Embedded Files
Materials
Materials
Area worksheet (one per person)
Right triangles from the area figures printout (one per person)
Scissors to cut out the right triangle figures (one per person)
Tape dispenser to attach rearranged shapes (one per group)
Area of a Right Triangle: Double and Divide
Area of a Right Triangle: Double and Divide
As with the parallelogram, the right triangle presents a challenge due to the partial square units that are cut by the hypotenuse. However, if we duplicate the triangle and arrange the pieces so the hypotenuses are aligned and the corresponding legs are opposite one another, the resulting shape is a rectangle. We know the shape formed by the two triangles together is a rectangle because:
the hypotenuses of the triangles have the same length and slope so they fit together exactly and form a quadrilateral,
the right angles of the triangles are now corners of the new shape and
the sum of interior angle measures of a triangle add to 180 degrees so the two remaining angle measures in the triangle add to 90 degrees. Since the two angles coming together at each corner are the non-right angles of the two triangles, they also must form right angles.
Note: The justification above for why the combined right triangles form a rectangle is provided for completeness. At the middle school level, this deductive proof is probably not expected from the students. However, students should be able to be convinced that because the hypotenuses and slopes of the triangles are equal there should be no overlap of the pieces. This allows students to explain why the area of the original triangle is half the rectangle. Further, if students are aware the sum of interior angle measures of a quadrilateral is 360 degrees, it should be easy to convince students why the two angles not marked as right must be right angles.
Since the two right triangles combined form a four sided figure with four right angles, we know we have a rectangle. The dimensions of the rectangle are equivalent to the lengths of the legs of the right triangle, each representing a base and the other, a height. In our example, the triangle has a base of 9 units and height of 5 units (or vice versa) so a rectangle with length 9 units and width 5 units has a total area of 45 square units.
We know the area of the rectangle is equal to twice the area of the triangle since it was formed by two triangles. Therefore, we know the area of the original right triangle has the same area half the rectangle. In our example, the right triangle has an area of 22.5 square units.
Area Formula: Right Triangle
Area Formula: Right Triangle
In general, the area of a right triangle is half the product of the lengths of the legs. Using one leg as the base b and the other as the height h:
Note: Every triangle has three possible bases and a corresponding height. In a right triangle, the legs naturally form a base and height pair because they intersect at a right angle. Students should recognize that the hypotenuse of a right triangle can also represent a base but that the height will not be represented by either of the remaining sides. If a student struggles to identify the height of a triangle when it is not the leg of a right triangle, the Height of Triangles activity might be helpful.
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