Instruction 1-2

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Q&A	Concepts in Geometry	Q&A

Measuring Angles | Areas of Triangles | Summary

AREAS OF TRIANGLE

CCSTD Trigonometry 14.0

Two Different Approaches

In geometry, we learned that the area of a triangle can be calculated using the lengths of its sides. We can use two methods, depending on the information we are given. These methods are called: Base-Altitude Method, and Two-Leg Method.

Base-Altitude Method: If we know the height and the base of a triangle, then we can use the formula below to find its area:

Two-Leg Method: If the lengths of the legs (sides) are given, the area of a right triangle can be calculated using this formula:

Now, we will explore the area of a triangle in using its sides and an interior angle with the Angle-Sides Method.
Angle-Sides Method (Formula): In real-life situations, it is not always possible to measure the base or height of a triangle.
For example, if we were to measure the area of a triangular pond, finding any height of the pond is not an easy task. In such situations, we can use the angle and side formula. If we know a triangle’s two sides and the angle included by these sides, we can find the area of the triangle. We use a ratio called “sin (pronounced like sign).”

Angle-Sides Method (Formula): In real-life situations, it is not always possible to measure the base or height of a triangle. For example, if we were to measure the area of a triangular pond, finding any height of the pond is not an easy task. In such situations, we can use the angle and side formula. If we know a triangle’s two sides and the angle included by these sides, we can find the area of the triangle. We use a ratio called "sin (pronounced like sign)." 

Figure 2.1

Properties of “sin” ratios: the “sin” ratio of an angle can be calculated by any scientific calculator. Name two sides of a triangle a and b and call the angle included by these sides m.

Then, the area of the triangle, S, is determined by the formula below:

Practice 3. Find the area of the triangle below:

Figure 2.2

Solution. We are given AB = 14, BC = 20, and mvB = 65. Replace these values in the formula. Use a calculator to find sin 65.

Practice 4. Find the area of the triangle ABC.

Figure 2.3

Solution. AB and AC are given. However, we do not have the measure of vA. To find it, use the Angle Sum Principle.
Angle Sum Principle : mvA + mvB + CÚm = 180
Now we solve using values from figure 2.3:
25 + 35 + mvA = 180
60 + mvA = 180
mvA = 180 – 60

A Real Life Application 2. The Area of a Lake

Figure 2.4 is a triangular-shaped lake. We can not measure any of its heights. We can determine the lengths of two sides AB and BC, and the angle mvB included by these sides. Calculate the area of the lake.

Solution. We are given the lengths of two sides and the angle included by them:
AB = 120 feet
BC = 164 feet
mvB = 75

Use these values in the formula:

Video Instruction
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• Area of Triangle [3:25]

for Students, Parents and Teachers

Now let's do Practice Exercise 1-2 (top).

Summary

We covered angles in this lesson. From the definition of an angle, to the area of a triangle, and everything in between, the applications and methods of measuring angles should be clear. To summarize:

• When two rays start from the same initial point they form an angle.
• The measurement of an angle is the magnitude of openness between its sides.
• A one-degree angle measurement is of the circumference of any circle.
• A central angle that intercepts an arc measures1 radian.
• The circumference of a circle is times a radius. Thus, an entire circle measures radians.