A square pyramid has a height of 9 centimeters (cm) and a volume of 108 text{ cm}^3. What is the...
GMAT Geometry & Trigonometry : (Geo_Trig) Questions
A square pyramid has a height of \(9\) centimeters (cm) and a volume of \(108 \mathrm{\text{ cm}}^3\). What is the area, in \(\mathrm{cm}^2\), of the base of the pyramid? (The volume of a pyramid is equal to \(\frac{1}{3}\mathrm{Bh}\), where \(\mathrm{B}\) is the area of the base and \(\mathrm{h}\) is the height of the pyramid.)
1. TRANSLATE the problem information
- Given information:
- Height of square pyramid: \(\mathrm{h = 9~cm}\)
- Volume of pyramid: \(\mathrm{V = 108~cm^3}\)
- Volume formula: \(\mathrm{V = \frac{1}{3}Bh}\)
- Need to find: Base area \(\mathrm{B}\)
- What this tells us: We have two of the three variables in the volume formula, so we can solve for the missing one.
2. INFER the approach
- Since we know volume and height, we can substitute these values into the formula and solve for base area \(\mathrm{B}\)
- Strategy: Use substitution followed by algebraic manipulation
3. SIMPLIFY through substitution and algebra
- Substitute known values into \(\mathrm{V = \frac{1}{3}Bh}\):
\(\mathrm{108 = \frac{1}{3}(B)(9)}\)
- Simplify the right side:
\(\mathrm{108 = 3B}\)
- Solve for \(\mathrm{B}\) by dividing both sides by 3:
\(\mathrm{B = 108 \div 3 = 36}\)
Answer: 36
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students make arithmetic errors when working with fractions and parentheses.
They might incorrectly calculate \(\mathrm{\frac{1}{3}(9)}\) as \(\mathrm{\frac{3}{9}}\) instead of 3, leading to the equation \(\mathrm{108 = (B)(\frac{3}{9})}\) or \(\mathrm{108 = \frac{B}{3}}\). This would give them \(\mathrm{B = 324}\), which doesn't match any reasonable expectation for a pyramid with this volume and height.
Second Most Common Error:
Poor TRANSLATE reasoning: Students confuse which variable represents what they're looking for.
Some students might think they need to find the height instead of the base area, or they might set up the equation incorrectly by mixing up the variables. This leads to confusion about what they're solving for and causes them to get stuck and guess.
The Bottom Line:
This problem tests whether students can work backwards from a volume formula - a reverse application that requires careful attention to which variable is unknown and systematic algebraic manipulation.