A three-dimensional shape built from two or more standard solids joined together.
- Hemisphere
- Half of a sphere, with volume two thirds π r cubed and curved surface area two π r squared.
- Seam
- The flat face where two solids in a composite shape are joined and hidden from the outside.
- Surface area
- The total area of all outer faces of a solid that are exposed to the outside.
Watch it explained
- How it worksHow do you split a composite solid into the standard pieces whose formulas you already carry?
- Worked exampleFind the volume of a silo built from a cylinder of radius 3 and height 10 topped by a hemisphere.
- Going furtherWhat changes about total surface area at the seam where two solids are joined together?
Check yourself
End of topic.