Depth · one concept from first principles · step 2 of 4

Equal overlap area yields a segment length

The figure at right shows rectangle MBCD and parallelogram ABCF placed so that they overlap. When the area of the shaded region is $132\,\text{cm}^2$ , what is the length of segment GF, in $\text{cm}$ ?

Figure description: The parallelogram ABCF on the left and the rectangle MBCD on the right overlap each other. Along the top edge, FM is $12\,\text{cm}$ and MD is $12\,\text{cm}$ ; the right side DC of the rectangle is $7\,\text{cm}$ . The point where the two figures cross is labeled G. The shaded region is the part where the two figures overlap.

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 rectangles Triangle height depends on chosen base

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▶ Practice — 12 problems