Perimeter & Area

Problem

Equal overlap area yields a segment length

A rectangle and a parallelogram are laid on top of each other so they share the bottom side and overlap in the middle. Both figures have the same base length (12 cm) and the same height (7 cm). The shaded picture is everything the two figures cover together (their combined region), and its area is 132 cm squared. I need the length of the segment GF, which is the height of the small triangle where the two figures overlap, measured straight down from the crossing point G to the bottom edge.
12 cm 12 cm 7 cm A F M D B C G
Measurement & data
Your answer
How to solve
Strategy Draw a Diagram — Draw and label the two overlapping figures so I can see that the shaded total equals the two areas added, then the doubly-counted overlap subtracted (Tool 1). Break the job into small pieces: rectangle area, parallelogram area, and overlap-triangle area (Tool 7). Because I know the final shaded total, I can work backwards to find the overlap area, then backwards again from the triangle's area formula to find its height GF (Tool 11). I avoid algebra (Tool 13) because each unknown comes out by a single clear arithmetic step a 4th grader can do.
1STEP 1

Area of the rectangle

The rectangle MBCD is 12 cm wide and 7 cm tall. Multiply width by height to get its area.

12 × 7 = 84 cm²
2STEP 2

Area of the parallelogram

The parallelogram ABCF sits on the same base of 12 cm and has the same height of 7 cm as the rectangle. Base times height gives its area.

12 × 7 = 84 cm²
3STEP 3

Find the overlap area by working backwards

Adding both figures counts the overlap twice, so 84 + 84 - overlap = 132 leaves overlap = 36.

84 + 84 - overlap = 132 → 168 - overlap = 132 → overlap = 36 cm²
4STEP 4

Use the triangle area to find GF

The overlap is triangle GBC on base 12, so 12 x GF divided by 2 = 36 makes GF = 6 cm.

12 × GF ÷ 2 = 36 → 12 × GF = 72 → GF = 6 cm
Answer
6 cm
The answer 6 cm is a length, which matches what GF should be, and it is less than the figures' full height of 7 cm, so the crossing point G sits below the top edge as the picture shows. Checking the overlap: triangle area = 12 x 6 / 2 = 36 cm squared, and 84 + 84 - 36 = 132 cm squared, exactly the shaded area given.
Takeaway

When two shapes overlap, add their areas and take away the shared part once, then reverse the triangle formula to find the missing height.

  • Area of the rectangle
  • Area of the parallelogram
  • Find the overlap area by working backwards
  • Use the triangle area to find GF
Where next?
Another one like thissuggested
12 practice problems
Step back and solidifysuggested
Decompose composite figures into rectanglesTriangle height depends on chosen base

▶ Practice — 12 problems