Expert: Sherry Wallin - 6/30/2009

Question
Given ΔABC, find the area of the triangle to the nearest tenth of a unit.

1.   a = 6 cm ,  b = 10 cm,  and  c = 8 cm.

2.   <A = 50°,  b = 12.5 cm, and  c = 18 cm.

3.   a = 16 cm ,  b = 18 cm,  and  c = 20 cm.

4.   <C = 80° ,  a = 42 cm ,  and  b = 38 cm.

5.   a = 25 cm,  b = 15 cm, and  c = 20 cm.

    6.   <B = 60° ,   a = 24 cm, and  c = 24 cm.

    7.   a = 6 cm,  b = 7 cm,  and  c = 8 cm.  

Answer
Hi Gwen~
     I see this as algebra or trigonometry and this is how I would approach finding the areas of these types of triangles:

#1 is a 3,4,5 triangle which is a right triangle so the area is (1/2)bh
(1/2)*6*8 = 24 sq cm

#5 as well is a right triangle since 25^2 = 15^2 + 20^2 or
625 = 225 + 400, so again (1/2)*15*20 = 150 sq cm

#'s 2,4, and 6 you an use the law cosines to find the 3rd side and then use Heron's Formula A = sqrt (s(s-a)(s-b)(s-c)) where s = (1/2)(a+b+c).
Here I will do #2 for you:  a^2 = b^2 + c^2 - 2bc*Cos A gives us
a^2 = 12.5^2 + 18^2 - 2*12.5*18*cos 50
a^2 = 156.25 + 324 - 450*cos 50
a^2 = 480.25 - 289.25
a^2 = 161
so a^2 = 161 thus c = sqrt 161 =~ 12.689 cm. I am using =~ to mean 'equals approximately'. Let's round it off to 12.7

Now us Heron's Formula: s = .5(12.5 + 18 + 12.7)= 21.6 so
sqrt(21.6(21.6-12.5)(21.6-18)(21.6-12.7))
= sqrt(21.6*9.1*3.6*8.9)
=~ sqrt 6297.8
=~ 79.4 sq cm

Note that  #6 is an isosceles triangle which means you have two 30-60-90 right triangles so figure the area of one and double it.
Two sides are 24cm so the hypotenuse of the two right triangles is 24
Let's look at just one of those right triangles. What you need is the altitude h which is the leg opposite the 60 degree angle so use:
sin 60 = h/24 so h = 24*sin 60 =~20.8. You also need the leg opposite the 30 degree angle, call it x, so use sin 30 = x/24 -> x = 24*sin 30 = 24(1/2) = 12. Now the area of a triangle is (1/2)bh but you need two of these so the area of both right triangles is the area of the whole triangle so you want 2*(1/2)bh = bh.  bh = 12*20.8 = 249.6 sq cm

I hope I have done enough explaining and examples for you to complete the rest. Good Luck.

Math Prof