Area, volume and scale factors — Digital SAT Math practice
Common volume formulas and how scaling changes area and volume.
Key terms
- Scale Factor: Ratio of corresponding linear dimensions between two similar figures.
- Base Area: The area of the flat surface at the bottom or top of a solid.
- Volume: Measure of the amount of three-dimensional space occupied by an object.
- Surface Area: Total area of all exterior faces of a three-dimensional shape.
Common traps
- Forgetting to cube the scale factor when calculating volume changes instead of squaring it.
- Using diameter instead of radius in sphere volume formulas leads to eight times the correct answer.
- Adding lateral area twice when calculating total surface area of a cylinder or prism.
Key ideas to know
Scaling volume
If a rectangular prism has its length, width, and height all multiplied by 4, by what factor does its volume increase?
64. When every length scales by k, volume scales by k3. Here 43 = 64.
Watch out: Multiplying the volume by the scale factor, or squaring it, instead of cubing it.
Picking Volume Formulas on the SAT
You need to find the volume of a solid with a circular base and a single apex point at the top. Which formula should you use?
Use one third times base area times height. Solids with a pointed top like cones or pyramids always require multiplying by one third.
Watch out: Confusing cone or pyramid formulas with prism or cylinder formulas leads to forgetting the one third multiplier.
Surface Area Face Counting Error
A student calculates the surface area of an open-top box by finding the area of all six faces and adding them together. What is wrong with this approach?
An open-top box only has five faces. You must only add the areas of the faces that actually exist in the shape.
Watch out: Assuming every geometric solid always has six faces or forgetting to remove missing faces like lids or bases.
Volume as Fill Rate Times Time
A pool fills at 2 cubic feet per minute for 3 hours. What is the total volume of water added?
The volume is 360 cubic feet. Convert time to minutes first: 3 hours equals 180 minutes. Then multiply rate by time: 2 times 180.
Watch out: Multiplying the rate by the raw hour or minute value without converting units leads to an answer that is off by a factor of 60.
Sample questions
Question 1
A right circular cone has radius 3 and height 9. What is its volume?
- 30π
- 81π
- 27π
- 54π
Show answer and explanation
Answer: 27π
V = (1/3)πr²h = (1/3)π(3²)(9) = 27π.