The Area of Acute Triangles Using Height and Base

Videos and solutions to help Grade 6 students find the area formula for a triangular region by decomposing a triangle into right triangles.

New York State Common Core Math Grade 6, Module 5, Lesson 3

Download lessons for 6th Grade

Lesson 3 Student Outcomes

Students show the area formula for a triangular region by decomposing a triangle into right triangles. For a given triangle, the height of the triangle is the length of the altitude. The length of the base is either called the length base or, more commonly, the base.

Students understand that the height of the triangle is the perpendicular segment from a vertex of a triangle to the line containing the opposite side. The opposite side is called the base. Students understand that any side of a triangle can be considered a base and that the choice of base determines the height.

Exercises

1. Work with a partner on the exercises below. Determine if the area formula A = 1/2 bh is always correct. You may use a calculator, but be sure to record your work on your paper as well.
2. Can we use the formula A = 1/2 x base x height to calculate the area of triangles that are not right triangles? Explain your thinking.
   
   
3. Examine the given triangle and expression.
   Explain what each part of the expression represents according to the triangle.
   
   
4. Joe found the area of a triangle by writing A = 1/2 (11 in.)(4 in.), while Kaitlyn found the area by writing A = 1/2 (3 in.)(4 in.) + 1/2 (8 in.)(4 in.). Explain how each student approached the problem.
   
   
5. The triangle below has an area of 4.76 sq. in. If the base is 3.4 in., let h be the height in inches.
   a. Explain how the equation 4.76 in² = 1/2(3.4 in.)(h) represents the situation.
   b. Solve the equation.

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Problem Set

1. • 4. Calculate the area of each shape below. Figures are not drawn to scale.
        
        
2. Immanuel is building a fence to make an enclosed play area for his dog. The enclosed area will be in the shape of a triangle with a base of 48 m. and an altitude of 32 m. How much space does the dog have to play?
   
   
3. Chauncey is building a storage bench for his son’s playroom. The storage bench will fit into the corner and against two walls to form a triangle. Chauncey wants to buy a triangular shaped cover for the bench.
   If the storage bench is 2 1/2 ft. along one wall and 4 1/4 ft. along the other wall, how big will the cover have to be to cover the entire bench?
   
   
4. Examine the triangle to the right.
   a. Write an expression to show how you would calculate the area.
   b. Identify each part of your expression as it relates to the triangle.
   
   
5. The floor of a triangular room has an area of 32 1/2 sq.m. If the triangle’s altitude is 7 1/2 m, write an equation to determine the length of the base, b, in meters. Then solve the equation.

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