Problem

Use shared base to find composite area

Two identical right triangles, ABC and DEF, overlap along one straight base line where B, E, C, F sit in that order. Triangle ABC has a 45 degree angle at A, so it is a right isosceles triangle. The top points A and D are 4 cm apart, and the vertical side DF of the right triangle is 8 cm. The two slanted sides cross at G, and the small triangle AGD is identical to the small triangle GEC. I need the total area of the shaded figure (the whole shape made by the two triangles together).

GeometryMeasurement & data

Your answer

How to solve

Strategy Identify Subproblems — The shaded shape is a compound figure built from two overlapping triangles, so I break it into pieces I can measure: each whole triangle, and the overlap they share. Drawing and labeling the picture with coordinates makes the equal legs and the overlap visible. Because the overlap is exactly triangle GEC, the shaded area equals one triangle plus the rest of the other triangle. I prefer plain subproblem-and-add-up reasoning over algebra because the lengths are all directly readable from the figure.

1STEP 1

Find each leg length

A right angle and a 45 degree angle force the third to 45, so the legs match and each is 8 cm.

angle B = 180° - 90° - 45° = 45°

2STEP 2

Area of one whole triangle

Each triangle has perpendicular legs of 8 cm, and a triangle's area is half of base times height.

1/2 × 8 × 8 = 32 cm²

3STEP 3

Area of the overlap

The overlap is triangle GEC, whose perpendicular sides are 4 cm each, so its area is 8 square cm.

1/2 × 4 × 4 = 8 cm²

4STEP 4

Add the two triangles and subtract the shared overlap

Adding both triangles counts the overlap twice, so take it off once: 32 + 32 - 8.

32 + 32 - 8 = 56 cm²

Answer

56 cm²

Each triangle alone is 32 square centimeters, so two of them with a small shared corner should be a bit less than 64. The answer 56 is just below 64, which fits, and the units are square centimeters as required for area.

Takeaway

When two shapes overlap, add their areas and take the shared middle away once so you do not count it twice.

Find each leg length
Area of one whole triangle
Area of the overlap
Add the two triangles and subtract the shared overlap

Where next?

Level upsuggested

Solve known areas first, step by step

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12 practice problems

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▶ Practice — 12 problems