SHSAT: Volume, capacity and displacement
Find volumes of boxes and prisms, convert cubic centimeters to liters, and solve displacement and packing questions.
How do you handle volume questions that involve water, liters or fitting boxes inside a container?
Short answer
Multiply base area by height, keep units on every number, and convert at the end. For displacement, the rise times the base area is the object's volume.
Know first: Multiplying three numbers, Area of rectangles and triangles
Step 1 of 5
Volume is base area times height
Every prism, whether a box or a triangular tent, has the same volume rule: area of the base × height (or length). For a box that's length × width × height.
Capacity questions add one more step: changing units. A cube 10 centimeters on each side holds exactly 1 liter, so 1 liter = 1000 cubic centimeters.
| Situation | Rule |
|---|---|
| Rectangular box | length × width × height |
| Triangular prism | (½ × base × height of triangle) × length; for example ½ × 6 × 4 × 10 = 120 |
| Cube with edge s | s³; tripling the edge multiplies volume by 27 |
| Cubic centimeters to liters | divide by 1000 |
| Water rises when an object sinks | object's volume = base area of tank × rise |
| Melting and reshaping | total volume stays the same |
Keep units on every number so you notice when centimeters need to become liters.
Step 2 of 5
Use the base area twice
Worked example: A stone in a tank
A rectangular tank has a base 30 centimeters by 20 centimeters. When a stone is fully submerged, the water level rises 4 centimeters and nothing spills. What is the stone's volume in cubic centimeters?
- A2.4
- B54
- C600
- D2400
- 1
Find the base area
30 × 20 = 600 square centimeters.
- 2
Multiply by the rise
600 × 4 = 2400 cubic centimeters. The extra water height is a thin box with the same base as the tank.
- 3
Read the units
The question asks for cubic centimeters, so the answer is 2400. In liters it would be 2.4.
- 4
See why the other choices are there
2.4 is the volume in liters, the wrong unit for this question. 54 adds the three lengths. 600 stops at the base area.
Answer
2400 cubic centimeters
Wrong: A box is 10 by 7 by 5 inches. 2-inch cubes: 350 ÷ 8 = 43.75, so 43 cubes fit.
Right: Along each edge: 10 ÷ 2 = 5, 7 ÷ 2 = 3 (round down), 5 ÷ 2 = 2 (round down). 5 × 3 × 2 = 30 cubes fit.
Solid cubes can't be split to fill the leftover strips, so count whole cubes edge by edge.
Step 3 of 5
Watch the units and the shape
Check yourself · Question 1
A rectangular prism has volume 324 cubic inches and a base 9 inches by 4 inches. What is its height in inches?
Answer: A
Base area = 9 × 4 = 36 square inches. Height = 324 ÷ 36 = 9 inches. Check: 9 × 4 × 9 = 324.
Check yourself · Question 2
A tank measures 40 centimeters by 25 centimeters by 30 centimeters. It is 3/4 full of water. How many liters of water are in the tank?
Answer: B
Full volume = 40 × 25 × 30 = 30,000 cubic centimeters = 30 liters. Three fourths of 30 is 22.5 liters.
Trap answer: 22,500
Stopping before the unit change
- Why it looks right
- The calculation is correct, and the number matches your scratch work exactly.
- Why it's wrong
- 22,500 is in cubic centimeters. The question asks for liters, and every 1000 cubic centimeters is 1 liter.
- The right answer
- 22.5 liters, from 22,500 ÷ 1000.
Check yourself · Question 3
Every edge of a cube is tripled. The new volume is how many times the old volume?
Answer: D
Volume uses three edges multiplied together, and each is 3 times as long: 3 × 3 × 3 = 27.
Common mistake
I multiply cubic centimeters by 1000 to get liters.
- Why it's tempting
- Converting units often means multiplying, and 1000 is the right number.
- Do this instead
- A liter is bigger than a cubic centimeter, so there are fewer liters. Divide by 1000.
Step 4 of 5
What to remember
Remember
- Volume of any prism is base area × height; for a triangular prism, include the ½.
- 1 liter = 1000 cubic centimeters, and a rise in water level times the tank's base area gives the sunken object's volume.
- For packing, count whole pieces along each edge and multiply. Scaling every edge by k scales volume by k³.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, which three measurements did you multiply, and what units was your answer in?