Math Background

Lesson: Area Formulas
Developing the Concept

Now that students know the formula for the area of a triangle, it's time to build on that knowledge to develop a formula for the area of a parallelogram.

Materials: Overhead transparency, an activity sheet with six parallelograms on it with base and height measures given for four of the parallelograms, rulers for students to use

Preparation: Draw a parallelogram on an overhead transparency.

  • Ask: Does anyone know what this figure is called?
    Students should say that it is a parallelogram. If they don't, tell them what it is.
  • Say: Let's list some of the properties of this figure on the board. Who can tell me something that is true about all parallelograms? Students will probably come up with several ideas such as the opposite sides are parallel and of equal length, the opposite angles are congruent, all rectangles are parallelograms, a rhombus is a parallelogram, the sum of the angles of a parallelogram is 360 degrees, and so on.

    Draw one of the diagonals of the parallelogram as shown below.

  • Ask: What can you tell me about the two triangles — triangle MNP and triangle OPN?
    Students should say that the two triangles are congruent. If they don't, suggest that they are. If you need to, cut out a parallelogram and then cut the parallelogram in half to show that the two triangles are congruent.
  • Say: That's right. They are congruent. Now we know how to find the area of a triangle, and since this parallelogram is made up of two congruent triangles, we can find the area of one of the triangles and double it.
  • Ask: How would I find the area of triangle MNP?
    Students should say you need to draw a line segment to show the height of the triangle. So draw the line segment PQ in triangle MNP as shown below.
  • Ask: What do I do now?
    Students will say you need to find the measure of line segments MN and PQ and substitute them into the formula A = one-half(b x h). So measure the sides and find the area.
  • Say: Now that I know the area of the triangle, how does that help me find the area of the parallelogram?
    Students should say that you need to double that value to find the area of the parallelogram.
  • Say: That's right, so the formula for the area of a parallelogram is twice the area for a triangle, A = one-half(b x h) + one-half(b x h), or A = b x h.

    Do another problem like this in which the measure for the height of a parallelogram is 8 feet and the base is 12 feet. Draw the picture of that parallelogram on the board. Have students find the area of the figure at their desks. Have someone come to the board and find the area for the class to see.

    Pass out an activity sheet for students to work on individually or in pairs.

Houghton Mifflin Math Grade 5