Finding the area of a quadrilateral using triangle decomposition

Key concept: This area problem is usually handled by splitting the quadrilateral into triangles and using trigonometric area formulas or angle relationships.

Step by step

Read the diagram as a triangle decomposition problem

The visible information shows quadrilateral $ABCD$ with an angle $\angle BDC=40^\circ$, a side of length 15 cm, another side of length 10 cm, and an angle of $115^\circ$ marked in the diagram. Problems of this type are usually solved by dividing the quadrilateral into two triangles and finding each triangle’s area.

Use the appropriate triangle area formula

If two sides and the included angle of a triangle are known, the area formula is

$

$A=\frac12 ab\sin C$

$

So if the 10 cm and 15 cm sides are the sides around the $40^\circ$ angle in one triangle, that triangle’s area would be

$

$A_1=\frac12(10)(15)\sin 40^\circ$

$

Then any second triangle formed by the diagonal and the remaining sides would need to be found using the remaining given angle information.

Why the layout matters

For quadrilateral area questions, the exact placement of each number in the diagram is essential. A side length printed away from the angle may belong to a different triangle than expected. The $115^\circ$ angle likely controls a second triangle or a supplementary angle relationship, so the full diagram determines the final numerical answer.

Key method

1. Split the quadrilateral into two triangles.
2. Match each given angle with the two sides that form it.
3. Use $\frac12 ab\sin C$ for each triangle.
4. Add the two triangle areas.

Final guidance

The correct approach is triangle decomposition with the sine area formula, but the exact final area depends on the full diagram placement of the 15 cm, 10 cm, $40^\circ$, and $115^\circ$ labels.

Pitfall alert

A common mistake is to try to use one triangle area formula with the whole quadrilateral at once. Quadrilaterals are usually not handled that way unless there is a special shape such as a rectangle or kite. Another error is assuming the 115° angle and the 40° angle are directly related without checking the diagram. In geometry, the position of each label matters as much as the number itself. Always identify which two sides actually surround the angle before using $\frac12 ab\sin C$.

Try different conditions

If the question were changed so that the quadrilateral was split by diagonal $BD$ and you knew $BD$, then each triangle could be found separately with Heron’s formula or the sine area formula. For example, if one triangle had sides 10 cm and 15 cm with included angle $40^\circ$, and the other triangle had side information tied to the $115^\circ$ angle, you would compute both areas and add them. If the figure were a kite or a parallelogram, there might also be a shortcut formula, but this depends on the exact shape.

Further reading

sine area formula, triangle decomposition, included angle