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AREA FORMULAS: TRIANGLE, PARALLELOGRAM, TRAPEZOID
Jump right to the exercises!
Take a vertex of a triangle, and drop a segment that is perpendicular to the opposite side (first picture below)
or (if needed) to an extension of the opposite side (second picture below).
The segment containing the chosen vertex is called a height
or altitude of the triangle,
and the side opposite this vertex is called the corresponding
base.
Since every triangle has three vertices, every triangle has three
height/base pairs.
If someone asks you for the "height and base of a triangle" there are three possible correct answers!
The phrase "height (or base) of a triangle" can refer to the actual line segment,
or to the length of the line segment—the context
will determine which is desired.
Look at the picture below, where the altitude has been cut in half.
Convince yourself that the yellow triangles are congruent, and the grey triangles are congruent.
This picture shows that the formula for the area of a triangle
is half the height, times the base.
Use the Geometer's Sketchpad to explore this picture!
AREA OF A TRIANGLE:
Let h and
b denote a height/base pair for a triangle.
The area of the triangle is given by the formula
12bh .
|
Of course, this formula can be written in a variety of ways:
12bh=
12hb=
hb2=
bh2=
h2⋅b and so on!
Use the Geometer's Sketchpad to explore the formula for the area of a triangle!
With the formula for the area of a triangle in hand,
it is now easy to find area formulas for
quadrilaterals that have at least one pair of parallel sides,
like parallelograms and trapezoids.
Whenever a quadrilateral has a pair of parallel sides,
then the (perpendicular) distance between these
parallel sides is called the height or altitude,
and the parallel sides are called the bases .
Since a parallelogram has two pairs of parallel sides,
it has two height/base pairs:
The secret to finding the area is now to put in a diagonal, and observe that
the area is broken into two triangles!
AREA OF A PARALLELOGRAM:
Let h and
b denote a height/base pair for a parallelogram.
The area of the parallelogram is given by the formula
bh .
|
Use the Geometer's Sketchpad to explore the area of a parallelogram!
Exactly the same idea is applied to find the area of a trapezoid:
AREA OF A TRAPEZOID:
Let h be the height of a trapezoid,
and let
b1 and
b2
denote the two bases.
The area of the trapezoid is found by averaging the two bases, and multiplying by the height:
area=
h(b1+
b22)
|
Of course, this formula can be written in a variety of ways:
h(b1+
b22)
=
h2(b1+
b2)
=
(b12
+b22
)h
=
hb12
+
hb22
=
12(
b1+b2
)h
and so on!
Use the Geometer's Sketchpad to explore the area of a trapezoid!
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.