How the Area of a Triangle Changes When One of Its Sides Gets Longer
When you change the length x of one side of a triangle, the area A of the triangle changes. How does the change in area depend on the change in x? For some values of x, a small change, dx, could make practically no change in A. But for some other values of x, the same difference dx can change A significantly.
In order to study this, we'll look at two questions:
1. How does A depend on the length x?
2. How does the rate of change of the area relative to x, namely, dA/dx, depend on x?
Let's look at a triangle with two sides, a and b, having lengths 3 and 4 inches.
Task 1.
Draw a few triangles, each with two sides of lengths 3 and 4 inches, with the third side (call it x) having different lengths.
What is the area of the triangle when all three sides are given, a, b, and x? Compute the areas of the triangles you draw. There is a nice formula for the area of a triangle when the lengths of its three sides are given. It is called Heron's formula, after Heron (or Hero) the Elder of Alexandria (c. 100 BC).
Heron's formula for the area A of a triangle with sides x, a, and b is
A = √(s(s-x)(s-a)(s-b))
where the semi-perimeter s = (x + a + b)/2.
For a = 3 and b = 4, note that possible values of x are 1 < x < 7. (Why?)
As you change x, how does A vary? For which x do you think the area A is the largest? the smallest?
Task 2.
Graph the area A of your 3-4-x triangle as a function of x. Enter into the TI-83/84 calculator
\Y_{1}=(A+B+X)/2
semiperimeter of triangle
\Y_{2}=√(Y_{1}*(Y_{1}-X)*(Y_{1}-A)*(Y_{1}-B))
Heron's formula for the area A
\Y_{3}=nDeriv(Y_{2},X,X)
derivative of area A
Set WINDOW to
Xmin=0
Xmax=8
Ymin=-1
Ymax=7
From the home screen, store 3 and 4 in A and B.
3→A:4→B
Set AxesOn, and graph Y_{2}. You will see
Can you explain the meaning of this curve? On the X-axis is the length of side X of a 3-4-X triangle. On the Y-axis is the area of the triangle with legs of the three lengths.
For what value of x is the area the largest? How large? We show three ways to find this.
1. Use CALC maximum on the calculator. You will be asked for a lower and an upper bound for X, and for a guess. Then you will see
So the area A takes on a maximum value of 6 square inches when side X = 5 inches. (Note that for X=5, we have a right triangle!)
2. Compute where the rate of change of the area function is zero. Namely, compute where Y_{3}, the first derivative of Y_{2}, equals zero.
Go to MATH and SOLVER. Enter the equation
eqn:0=Y_{3}
and press ENTER. You will see
Y_{3}=0
X=(some value)
A=3
B=4
bound={-1E99,1E99}
Since X cannot be negative, you may set the bound to look as follows:
bound={0,1E99}
Set the cursor on the X= line, enter a guess, and press ALPHA SOLVE, and you will see
X = 4.99999990000
So the area is a maximum when X=5 inches.
From the home screen, enter
Y_{2}(5) ENTER,
and you will see 6, the maximum area.
3. Solve the equation nDeriv(Y_{2}) = 0.
Again, go to MATH and to SOLVER. Enter
eqn:0=nDeriv(Y_{2},X,X) ENTER
and you will see
nDeriv(Y_{2},X,X)=0
X= (some value)
A=3
B=4
bound={0,1E99}
Set the cursor on the X= line, enter a guess, and press ALPHA SOLVE
You will see
X=4.9999998999261
as we have seen before.
Task 3. Explain why the curve above is not symmetric.
Hint: You can investigate this by setting Float to 2 and entering Y_{2}({1,2,3,4,5,6,7}). You will see
{0.00 2.90 4.47 5.56 6.00 5.33 0.00}
Task 4. If we vary the angle X (not the side!) between sides a and b, how does the area vary?
The equation for the area of a triangle in terms of an angle X and its two adjacent sides a and b is A = a*b*sinX/2.
Here, X can vary between 0 and 180 degrees. Enter
\Y4=ABsin(X)/2
Set the window for X between 0 and 180 degrees, and set Xscl to 30. Now graph Y4.
You will see
What is the meaning of this curve? Why is it symmetric?
For what angle X is the area the biggest? How big?
Task 5. Investigate how the area of the triangle changes as a function of the length of side x for different values of A and B.
The cosine theorem says that in any triangle with sides a, b, c, and angle C opposite to side c,
a^{2} + b^{2} - 2*a*b*cos(C) = c^{2}.
So for a right triangle, cos (90°) = 0, and we have the Pythagorean theorem.
Is it true that the maximum area occurs when the angle across from side X is a right angle?
Task 6. Make a set of 3-4-x triangles, for x varying from 1.5 inches through 6.5 inches, as shown at the beginning of the unit.
Task 7. (I may or may not include this) We have written a program, AREA, that takes as input side lengths A and B, B>A, and a "step size" for the third side X. It will draw a sequence of triangles with sides A, B, and X, and also a square that has the same area as the triangle it draws. Let's see if we can get a better visual idea of how the area varies as a function of the length of the third side. (Program code not included here)
Remark. The "triangle fan" shown above can use the shortest side as its base, as in the first picture below, or can use a longer side in its base, as in the second picture below.