Volume of composite solids, step by step
Two boxes stuck together hold the sum of what each holds alone.
The question
cm³
Volume of a 5 × 9 × 9 stepped solid
5 × 9 × 9with a step on top
slab:5 × 9 × 9 =405
block:3 × 9 × 9 =243
slab:405
block:243
405 + 243 =648 cubic cm
filled box:5 × 9 × 18 =810
notch:2 × 9 × 9 =162
filled box:810
notch:162
810 − 162 =648 cubic cm
- A block standing on a slab. There is no formula for this shape — the job is to turn it into boxes, which do have one, and there is more than one way.
- First way: cut it into the two boxes it already is. The slab underneath is 5 × 9 × 9 = 405, and the block standing on it is 3 × 9 × 9 = 243.
- 405 + 243 = 648 cubic cm.
- Second way: fill the notch in to make one big box. That box is 5 × 9 × 18 = 810, and the notch taken back off is 2 × 9 × 9 = 162.
- 810 − 162 = 648 cubic cm. Two methods agreeing is how you know.
How it works.
- 01
A block standing on a slab. There is no formula for this shape — the job is to turn it into boxes, which do have one, and there is more than one way.
- 02
First way: cut it into the two boxes it already is. The slab underneath is 5 × 9 × 9 = 405, and the block standing on it is 3 × 9 × 9 = 243.
- 03
405 + 243 = 648 cubic cm.
- 04
Second way: fill the notch in to make one big box. That box is 5 × 9 × 18 = 810, and the notch taken back off is 2 × 9 × 9 = 162.
- 05
810 − 162 = 648 cubic cm. Two methods agreeing is how you know.