Separate composite volume from exposed surface area

Learning goal: Solve volume and surface-area problems for composites of right prisms and cubes. Add nonoverlapping volumes, identify internal contact faces and distinguish three-dimensional capacity from two-dimensional covering. This lesson builds on the composite-area lesson "Find composite areas using nonoverlapping parts and holes", which covers the standard's area part.

Volume measures the space occupied by a solid, in cubic units. A right prism has volume equal to its base area times its perpendicular height. For a rectangular prism, V = lwh. Surface area measures the total exposed face area, in square units. A composite can be built from simpler solids, but the rules for combining volume and exposed area are different.

Consider a lower rectangular block four units long, three units deep and two units high. Its volume is 4 × 3 × 2 = 24 cubic units. Place an upper block two units long, three units deep and one unit high on one end, with its base fully contacting the lower block and no interior overlap. The upper volume is 2 × 3 × 1 = six cubic units. Composite volume is thirty cubic units.

The blocks meet along a face but do not occupy the same three-dimensional interior. Adding their volumes counts each cubic region once. If one solid penetrated another rather than resting on it, the overlap volume would require separate attention. The diagram's stated placement makes simple addition valid.

The lower block's surface area is 2(4×3 + 4×2 + 3×2) = 2(12 + 8 + 6) = 52 square units. The upper block's area is 2(2×3 + 2×1 + 3×1) = 2(6 + 2 + 3) = 22. Adding gives 74, but that includes faces now hidden at the contact: the lower block's covered top patch and the upper block's covered bottom.

The contact area is 2 × 3 = six square units. It appears once in each separate block's surface-area total, so subtract it twice: 52 + 22 − 2(6) = 62 square units of exterior surface. Subtracting only six would leave one hidden copy counted as exposed. Internal joining faces do not contribute to a coating applied only to the outside.

Verify by exposed faces. The composite bottom contributes twelve. The top surfaces together contribute twelve: six from the upper top and six from the uncovered lower top. The front and back each have area ten, formed from a four-by-two rectangle plus a two-by-one upper rectangle. The left side has area nine and the right side six. The vertical step contributes three. Total: 12 + 12 + 20 + 9 + 6 + 3 = 62.

The question may exclude some exterior faces. If a coating covers every exposed surface except the bottom resting on a table, subtract the bottom's twelve from 62, giving fifty square units. That is a different task, not a correction to the full exterior surface-area calculation. State which faces count before choosing a formula.

A cube is a rectangular prism with all three dimensions equal. Joining two unit cubes face to face gives volume two cubic units. Their separate surface areas total twelve square units, but two copies of the one-square-unit shared face become internal, leaving ten. This small model makes the double subtraction visible and can be checked by counting actual exposed square faces.

Right prisms can also have triangular bases. A prism with a right-triangle base of perpendicular legs three and four, hypotenuse five and prism length six has volume [(1/2)(3)(4)] × 6 = 36 cubic units. Its surface area is two triangular bases totaling twelve plus lateral rectangles 3×6, 4×6 and 5×6, totaling 72; full surface area is 84 square units. The three boundary sides of the base determine the lateral rectangles.

When combining a triangular prism with another solid, identify both the contact shape and its dimensions. A shared rectangular face contributes its area twice to separate surface totals, just as in the block example. Avoid assuming every joint is one square unit or that a volume dimension can be substituted directly for a contact area.

Use consistent units and precise labels. The composite's thirty cubic units cannot be compared numerically with 62 square units as if one were simply more of the same thing. A filling problem asks about volume; a wrapping or coating problem asks about selected surface area. A diagram can support both calculations, but the desired measure determines which parts to count.

Worked contact check: Two identical two-by-two-by-two cubes are joined along a full face. Each has volume eight and surface area 24. Composite volume is sixteen; exterior surface area is 48 − 2(4) = forty square units. The contact area is four, not the cube's volume eight.

Worked hollow exclusion: If a problem asks only for the open-top material area of a single four-by-three-by-two box of negligible thickness, start with its closed surface area 52 and remove the top twelve, leaving forty. Do not apply that open-top interpretation to the solid composite unless the problem actually describes an opening.

Worked reasonableness: The two stacked blocks have a separate surface total of 74, so their joined exterior must be less than 74 when a positive-area contact is hidden. A result of 86 suggests contact area was added rather than removed. The exact correction is twelve because both face copies are internal.

Model investigation: Build the stacked shape from unit cubes or a dimensioned digital model. Count or partition the exposed faces and compare with the subtract-two-contact-areas method. Mark the internal contact in a separate exploded sketch so it is not mistaken for a coated exterior face.

Independent performance: Create and solve three composite-prism problems with specified nonoverlapping placements, including one cube combination and one triangular-prism component. Compute volume and full exterior area with units, then answer a covering task that excludes a named exterior face. For one composite, verify surface area by an independent face-by-face count. A partner checks that every hidden contact is accounted for twice and that any omitted exterior face is excluded by the task's wording.