Lesson 6: Apply Volume Knowledge

Property

The volume of a composite solid made of rectangular prisms can be found using two main methods:

• Addition: Decompose the figure into non-overlapping rectangular prisms and add their individual volumes. $V_{total} = V_1 + V_2$
• Subtraction: Enclose the figure within a larger rectangular prism and subtract the volume of the missing portion. $V_{total} = V_{large} - V_{missing}$

Examples

• Addition Method: For an L-shaped prism, you can split it into two smaller prisms. If one prism is $2 \times 4 \times 5$ and the other is $3 \times 4 \times 2$, the total volume is $(2 \times 4 \times 5) + (3 \times 4 \times 2) = 40 + 24 = 64$ cubic units.
• Subtraction Method: For the same L-shaped prism, you can imagine a large rectangular prism and subtract the empty space. If the large prism is $5 \times 4 \times 5$ and the empty space is $3 \times 4 \times 3$, the total volume is $(5 \times 4 \times 5) - (3 \times 4 \times 3) = 100 - 36 = 64$ cubic units.

Explanation

A composite prism is a 3D figure made up of two or more rectangular prisms. To find its volume, you can use the addition method by breaking the shape into smaller, familiar prisms and summing their volumes. Alternatively, the subtraction method involves calculating the volume of a larger, simpler prism that encloses the shape and then subtracting the volume of the parts that are not part of the figure. Both methods apply the standard volume formula, $V = l \times w \times h$, and will yield the same final answer.

Property

Real-world objects often require both volume and surface area calculations. Always read carefully to decide which one you need:

• Filling an object (water, dirt, air) = Volume.
• Covering an object (paint, wrapping paper, tile) = Surface Area.

Examples

• Swimming Pool (Volume): An L-shaped pool has a shallow end (10 ft by 12 ft, depth 4 ft) and a deep end (15 ft by 12 ft, depth 9 ft).

Volume of shallow end: 10 x 12 x 4 = 480 cubic ft.
Volume of deep end: 15 x 12 x 9 = 1620 cubic ft.
Total water needed: 480 + 1620 = 2100 cubic ft.

• Skate Park Ramp (Surface Area): A wooden ramp is built from a rectangular prism and a triangular prism. To figure out how much special grip paint to buy, you must find the Surface Area. Remember to exclude the bottom faces touching the ground and the faces where the two prisms connect!

Explanation

Out in the real world, nobody hands you a perfectly simple cube! Pools, couches, houses, and skate ramps are all composite figures. By breaking these complex shapes into easy rectangular and triangular prisms, you can calculate exactly how much water fills a pool or how much paint covers a building. Just take it one block at a time!