find the area of a triangle that the sides are 12cm, 8cm and 6cm.
Layout the triangle with point A at the origin, point B 8cm to
the right, and point C somewhere above, 6cm from B and 12cm
from the origin. If this were a right-triangle, the hypotenuse
would be 10cm, but in this triangle, what would be the
hypotenuse is longer than that. We need another point, D,
directly below C, forming two additional triangles, both of
which will be right-triangles. They will be triangle ADC
(which encloses triangle ABC) and triangle BDC (which is
outside of triangle ABC). We will then find the area of
triangle ADC and subtract the area of triangle BDC,
leaving the area of triangle ABC.
The endpoints are layed out similar to this:
C
A B D
Call side AB a.
Call side BC b.
Call side AC c.
Call angle ABC theta.
The length of c is given by the general equation
a^2 + b^2 - 2ab(cos theta) = c^2
We know a, b and c, so all we need to do is solve for
the cosine of angle theta. Subtract a^2 from both sides,
subtract b^2 from both sides, and divide both sides by
-2ab.
cos theta = (c^2 - a^2 - b^2) / (-2ab)
cos theta = (12^2 - 8^2 - 6^2) / (-2 * 8 * 6)
cos theta = (144 - 64 - 36) / (-96)
cos theta = 44 / -96 = -0.458333....
Using the inverse cosine, we find that angle theta
is 117.2796 degrees. That's the angle inside the
given triangle, facing the 12cm side. We want the
supplement, the angle CBD, facing away from the origin.
180 - 117.2796 = 62.72 degrees
With that, we can find the length of CD. Call that d.
d / 6cm = sin 62.72
d = 6cm * (sin 62.72) = 6cm * 0.88878 = 5.3327cm
We need the length of side BD. Call it e.
e / 6cm = cos 62.72
e = 6cm * (cos 62.72) = 6cm * 0.4583 = 2.75cm
Those two lengths can be used to find the area
of this small triangle:
area1 = 1/2 * d * e
area1 = 1/2 * 5.3327cm * 2.75cm = 7.3325cm^2
Area2, the area of triangle ADC is found by first
adding the length of BD to the length of AB. Call that f.
f = a + e = 8cm + 2.75cm = 10.75cm
area2 = 1/2 * d * f
area2 = 1/2 * 5.3327cm * 10.75cm = 28.66327cm^2
Area3, the area of the original triangle, is found by
subtracting the area of the smaller triangle from
the area of the larger triangle.
area3 = 28.66327cm^2 - 7.3325cm^2 = 20.33cm^2