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This lesson presents the idea that the area of any triangle is exactly half of a certain parallelogram -- thus we get the familiar formula of multiplying the base and the altitude, and taking half of that. The lesson contains varied exercises for students.

Area of Triangles

We can always put any triangle together with
a copy of itself to make a parallelogram.

Therefore, the AREA of any triangle must be
exactly half of the area of that parallelogram.

   

1. Find the area of the shaded triangle in the picture above.
 

2. Draw the corresponding parallelograms for these triangles, and find their areas.
    Hint: draw a line that is congruent to the base of the triangle, starting at the top vertex.

    a. _______ square units

b. _______ square units

c. _______ square units

   

Again, we use a base and an altitude.
The base can be any side of the triangle,
though people often use the “bottom” side.

The altitude is perpendicular to the base,
and it goes from the opposite vertex to the
base (or to the continuation of the base).

Since the area of a triangle is half of the area of
the corresponding parallelogram, we can calculate
the area as half of the base times the altitude, or:

A =

BASE × ALTITUDE


2

You can choose any side to be
the base. Here, it makes sense to
choose the vertical side as the base.

The area is  

4 × 6

2

  = 12 square units.

3. Draw an altitude in each triangle, and mark the base. Find the area of each triangle.

    a. _______ square units

b. _______ square units

c. _______ square units



Example 1. The altitude of a triangle may fall outside of the triangle itself. It is still perpendicular to the base, and starts at a vertex.

The corresponding parallelogram is seen if you follow the dotted lines.

The area is  

4 × 3

2

= 6 square units.

Example 2. Here it is easiest to think of the base being on the “top.”
Again, the altitude falls outside the actual triangle.

The area is  

5 × 3

2

= 7 1/2 square units.

 

5. This figure is called a _________________________.

   Calculate its area.

      

6. Draw as many different-shaped triangles as you can that have an area of 12 square units.



How to find the area of a triangle not drawn on grid

  1. First, choose one of the sides as the base. It can be any of the sides!
  2. Draw the altitude. Use a protractor or a triangular ruler to draw the
    altitude so that it goes through one vertex, and is perpendicular to
    the base. See the illustration.

    Place the protractor so that the line you will draw will pass through
    the vertex
    and so that the 90°-mark is lined up with the base of
    the triangle.

  3. Measure the altitude and base as precisely as you can with a ruler.
  4. Calculate the area.


8. Draw your own triangle, and find its area!



9. Draw a triangle with an area of 3 square inches.
    Is it only possible to draw just one triangle with
    that area, or is it possible draw several, with
    varying shapes/sizes?


 

10. Draw a triangle, without measuring anything, so that its area is close to 20 cm2. Check by drawing
      the altitude and measuring! Practice until you get a triangle with an  area of approximately 20 cm2.

      You can even make a game out of this: whoever gets the area the closest to the given area is the
      winner or gets the most points. Or, take turns with your friend, asking the friend to draw a triangle with
      a specific area. Perhaps add a condition that the triangle has to be obtuse/acute/right, or
      equilateral/isosceles/scalene.




This lesson is taken from Maria Miller's book Math Mammoth Geometry 2, and posted at www.HomeschoolMath.net with permission from the author. Copyright © Maria Miller.



Math Mammoth Geometry 2

A self-teaching worktext for 6th-7th grade that covers the area of triangles, parallelograms, and polygons, pi, area of circle, nets, surface area, and volume of common solids.

Download ($5.70). Also available as a printed copy.

=> Learn more and see the free samples!

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