We can reason about the area of a triangle by using what we know about parallelograms. Here are three general ways to do this:

• Make a copy of the triangle and join the original and the copy along an edge to create a parallelogram. Because the two triangles have the same area, one copy of the triangle has half the area of that parallelogram.

  

  Two figures labeled A, and B. Figure A is a triangle. Figure B is the same triangle as figure A with a copy along the edge of the original to create a rectangle. The right side of the rectangle is labeled 2 units, and the bottom is labeled 8 units. 

  The area of Parallelogram B is 16 square units because the base is 8 units and the height 2 units. The area of Triangle A is half of that, which is 8 square units.

  

  Two figures labeled C, and D. Figure C is another triangle, and figure D is the same triangle as figure C, with a copy along the edge of the original to create parallelogram. The left base of the parallelogram is labeled 4 units, and the height is labeled 6 units.

  The area of Parallelogram D is 24 square units because the base is 4 units and the height 6 units. The area of Triangle C is half of that, which is 12 square units.

• Decompose the triangle into smaller pieces and compose them into a parallelogram.

  

  Two images of a triangle. Image on right has a dashed line cutting off the top portion. Image on left has the cut off portion moved next to the bottom of the triangle to create a parallelogram. An arrow indicating that the cut off portion from other image was moved.

  In the new parallelogram, , , and , so its area is 12 square units. Because the original triangle and the parallelogram are composed of the same parts, the area of the original triangle is also 12 square units.

• Draw a rectangle around the triangle. Sometimes the triangle has half of the area of the rectangle.

  

  The large rectangle can be decomposed into smaller rectangles. Each smaller rectangle can be decomposed into two right triangles.

  • The rectangle on the left has an area of , or 12, square units. Each right triangle inside it is 6 square units in area.

  • The rectangle on the right has an area of , or 6, square units. Each right triangle inside it is 3 square units in area.

  • The area of the original triangle is the sum of the areas of a large right triangle and a small right triangle: 9 square units.

  Sometimes, the triangle is half of what is left of the rectangle after removing two copies of the smaller right triangles.

  

  Three images of the same triangle. The first image is the triangle alone. The second is the triangle surrounded by a rectangle. The third image is of the triangle now with a copy composed into a parallelogram within the rectangle, with arrows drawing the remaining parts of the rectangle into a smaller rectangle.

  The right triangles being removed can be composed into a small rectangle with area square units. What is left is a parallelogram with area , which equals , or 9, square units.

  Notice that we can compose the same parallelogram with two copies of the original triangle! The original triangle is half of the parallelogram, so its area is , or 4.5, square units.