Students derive the formulas for the area and perimeter of a triangle, and practice using these formulas.
After completing this tutorial, you will be able to complete the following:
Recall that the perimeter of a region is the length of the path that surrounds a region (or the sum of the lengths of the sides of the region), and that the area of this region is the number of square units covered by the region. Since the perimeter of any polygon is the sum of its side lengths, the perimeter of a triangle with side lengths a, b, and c is P = a + b + c.
The formula for the area of a triangle can be derived either from the formula for the area of a rectangle or from the formula for the area of a parallelogram. For the first derivation, consider a triangle drawn in a rectangle in the following way.
Obviously, the area of the triangle is one half of the area of the rectangle. If the length of one leg of this triangle is b and the length of the other is h, then the area of the rectangle is Thus, the area of the triangle is
However, this right triangle was drawn so that the legs of the triangle coincide with two sides of the rectangle. What if the triangle was drawn in the rectangle in a different manner? For example, how much of the rectangle is taken up by the triangle below?
Once again, we will see that if the length of the rectangle is b and the width of the rectangle is h, then the area of the triangle is First, divide the rectangle into two smaller rectangles in the following way:
Clearly, each smaller rectangle is divided in half by the sides of the triangle. It follows that the area of the triangle is half the area of the rectangle. Again, since the area of the rectangle is the area of the triangle is
Additionally, the area of a triangle can be derived from the area of a parallelogram. Consider a triangle with base b and height h. By rotating the triangle as shown below, we obtain a parallelogram whose area is twice that of the triangle. Notice that a base of the parallelogram has length b and the corresponding height is h. Since the area of a parallelogram with base b and height h is , the area of the triangle is
|Approximate Time||30 Minutes|
|Pre-requisite Concepts||Students should know the formula for the area of a parallelogram; know the perimeter of a polygon, know the definition of a triangle and its properties; be able to calculate the distance between two points on the coordinate plane; and be familiar with the Pythagorean theorem.|
|Type of Tutorial||Visual Proof|
|Key Vocabulary||perimeter, triangle, Pythagorean theorem|