Area Formulas

Geometry and measurement ideas are usually taught together in elementary school. The connection between geometry and measurement is very evident in the development of formulas. Formulas are equations which use measures that are easy to determine such as length or height, to find measures that are more difficult to determine such as area and volume. For example, it is much easier to measure the length and width of a rectangle than it is to find the number of square units that cover the rectangle.

It is important that students participate in the development of formulas rather than just being given the formulas and asked to apply them. Participating in the development will help students make connections between the formulas for different figures. For example, your students already know and understand the formula for finding the area of a rectangle, A = l x w, where A represents the area, l the length, and w the width of the rectangle. They will see the relationship between this formula and the formula for finding the area of a triangle from examples like the following.

1. Start with a triangle. Draw a line parallel to one side of the triangle through the vertex that is opposite that side.
2. Draw line segments that form right angles from the other two vertices of the triangle up to the parallel line. Now your triangle is in a rectangle.
3. Draw a perpendicular line from the vertex of the triangle that is opposite the side that is shared by both the triangle and rectangle. Notice that two sets of congruent triangles are formed.

Look at the relationship between the different parts of the triangle and rectangle above. The height of the triangle, line segment CD, is congruent to the width of the rectangle, sides AX and BY. The base of the triangle, line segment AB, is the same as the length of the rectangle, line segment AB. Triangles ACX and ACD are congruent, and triangles BYC and CDB are also congruent. If you took away one triangle from each pair of congruent triangles, you would remove half the area of the rectangle. So, the area of triangle ABC is half the area of rectangle ABYX. The formula for finding the area of any triangle is A = 1/2 (b x h), where b is the base of the triangle and h is the height. To reinforce students' understanding of the formula for finding the area of a triangle, repeat steps 1-3 above with other triangles.

A similar kind of experience can also come from taking any rectangle and drawing a triangle in it. Look at the examples below. Use one side of the rectangle as the base of the triangle and locate the other vertex of the triangle anywhere on the opposite side of the rectangle. Draw a perpendicular line from the base of the triangle to the vertex to show the height of the triangle. Any triangle drawn this way in the same rectangle will have the same area, since the formula for finding the area of a triangle is A = 1/2(b x h), and the base and height will always be equal to the length and width of the rectangle.

A similar kind of hands-on experience can be provided in the development of the formula for the area of a parallelogram. A line segment joining two vertices of a polygon that is not a side of the polygon is called a diagonal. Two diagonals can be drawn in a parallelogram. Either diagonal divides the parallelogram into two congruent triangles. Line segment NP divides the parallelogram below into two congruent triangles, triangle MNP and triangle PON. The formula for finding the area of each triangle is A = 1/2 (b x h), so the formula for finding the area of the parallelogram is A = 1/2 (b x h) + A = 1/2 (b x h) or A = (b x h).

Another way to see this formula is to cut off the right triangle from one of the corners as shown below. By placing triangle MQP over to the right, shown as triangle NRO, we can see that parallelogram MNOP has the same area as rectangle QROP. So we can use the formula A = (b x h) to find the area of the parallelogram.

It is important to note that by the height we are talking about the perpendicular distance between two parallel sides in a parallelogram. In a triangle, when we refer to the height, we are talking about the perpendicular distance from a vertex to its opposite side.

• Introducing the Concept
• Developing the Concept