L1 · Surface area of right composite 3-D shapes (decompose and add)

Learning goal

“I can find the surface area of a composite solid by listing every EXPOSED face - and I know the faces where two parts join are hidden, so they are counted zero times, not twice.”

Composite 3-D shape — a solid built by joining (or cutting into) two or more simpler solids, e.g. a pyramid sitting on a cube.

Surface area — the total area of the surface of a solid, in SQUARE units (cm2, m2): the amount of wrapping that would cover it exactly.

Join (interface) — the face where two parts touch. Once the parts are joined, that face is inside the solid: it is NOT part of the surface.

Slant height — for a pyramid, the height of one triangular face, measured up the face itself (not the pyramid's vertical height). It is given whenever needed.

Worked example

A step-shaped block is made from two rectangular prisms. The bottom prism measures 10 cm long x 6 cm deep x 4 cm high. The top prism measures 4 cm long x 6 cm deep x 3 cm high and sits on top of the bottom prism at the left end, flush on the left side and flush front-to-back. Find the surface area of the block. Step 1: Name the join. The top prism sits on a patch of the bottom prism's top: 4 x 6 = 24 cm2. That patch is hidden on BOTH solids - it contributes 0. Step 2: List the exposed faces by direction. Bottom: 10 x 6 = 60. Top: top of the small prism 4 x 6 = 24, plus the exposed part of the big top (10 - 4) x 6 = 36, total 60. Front: big front 10 x 4 = 40, plus small front 4 x 3 = 12, total 52. Back: also 52. Left end: big 6 x 4 = 24, plus small 6 x 3 = 18, total 42. Right end: big 6 x 4 = 24, plus the small prism's right face 6 x 3 = 18, total 42. (That right face of the small prism hangs above the big prism's top - it is exposed to the air, so it counts.) Step 3: Add: 60 + 60 + 52 + 52 + 42 + 42 = 308 cm2. Step 4 (check by the second method): SA(bottom prism) = 2(60 + 40 + 24) = 248; SA(top prism) = 2(24 + 12 + 18) = 108; subtract the join TWICE (once for each solid that lost it): 248 + 108 - 2 x 24 = 308. Both methods agree. Say it plainly: SUM OF PARTS - 2 x JOIN. The join is never counted once, and never left in twice.

Check yourself — explain it

Two prisms touch on a face of 20 cm2. Compared with the two separate surface areas added together, the composite's surface area is...

Paired problem — example still visible

A block: bottom prism 8 x 5 x 2, top prism 8 x 5 x 3 stacked on it with the SAME footprint (a straight stack). -> Compute by parts: SA(bottom) = 2(40 + 16 + 10) = 132; SA(top) = 2(40 + 24 + 15) = 158; join 40: 132 + 158 - 80 = 210 cm2. (Sanity: it is just one 8 x 5 x 5 prism: 2(40 + 40 + 25) = 210.)

Faded completion — last step is yours

"A cube of edge 5 cm has a second cube of edge 5 cm placed on top of it. SA(one cube) = 150. Join = ___ cm2. Composite SA = 150 + 150 - 2 x ___ = ___ cm2" -> 25; 25; 250 cm2

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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