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Word Problems: Area and Perimeter of Triangles
In order to solve problems which require application of the area and perimeter for triangles, it is necessary to
 
 
A typical problem involving the area and perimeter of a triangle gives us the area, perimeter and/or side lengths and altitude (we shall use the variableh” for height) of the triangle. We may also be given a relationship between the area and perimeter or between the sides and altitude of the triangle. We may know one or more of the angles of the triangle. We need to calculate some of these quantities given information about the others. Two examples of this type of problem follow.
 
Example #1:
 
Suppose in a right triangle one of the legs is of length 5 and the angle formed by the hypotenuse and this leg is 28°. What are the area and perimeter of this triangle? A diagram is shown below.
 
 
A right triangle is a special case of this type of problem since the legs are perpendicular. One leg is the base and the other is the height or altitude of the triangle. In order to find h in the diagram above we use
 

h » 2.6585
 
We can now find the area from
 
 A = (1/2)bh = (1/2)(5)(2.6585) = 6.64625
 
To find the perimeter, we need the hypotenuse so we can add all three sides. We find the hypotenuse using the Pythagorean Theorem:
 
 
We add the lengths of the three sides and get
 
perimeter = 13.3213
 
Example #2:
 
Suppose a triangle with area 28 has a base of 7x and a height of 4x as shown in the diagram below. What are the area, the perimeter, and the dimensions of this triangle?
 
 
We know area is 28 and equals ½bh. We write
 
 
Next we solve the equation 28 = 14x2 for x
 
x =
 
We now know that
 
 
We can check our work by recalculating the area
 
A =
 
The frustrating part of this analysis is that we cannot find the measures of the other two sides with the information that we have been given. Therefore, we can solve for x, and get the lengths of the altitude and base, but we will never be able to determine either the dimensions or the perimeter of this triangle.
 

Let's Practice

Question #1
AudioThe altitude of a right triangle is twice the base. If the area is 36, what are the dimensions and the perimeter of this triangle?


Question #2
AudioA equilateral triangle has a base of 7. What are the altitude, the area and the perimeter of the triangle?


Question #3
AudioA right triangle has a hypotenuse of 10 and one acute angle of 42°. Find the area and perimeter of this triangle.



Try These
Question #1
AudioA triangle has a base of b = 20 cm. If the height equals 2x cm and its area equals 80x2 cm2, find the values of its height and area.


Question #2
AudioAn equilateral triangle has an altitude of 6 inches. What are the perimeter and area of this triangle?


Question #3
AudioA right triangle has one leg of length 7 which makes an angle of 36° with the hypotenuse. What are the area and perimeter of this triangle correct to one decimal place?



This type of problem involves relationships among the lengths of the sides and altitude, area, and perimeter of a triangle. We need to focus on the area formula which requires a measurement of the base and altitude, which we call h for “height.” Finding either of these quantities requires different approaches depending on whether the triangle is equilateral, right, or neither. There are formulas for an equilateral triangle that relate h to the length of a side and that relate the area to the length of a side. In a right triangle, we may have to apply the Pythagorean Theorem or use basic trigonometry with sine, cosine, or tangent in order to calculate h or b.
 
 
When using the Pythagorean Theorem or basic trigonometry, there may be round off errors during the course of solving the problem. Be aware that repeated calculations using rounded off values will result in slightly differing answers among students.

M Ransom

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