TriangLE

Materials

Area worksheet (one per person)

Triangles from the area figures printout (one per person)

Scissors to cut out the triangle figures (one per person)

Tape dispenser to attach rearranged shapes (one per group)

Area of a Triangle: Double and Divide

As with the right triangle, the triangle presents a challenge due to the partial square units that are cut by the sides. However, if we duplicate the triangle and arrange the pieces so the longest sides are aligned and the other two corresponding sides are opposite one another, the resulting shape is a parallelogram. We know the shape formed by the two triangles together is a parallelogram because:

the hypotenuse of the triangles have the same length and slope so they fit together exactly and form a quadrilateral,
the largest angles of the quadrilateral are congruent and opposite one other
the sum of interior angle measures of a triangle add to 180 degrees so the two remaining angle measures in the triangle add to 180 degrees less the largest angle. Since the two angles coming together at each corner are the non-corresponding smallest angles of the two triangles, they also must form congruent angles.

Note: The justification above for why the combined triangles form a parallelogram 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 longest sides 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 parallelogram. 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 corresponding to the largest angles in the parallelogram are equal.

Since the pairs of opposite angles in the quadrilateral are congruent, we know we have a parallelogram. The base of the new parallelogram is congruent to the base of the triangle, as are the corresponding heights. In our example, we take the base to be the shortest side of the parallelogram and the height of the parallelogram corresponds to the height of the triangle when the shortest side is taken to be the base of the triangle. In our example, the parallelogram has a base of 4 units and height of 9 units (or vice versa if we take the second longest side to be the base) and a total area of 28 square units.

We know the area of the parallelogram is equal to twice the area of the triangle since it was formed by two triangles. Therefore, we know the area of the original triangle has the same area as half the parallelogram. In our example, the triangle has an area of 14 square units.

Area Formula: Right Triangle

In general, the area of a triangle is half the product of the lengths of the the base b and height h:

Note: The area of every triangle, including right triangles, can be found using this formula; a right triangle is simply a special case where the base and height correspond to the legs of the triangle. This activity splits the cases in two so that if a student struggles to identify the height of a triangle when it is not the leg of a right triangle, they can still understand the simpler case for right triangles. 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.

Area of a Triangle: Decomposing into Right Triangles

Another way to calculate the area of a triangle is to identify a base and corresponding height then decompose it into two right triangles where the sum of the two bases is equivalent to the original base and the heights are equivalent.

Using algebra, we can add the areas of the two triangles to find the area of the original triangle:

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