Circle K has a radius of 4 millimeters (mm). Circle L has an area of 100pi mm². What is the...

1. TRANSLATE the problem information

• Given information:
  • Circle K has \(\mathrm{radius = 4\ mm}\)
  • Circle L has \(\mathrm{area = 100π\ mm^2}\)
  • Need to find total area of both circles

2. INFER the approach needed

• To find total area, I need the individual area of each circle
• Circle L's area is already given
• For Circle K, I need to calculate area using the radius

3. TRANSLATE Circle K's radius to its area

• Use the circle area formula: \(\mathrm{A = πr^2}\)
• Area of Circle K =
  \(\mathrm{π(4)^2}\)
  \(\mathrm{= π(16)}\)
  \(\mathrm{= 16π\ mm^2}\)

4. SIMPLIFY to find the total

• Total area = Area of K + Area of L
• Total area =
  \(\mathrm{16π + 100π}\)
  \(\mathrm{= 116π\ mm^2}\)

Answer: D. \(\mathrm{116π}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students add the given measurements directly without converting to comparable units.

They might think: "Circle K has radius 4, Circle L has area 100π, so I add \(\mathrm{4 + 100 = 104}\)." Or they might notice that if Circle L has area 100π, then its radius is 10 (since \(\mathrm{πr^2 = 100π}\) means \(\mathrm{r = 10}\)), and then incorrectly add the radii: \(\mathrm{4 + 10 = 14}\).

This may lead them to select Choice A (\(\mathrm{14π}\)).

Second Most Common Error:

Conceptual confusion about radius vs diameter: Students might convert Circle L's area to its radius (\(\mathrm{r = 10}\)), then think about diameters instead of radii, calculating \(\mathrm{(8 + 20)π}\).

This may lead them to select Choice B (\(\mathrm{28π}\)).

The Bottom Line:

This problem tests whether students can work systematically with circle measurements, converting everything to the same type (areas) before combining. The key insight is recognizing that you can't directly combine a radius with an area - you must convert the radius to an area first.