Volume as additive
Recognise volume as additive; find volumes of composite solid figures made of two or more non-overlapping right rectangular prisms
What a learner can do afterwards
- Decompose an L-shaped solid into two cuboids and calculate total volume
- Solve a real-world problem requiring the volume of a composite figure (e.g. a step-shaped structure)
- Explain why splitting a composite solid into rectangular prisms allows calculation of total volume
1 · Read
A composite solid is a shape made from two or more boxes joined together, like an L-shape or a staircase. You already know how to find the volume of one rectangular prism: multiply length, width, and height. For a composite solid, split it into separate prisms first.
An L-shaped pool splits into two rectangular sections. Section 1 is 6 m long, 3 m wide, and 2 m deep: that's 6 x 3 x 2 = 36 cubic meters. Section 2 is 4 m long, 2 m wide, and 2 m deep: that's 4 x 2 x 2 = 16 cubic meters. Total volume: 36 + 16 = 52 cubic meters.
Split a composite solid along its natural edges so the pieces don't overlap and don't leave gaps. Then every cubic unit gets counted exactly once.
Split a composite solid into rectangular prisms, find each volume, then add them together for the total.
More to read
9.3 Measuring Volume
Volume counts the unit cubes that fill a solid. A cuboid 5 by 4 by 3 holds 60 cubes. Composite solids just add two or more such counts. An L-shape splitting into 60 and 24 holds 84 in total. Splitting never changes the total: it only makes counting possible.
9.6 Solve Geometry Applications: Volume and Surface Area
Real structures stack boxes together. A step with a 6 by 4 by 2 bottom block and a 4 by 4 by 2 top block holds 48 plus 32, which is 80. Compute each cuboid with length times width times height. Then add. Sketching the split with dashed lines keeps the parts clear and stops double counting.
3 Ways to Calculate Volume of a Box
Every box volume starts from three edge lengths. Multiply all three: a 3 by 3 by 3 cube holds 27. A 6 by 3 by 2 block holds 36. Stacked together they hold 63. Measure edges carefully first, since one wrong edge spoils every later step.
Key ideas: Volume of composite solids
Splitting works because volumes add when parts do not overlap. If two cuboids share a region, adding both counts that region twice, so subtract the overlap once. Parts that only touch at faces are safe to add directly. Overlap is the one trap: check for it before you add.
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2 · Watch
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Where this leads
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.