8.1
What is Area?
You have learned a good deal about area in the past. There is an array of formulas with which one computes area; you are familiar with some of these. In this section, we approach area in the same formal way that we have earlier dealt with segments, angles, and arcs. We need postulates that describe how this new notion of area works, and then we can use those postulates to develop ways to compute area — including the formulas that you have encountered in the past.
- Postulate: For each closed region of the plane, there is a unique positive real number called the area of the region.
We will denote the area of region $R$ by the symbol $a(R)$.
- Postulate: If two regions are congruent, their areas are equal.
- Postulate: The Area Addition Postulate
- If two regions have no interior points in common, then the area of the union of those two regions is the sum of the areas of the separate regions.
We have encountered postulates of this form before. The Segment Addition Postulate, the Angle Addition Postulate, and the Arc Addition Postulate all tell us when we can add measures of the parts to obtain the measure of the whole thing.
- Postulate: The area of a square is the square of the length of a side.
This is the first area "formula;" now we know that the area of a square of side $s$ is $s^2$.
- Theorem 8.1: The area of a rectangle is the product of the lengths of two consecutive sides.
Restatement:
If a rectangle has length $l$ and width $w$, then the area of the
rectangle is $l \cdot w$. Given: Rectangle $ABCD$ with $AB=l$ and $BC=w$. |
Prove: $a(ABCD)=l \cdot w$
Proof: We
start
with a square of side $l+w$, and subdivide it into four parts as
shown. The four pieces have no interior points in common, so the
area
of the whole square is the sum of the areas of the four separate
pieces. Two of the smaller pieces are squares; the other two are
congruent rectangles. The congruent rectangles have equal areas,
and
the squares have areas that we know (all of this follows from our
postulates about area). Thus we have the following. \begin{align*} a(\text{whole square})&=a(\text{square of side }l)+a(\text{square of side }w)+2\cdot a(\text{rectangle ABCD})\\ (l+w)^2&=l^2+w^2+s\cdot a(ABCD)\\ l^2+2l\cdot w+w^2&=l^2+w^2+2\cdot a(ABCD)\\ 2l\cdot w&=2a\cdot a(ABCD)\\ a(ABCD)&=l\cdot w \end{align*} |
Example: Find the area of the figure pictured, assuming all the angles are right angles. |
Solution:
We break the
figure down into three non-overlapping pieces. $A_1$ is a
rectangle of length 6 and width 3. $A_2$ is a rectangle of length
7 and width 2. $A_3$ is a square of side 3. The area $A$ is
given by \begin{align*} A&=a(A_1)+a(A_2)+a(A_3)\\ &=(6\cdot 3)+(7\cdot 2)+3^2\\ &=18+14+9\\ &=41 \end{align*} |