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Areas and Perimeters of Triangles and Special Quadrilaterals
Introduction: Area is a measure of the amount of space contained insides a closed figure. Perimeter is a measure of the distance around a closed figure. We examine these concepts for triangles and several special quadrilaterals.

The Lesson:
We begin with the area and perimeter of a rectangle. A diagram below illustrates these concepts. To find the perimeter, we add the lengths of the sides. To find the area, we count the squares inside the rectangle.



The dimensions of this rectangle are 2 x 4. We say that the length l is 4 and the width w is 2. Notice that there re 8 squares inside the rectangle. This is the area and is found by multiplying 2 x 4. the perimeter is the sum of the sides and is 12.

We generalize this result for rectangles: A triangle can be thought of as half of a rectangle when there is a right angle.



The area of the triangle will be .

Two special quadrilaterals that we will examine are the parallelogram and the trapezoid. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A trapezoid is a quadrilateral with only one pair of opposite sides parallel. The perimeters of each are the sum of the lengths of the sides.

In the diagram below, we have divided the parallelogram into two triangles. The area of each triangle is ½bh. Since there are two triangles, the area of the parallelogram is bh. Remember that the height of a triangle equals the length of a perpendicular line segment that connects the angle opposite the base to the "line of the base."

 


Another way of visualizing this formula is shown in the next two diagrams where the "left corner triangle" is "moved to the right" and completes a rectangle whose area is found by multiplying base x height.



A trapezoid is shown below. It is also divided into two triangles. The parallel bases are labeled b and B. The height of each triangle is the height of the trapezoid and is labeled h.



The area of the triangle with base b is . The area of the other triangle is . Adding these two areas, we get the area of the trapezoid as .
We summarize our results for areas and add one definition:
Definition: An isosceles trapezoid is a trapezoid which has equal base angles and equal non-parallel sides.

Let's Practice:
  1. Suppose a parallelogram has an area of 36 square feet and the height is 4 feet. What is the measure of the base?
The area of a parallelogram is bh = 4b = 36.
Therefore b = 9 feet.
  1. A rectangle has a perimeter of 14 inches. The length is 5 inches. What is the area of this rectangle and what is the width?
Since the perimeter = 2l + 2w = 10 + 2w = 14, we must have 2w = 4.
Therefore the width is 2 inches.

The area is lw = 5 x 2 = 10 square inches.
  1. A trapezoid has parallel bases of length 5 feet and 9 feet. The height is 4 feet. What is the area? What is the perimeter?
The area of a trapezoid is
ft2
We cannot find the perimeter unless we know more about its base angles in order to measure the non-parallel sides.

Examples
Example A right triangle has legs of length 4 and 7 feet. What is the area?
What is your answer?
 
Example
What is the area and perimeter of the parallelogram shown below?
 
 
What is your answer?
 
Example An isosceles trapezoid has bases of 9 and 16 feet. The height is 8 feet. The base angles are 39º. What is the area and what is the perimeter?
What is your answer?
 



M Ransom

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