The area of a triangle is 270 square inches. If the base of the triangle is 30 inches, what is...

1. TRANSLATE the problem information

• Given information:
  • Area of triangle = 270 square inches
  • Base = 30 inches
  • Height = ? (what we need to find)

2. INFER the approach

• We need the area formula for a triangle: \(\mathrm{A = \frac{1}{2}bh}\)
• Since we know area and base, we can substitute and solve for height
• This becomes a simple linear equation to solve

3. TRANSLATE into mathematical equation

• Substitute the known values into \(\mathrm{A = \frac{1}{2}bh}\):
  \(\mathrm{270 = \frac{1}{2}(30)(h)}\)

4. SIMPLIFY the equation step by step

• First, multiply \(\mathrm{\frac{1}{2}(30)}\):
  \(\mathrm{270 = 15h}\)
• Divide both sides by 15:
  \(\mathrm{h = 270 ÷ 15 = 18}\)

Answer: 18 inches

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Students forget that triangles use \(\mathrm{A = \frac{1}{2}bh}\) and instead use the rectangle formula \(\mathrm{A = bh}\).

Using \(\mathrm{A = bh}\) gives: \(\mathrm{270 = 30h}\), so \(\mathrm{h = 270 ÷ 30 = 9}\) inches. This leads to an incorrect answer of 9 instead of 18.

Second Most Common Error:

Weak SIMPLIFY skills: Students make arithmetic errors when dividing 270 ÷ 15, perhaps getting 16 or 20 instead of 18.

This causes them to arrive at an incorrect height value even though their approach was correct.

The Bottom Line:

This problem tests whether students remember the correct area formula for triangles (which includes the 1/2 factor) and can perform accurate arithmetic division. The conceptual distinction between triangle and rectangle area formulas is crucial.