We have several different ways to get the area of a triangle. The simplest is Area = ½bh,which assumes we know the length of one of the sides and the length of the perpendicular line segment that connects the opposite vertex to the known side length. If we know all three side lengths, we have Heron's Formula, which we have discussed in previous blog posts and in class. Let's assume instead we know two side lengths AND the measure of the angle that lies between them. In the picture above, it is assumed we know the length of side b, and either we know angle A and side c, or we know angle C and side a. In either case, the height is opposite the known angle in the right triangle formed by the dotted line. Since sin = opp/hyp we get
sinA = height/c
or
sinC = height/a
This means we have two ways to represent this height,
height = csinA
or
height = asinC
What this gives us is two ways to represent the area that assumes b is the base
Area = ½basinC
or
Area = ½bcsinA
By rearranging the letters we have a similar formula with sin B
Area = ½acsinB
From here, the algebraic manipulation goes as follows.
sinA = height/c
or
sinC = height/a
This means we have two ways to represent this height,
height = csinA
or
height = asinC
What this gives us is two ways to represent the area that assumes b is the base
Area = ½basinC
or
Area = ½bcsinA
By rearranging the letters we have a similar formula with sin B
Area = ½acsinB
From here, the algebraic manipulation goes as follows.
In the pictures I found online the angles are labeled A, B and C. In class, I usually use alpha, beta and gamma. The math hasn't changed, just the labels.
An angle in a triangle has to be between 0° and 180°, which means in Quadrant I or Quadrant II, or possibly on the x or y axis. The value of sine is greater than or equal to zero in that range of angles, so using this method to find the area will always give us a non-negative number.
Examples: Let's say a = 3 and b = 4. If we have the measure of angle C, we can use our formula to find the area. Round the answer to the nearest thousandth.
a) Angle C = 0°
b) Angle C = 10°
c) Angle C = 20°
d) Angle C = 30°
e) Angle C = 40°
f) Angle C = 50°
g) Angle C = 60°
h) Angle C = 70°
i) Angle C = 80°
j) Angle C = 90°
Answers in the comments.
Examples: Let's say a = 3 and b = 4. If we have the measure of angle C, we can use our formula to find the area. Round the answer to the nearest thousandth.
a) Angle C = 0°
b) Angle C = 10°
c) Angle C = 20°
d) Angle C = 30°
e) Angle C = 40°
f) Angle C = 50°
g) Angle C = 60°
h) Angle C = 70°
i) Angle C = 80°
j) Angle C = 90°
Answers in the comments.
a) Angle C = 0°
ReplyDelete6*0 = 0
b) Angle C = 10°
6*sin(10°) ~= 1.042
c) Angle C = 20°
6*sin(20°) ~= 2.052
d) Angle C = 30°
6*sin(30°) = 3
e) Angle C = 40°
6*sin(40°) ~= 3.857
f) Angle C = 50°
6*sin(50°) ~= 4.596
g) Angle C = 60°
6*sin(60°) ~= 5.196
h) Angle C = 70°
6*sin(70°) ~= 5.638
i) Angle C = 80°
6*sin(80°) ~= 5.909
j) Angle C = 90°
6*sin(90°) = 6
If you look at the the angles between 90° and 180°, you'll see that sinA = sin(180° - A), so the area reaches a maximum when the angle in question is 90°.
I'm a little confused because it says "the height is opposite the known angle" and it is assumed that we do not know angle B.
ReplyDeleteIf we know either angle A and side c or angle C and side a, shouldn't the height be from angle A perpendicular to side a or from angle C perpendicular to side c? as opposed to from angle B perpendicular to side B since we do not know angle B
In the second paragraph, you leave out the statement that we know side b as well. We are assuming we know side-angle-side, and we could use either side as the base and use the other side to find the height by multiply by sine of the known angle. The difference in the formulas is simply a matter of order.
ReplyDeleteArea = ½absinC
or
Area = ½basinC