L2 · Volume of right composite 3-D shapes (formulas given)

Learning goal

“I can find the volume of a composite solid by splitting it into prisms, pyramids and cylinders, then ADDING the parts' volumes - or SUBTRACTING a part that was cut away. Unlike surface area, volume never needs a 'join correction': the parts fill space exactly once each. (Formulas provided: rectangular prism V = l x w x h; triangular prism V = (area of triangular base) x length; cylinder V = pi x r^2 x h, with pi = 3.14; pyramid V = (1/3) x (area of base) x height.)”

Volume — the amount of space a solid fills, in CUBIC units (cm3, m3).

Decompose — to split a complicated solid into simpler solids whose measures you can compute, then combine the results.

Height of a pyramid — the PERPENDICULAR distance from the pyramid's base to its apex (top point). It is not the slant height from L1; volume uses this one.

Worked example

The step block from L1: bottom prism 10 cm x 6 cm x 4 cm, top prism 4 cm x 6 cm x 3 cm. Find its volume. Step 1: Volume counts filled space. The two prisms fill DIFFERENT space - where they touch, they share a boundary, not a region. Nothing is double-counted. Step 2: V(bottom) = 10 x 6 x 4 = 240 cm3. V(top) = 4 x 6 x 3 = 72 cm3. Step 3: V = 240 + 72 = 312 cm3. Say it plainly: in L1 the join patch had to be REMOVED TWICE from surface area, because it was surface that stopped existing. For volume, the patch is just the border between two filled regions - you add the parts and you are done.

Check yourself — explain it

For which measure do you subtract twice the join: surface area, volume, or both?

Paired problem — example still visible

Two stacked boxes with the same footprint: lower 8 x 5 x 2, upper 8 x 5 x 3. -> V = 80 + 120 = 200 cm3 (check: one 8 x 5 x 5 prism = 200)

Faded completion — last step is yours

"A channel is cut through a block 10 cm x 8 cm x 6 cm. The removed piece is 4 cm x 8 cm x 3 cm. V(block) = 480; V(removed) = ___; V(remaining) = 480 - ___ = ___ cm3" -> 96; 96; 384 cm3

Ready to practise?

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

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