# Using Trigonometry to Determine Area

### New York State Common Core Math Geometry, Module 2, Lesson 31

Worksheets for Geometry, Module 2, Lesson 31

Student Outcomes

• Students prove that the area of a triangle is one-half times the product of two side lengths times the sine of the included angle and solve problems using this formula.
• Students find the area of an isosceles triangle given the base length and the measure of one angle.

Using Trigonometry to Determine Area

Classwork

Opening Exercise

Three triangles are presented below. Determine the areas for each triangle, if possible. If it is not possible to find the area with the provided information, describe what is needed in order to determine the area.

Is there a way to find the missing information?

Example 1

Find the area of △ 𝐺𝐻I

Example 2

A farmer is planning how to divide his land for planting next year’s crops. A triangular plot of land is left with two known side lengths measuring 500 m and 1,700 m.

What could the farmer do next in order to find the area of the plot?

Exercise 1

A real estate developer and her surveyor are searching for their next piece of land to build on. They each examine a plot of land in the shape of △ 𝐴𝐵𝐶. The real estate developer measures the length of 𝐴𝐵 and 𝐴𝐶 and finds them to both be approximately 4,000 feet, and the included angle has a measure of approximately 50°. The surveyor measures the length of 𝐴𝐶 and 𝐵𝐶 and finds the lengths to be approximately 4,000 feet and 3,400 feet, respectively, and measures the angle between the two sides to be approximately 65°.
a. Draw a diagram that models the situation, labeling all lengths and angle measures.
b. The real estate developer and surveyor each calculate the area of the plot of land and both find roughly the same area. Show how each person calculated the area; round to the nearest hundred. Redraw the diagram with only the relevant labels for both the real estate agent and surveyor.
c. What could possibly explain the difference between the real estate agent’s and surveyor’s calculated areas?

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