where s = [1/2](a+b+c) is the semiperimeter of the triangle.
Find the area of a triangle that has two sides of lengths 8 and 12 with an included angle of 45°
Find the area of a triangle with side lengths 10, 14, and 16.
Without using Heron's formula, find the area of a triangle whose side lengths are 7, 9, and 14.
Find the area of a triangle that has two sides of lengths 7 and 13 with an included angle of 32°
A triangle has two sides of length 5 and 6 with an included angle of 60°
. Find the area of the triangle.
Finding the Area of a Triangle
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
Let us recap what we did there, I drew out what I knew on the triangle.0122
I knew two sides and an included angle, I wanted to find an area.0126
I wanted to find base and height of the triangle so I dropped this altitude down from the top corner and then I found out the length for the altitude using SOHCAHTOA and the altitude and hypotenuse that I already knew.0130
Remember SOHCAHTOA does not work in evry triangle, it only works in right triangles.0144
What I did here, by dropping the altitude, the perpendicular, was I made a little right triangle inside the triangle that we are given so it is ok to use SOHCAHTOA there.0149
Once I found the length of that altitude, I just used area is 1/2 base x height.0160
I knew the base, height, and I plugged down it into the formula and simplified it down to 24.1.0167
If you talk about one of the other angles, θ, then you label all the sides as the hypotenuse, the side opposite θ and the side adjacent θ.0036
SOH CAH TOA stands for the sin(θ) is equal to the length of the opposite side over the hypotenuse, the cos(θ) is the length of the adjacent side over the hypotenuse, the tan(θ) is equal to the opposite side over the adjacent side.0049
Remember, we have a little mnemonic to remember that, if you can't remember SOH CAH TOA, you remember Some Old Horse Caught Another Horse Taking Oats Away.0065
Remember, SOH CAH TOA only works in right triangles, you have to have a right angle to make that work.0078
The law of cosines works in any triangle, I'll remind you how that goes.0083
We're assuming here that your sides are labeled a, b, and c, little a, little b, and little c, then you label the angles with capital letters opposite the sides with the same letter, that would make this capital A, capital B, and this, capital C.0090
The law of cosines relates the lengths of the three sides, little a, b and c, to the measure of one of the angles, c2=a2+b2-2abcos(C).0107
This works in any triangle, it does not have to be a right triangle.0121
In fact, if it happens to be a right triangle then the cos(C), if C is a right angle, the cosine of C is just 0, that whole term drops out and you end up the Pythagorean theorem for right triangle, c2=a2+b2.0124
Finally, the important formula that we're going to be using for areas is Heron's formula.0141
That's very useful when you know all the three sides, a, b and c, of a triangle ABC.0148
First, you work out this quantity s, the semi-perimeter, where you add up a, b and c, that's the perimeter, divide by 2, so you get the semi-perimeter, then you drop that into this area formula, and you drop in the lengths of all three sides.0158
Heron's formula gives you a nice expression for the area of a triangle without ever having to look at the angles at all.0174
Now we can use the old formula from geometry for the area of a triangle, just 1/2 base times height.0300
The h stands there for height instead of hypotenuse.0309
I know that's a little confusing to be using h for two different things, but we're kind of stuck with that in English that hypotenuse and height both start with the same letter.0311
One-half the base here is 12, and the height we figured out was 4 square root of 2.0320
We multiply those out, that's 6 times 4 square root of 2, that's 24 square root of 2 for my area there.0329
That one came down to drawing an altitude in the triangle and then using SOH CAH TOA.0341
We didn't really have to use anything fancy like the law of cosines or Heron's formula, although we could have.0347
You'll see some examples later where we use the law of cosines and Heron's formula instead.0352
In this one, we just drew this altitude, we used SOH CAH TOA to find the length of the altitude, then we used the old-fashioned geometry formula, 1/2 base times height, to get the formula of the triangle.0358
In the next example, we're given a triangle with side lengths 10, 14 and 16.0372