The following discussion will focus on calculating
the areas of rectangles, circles and triangles.
(Review
Measurement
before proceeding.)
The area of a rectangle, shown below, is
its length multiplied by its width. For example if its length L is 10
meters and its width is 5 meters, then its area A = 10 m * 5 m = 50 m2
.
The unit square meters is important; one must be
careful to multiple lengths in the same unit of measure. For
example, if the length of the rectangle is 6 inches and the width is 1½
feet, then we must either convert the inches to feet or the feet to inches.
We'll do both.
First, 6 inches is ½ foot; so we
have 1/2 ft * 3/2 ft = 3/4 ft2
Second, 1½ feet is 18 inches; so we have 6 in* 18
in = 108 in2
We have very differently looking answers, are they
the same?
Well a square foot is 12in * 12in =
144 in2 , so 3/4 of this
is 3*144 / 4 = 432 /4 = 108.
One last comment about rectangles is if
all sides are equal in length then we have
a square. the area of a square is
calculated the same way except the L = W
so, the area is L2.
The area of a circle, shown below, is
found by multiplying the constant pi
by its radius squared. For a
discussion on pi refer to pi.
This calculation is straight forward, for
example, using the value 3.14 for pi and
given a circle with radius 4 in then its
area A is
3.14 * (4 in)2 = 3.14 * 16 in2
= 803.84 in2
As another example, suppose we're give a
circle whose diameter is 86 cm.
W need to first recall that the diameter
of a circle is twice the radius. So
we use half our diameter 86 cm to get
A = 3.14 * (43 cm)2 =
3.14 * 1849 cm2 = 5805.86 cm2
One final note, as an approximation to pi
one can use the fraction 22/7.
If you regard a triangle as a rectangle or
parallelogram cut in half along a diagonal, then the area of a triangle would be
half of the rectangle or parallelogram from which it belongs. For example,
in triangle T1 above, if you drew in the top and the right-hand side you'd have
a rectangle whose area A = b * h. Since the triangle is half that
rectangle we have A = ½ b*h.
For now let's focus on the meaning of 'base' and
'height' (or altitude) of a triangle. The base of a triangle is any one of
its sides. Once you've selected this side, the height is the line segment
drawn from this base to the vertex opposite this base in such a way that the
segment is perpendicular to the base. In triangle T1 above you see a blue
square against the height and the base. This square indicates that the
height is perpendicular to the base. In the rest of the triangles this
square is shown hyphenated and not colored in.
Triangle T2 is a typical example of the height
lying outside the triangle. Since there is now way to draw the height
inside the triangle to intersect the base at right angles, we must extend the
base until we can draw the height perpendicular to the base.
In all the examples above the altitude of each
triangle is shown starting from the vertex with the heavy dot. You should
strive to become comfortable with any triangle in any position. Draw
a triangle on a piece of heavy paper, cut it out from the paper and rotate this
triangle from side to side marking the altitude from each side it rests on. Use
a ruler or yardstick to measure each side and calculate the area 4 different
ways: with 3 different bases and altitudes and the equation for 's'.
Convince yourself that the area is the same in all for calculations. Use
the shapes of triangle T1 and T2 for this exercise.
This is another FREE ALGEBRA PRINTABLE presented to you from the
Algebra section of
K12math.com