L4 · Composite solids clinic

Learning goal

“I can take an unfamiliar composite solid and choose the right routine on my own: decompose it, correct surface area for joins, add or subtract volumes, and use SA-volume relationships to make and defend a design decision. (Formulas provided: rectangular prism V = lwh, SA = 2(lw + lh + wh); triangular prism V = (area of triangular base) x length; cylinder V = pi x r^2 x h, lateral surface 2 x pi x r x h, one end pi x r^2 (pi = 3.14); pyramid V = (1/3) x base area x (perpendicular) height; pyramid surface = base + triangular faces (slant height given).)”

Composite 3-D shape — built from prisms, pyramids, cylinders (L1).

Join — the hidden touching face; counted zero times in surface area, irrelevant to volume (L1-L2).

Minimize — for a fixed volume, the cube is the rectangular prism with the smallest surface area (L3).

Worked example

A garden shed is a rectangular body 6 m wide x 8 m deep x 4 m high, with a triangular-prism roof: the triangular ends have base 6 m and height 4 m, each sloped side is 5 m long, and the roof runs the 8 m depth of the shed. Find (a) the surface area (for cladding) and (b) the volume (for storage). (a) Surface area - list exposed faces: Body bottom: 6 x 8 = 48. Body front/back: 6 x 4 = 24 each, total 48. Body ends: 8 x 4 = 32 each, total 64. (Body top: 0 - the roof sits on it: join.) Roof triangles: 2 x ((1/2) x 6 x 4) = 24. Roof slopes: 2 x (5 x 8) = 80. SA = 48 + 48 + 64 + 24 + 80 = 264 m2. Check: SA(body) = 2(48 + 24 + 32) = 208; SA(roof, including its underside) = 24 + 80 + 48 = 152; join 48: 208 + 152 - 2 x 48 = 264. (b) Volume - no join correction, just add: Body: 6 x 8 x 4 = 192. Roof: ((1/2) x 6 x 4) x 8 = 96. V = 192 + 96 = 288 m3. Say it plainly: SAME decomposition, two different rules. Surface area asks "what is exposed?" and punishes hidden faces; volume asks "what is filled?" and never does.

Check yourself — explain it

In the shed problem, the join (the body's top, 48 m2) was...

Paired problem — example still visible

A dog house: body 4 m long x 3 m deep x 3 m high; roof is a triangular prism with triangular ends of base 4 m and height 1.5 m, sloped sides 2.5 m, running the 3 m depth. Find SA and V. -> SA: bottom 12; front/back 12 x 2 = 24; ends 9 x 2 = 18; triangles 2 x ((1/2)(4)(1.5)) = 6; slopes 2 x (2.5 x 3) = 15; total 75 m2. V: body 36; roof ((1/2)(4)(1.5)) x 3 = 9; total 45 m3.

Faded completion — last step is yours

"For the shed: SA(body) = 208, SA(roof) = 152, join = 48. Composite SA = 208 + 152 - ___ = ___ m2. Composite V = 192 + 96 - ___ = ___ m3 (volume takes NO join correction)." -> 96; 264; 0; 288

Ready to practise?

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

Start practice →Take the unit quiz