Analyzing The Formula of A Triangle

Analyzing The Formula of A Triangle

About Base and Height

Opening

About Base and Height

Discuss the following statements.

  • The base of a triangle can be any of the three sides.
  • The height of a triangle is the perpendicular distance from the base to the vertex opposite the base.
  • As shown in the diagram, the height can be inside or outside the triangle, or it can be one of the sides.

Introduction to Triangles

Opening

Introduction to Triangles

Can you take any triangle, copy it, and then combine the two triangles so that they form a parallelogram?

Try it with triangles like the ones in the diagram.

  • What do your results tell you about the area of a triangle?
  • Write a formula for the area of a triangle.

Math Mission

Opening

Explore the formula for the area of a triangle.

Explore the Area of Triangles

Work Time

Explore the Area of Triangles

The formula for the area of a triangle is
area = 12 •  base • height, or A = 12bh

Use the Triangle interactive to explore the area of a triangle. Move the vertices of the triangle and explore what happens to the area.

  • What happens if you keep the height and base constant and move the vertex parallel to the base?
  • What happens if you keep the base constant and change the height?
  • Try to discover one more interesting fact about a triangle and its area that you can share with the class.

INTERACTIVE: Triangle

Hint:

  • How does knowing the formula for the area of a parallelogram help you understand the formula for the area of a triangle?
  • There are two variables, base and height, that determine the area of a triangle. A triangle also has angle measures and side lengths for the two “non-base” sides. Try experimenting with all of these measures.

Prepare a Presentation

Work Time

Prepare a Presentation

  • Select one of your conclusions about what happens to the area of a triangle when you change one or more variables.
  • Be prepared to demonstrate your conclusion using the Triangle interactive, and to support your thinking mathematically.

Challenge Problem

Suppose the base of a triangle lies on one of two parallel lines, and the vertex opposite the base lies on the other parallel line.

  • If you slide the vertex along the line, what do you think will happen to the area of the triangle? Use the Triangle interactive to test your prediction.
  • Explain your results.

INTERACTIVE: Triangle

Make Connections

Performance Task

Make Connections

  • Take notes about your classmates’ conclusions concerning what happens to the area of a triangle when you change one or more variables.

Hint:

As your classmates present, ask questions such as:

  • What surprised you in your exploration of the area of a triangle?
  • How do your conclusions about the area of a triangle compare with those of other presenters?

Area of Trapezoid

Work Time

Area of Trapezoid

  • Find the area of this trapezoid.

Area of Triangle

Work Time

Area of Triangle

  • Find the area of this triangle.

Area Formulas

Formative Assessment

Area Formulas

Read and Discuss

  • The area of a rectangle is equal to its base times its height.
    A = bh
  • The area of a parallelogram is equal to its base times its height.
    A = bh
  • The area of a trapezoid is equal to one half times the sum of the bases times the height.
    A = 12(b1 + b2)h
  • The area of a triangle is equal to one half the base times the height.
    A = 12bh

Hint:

Can you:

  • Calculate the area of a triangle, parallelogram, or trapezoid given the values of the base(s) and height?
  • Calculate the height of a triangle, parallelogram, or trapezoid given the values of the base(s) and area?

Reflect On Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

The way I remember the formula for the area of a triangle is …