Area of a Triangle Given 3 Sides Herons Formula

In this video, we're going to focus on the many different ways of finding the area of triangle. So, in our first example, let's say that we have right triangle. Let's say the base is 10 and the height is 12. Go ahead and find the area. So the first thing you need to know is the formulas to use for any right triangle or any triangle that you know the base and the height of the area is simply 12 base time height. So for this example it's 12 * the base of 10 * height of 12. Half of 10 is 5 and 5 * 12 is 60. So that's the area of the right triangle on the left. It's 60 square units. Now, what about finding the area of triangle that looks like this? Let's say if you're given base of eight and height of six, what is the area of the shader region? Now you can use the same formula to find the area of that triangle. It's 12 base time height. So in this example, the base is 8 and the height is 6. Half of 8 is 4. 4 * 6 is 24. So that's the answer for this example. Now let's say if we have an isosles triangle that's where two sides are congruent to each other and let's say the base is 8 units long. Find the area of this triangle using the same formula. = 12 base time height. Now I'm going to redraw the triangle because it's an isosesles triangle. We can split it right in the middle. So this side is four and the other side is four as well. And this is five. Our goal is to find the height of the triangle. If we could find the height, then we can easily find the area. So, I'm going to focus on the right side of this triangle. If we just focus on the right side, notice that it forms right triangle with the hypotenuse being five and the base is four. So, we got to find the missing side. Whenever you have right triangle, you can use the pagorean theorem to find the missing side. Let's call this and So in this example, we know that is 4, is 5. So we can use this equation. A^2 + b^2 is equal to c^2. Our goal is to find the value of 42 is 16. 5^2 is 25. 25 - 16 is 9. And to find the value of we need to take the square root of both sides. The square root of 9 is three. So therefore, we have the height of the triangle. is three. So now let's draw the original picture. So we have height of three and base of eight. So really all you need in triangle that looks like this is the base and the height. Once you have it, you can find the area of the whole triangle. So it's 12 base time height. The base is 8, the height is three. Half of 8 is 4. 4 * 3 is 12. So it's 12 square units. Now what about this example? Let's say if we have an equilateral triangle, all sides are the same. Go ahead and find the area of this triangle. Now, if you decide to use the same method, we know this side is going to be 10. And we're looking for the height. So, this has to be five. The left side is five and the right side is five. Has to add up to 10. So, we could find the missing side. So, using this equation, a^2 + b^2 is equal to c^2. is 5. We're looking for is 10. 5^2 is 25. 10^ squar is 100. 100 - 25 is 75. So is the square root of 75. How can we simplify the square root of 75? 75 is basically 3 * 25 and the square root of 25 is 5. So this reduces to 5 3. So now we have the height of the triangle is 5 3. So now we can use this equation. is equal to 12 base time height. So we have base of 10 height of 5 3. Half of 10 is five and 5 * 5 is 25. So the area is 25 3. Now that's one way to get the answer. There is another way. Whenever you have an equilateral triangle where all sides are the same, you can use this formula. The area is the 3 / 4 * s^ 2. So in our example, is 10. So if you know this formula, you could just go ahead and use it to get the area. Just plug in the length of the side. So it's 3 over 4 * 10^ 2. 10 * 10 is 100 and 100 / 4 is 25. So this will give us the same answer 25 3. Now let's say if you have triangle where you have the value of two sides and the included angle, what can you do to find the area of this triangle? Sometimes you might see it as side angle side. You have this side and then you have the angle and then you have the side. Before we can use the formula, we need to define few things. Let's call this angle angle angle Across angle you have side Across angle is side Across angle is side So you need to use basically an equation that deal with signs. So the area is 12 * the side * side time of angle So that's the formula that you want to use. In our example, is 10 and is 15. So it's going to be 12 15 * 10 time of the angle on the inside, which is 30°. Now 15 * 10 is 150. So we have half * 150 and sin 30 if you type it in your calculator make sure you put it in degree mode sin 30 is 12. Now 12 of 150 is 75. So now we have 75 * half which is going to be 75 over 2 or you can write it as 37.5 as decimal. So that's the answer for this example. Now let's say if we have right triangle where this is the base, this is the height and the included angle is 90°. Let's start with the same formula that we used in the last example. 12 * sin of So in this example, we could say that is the height. Let's call the height. So I'm going to replace with and let's call it the base. And then we have of the angle, which the angle is 90°. of 90 is 1. So therefore you get the area of right triangle which is 12 base time height. So this equation comes from this equation. Sin 90 is simply one. But if the angle is anything different then you want to use this equation. Now let's try one more example. So this time we're going to have triangle with three different sides. So it's SSS side. What can we do to find the area of this triangle? Now there's something called Heron's formula. And to apply it, you need to find which is basically 12 of the perimeter of the triangle. It's + + / 2. So we're going to add 9, 10, and 11, and then divide the whole thing by two. 9 + 10 is 19. 19 + 11 is 30. And 30 / 2 is 15. So now we have the value of Once you find the value of now you can use that to find the area of the triangle. So here's the formula. The area is going to be the square of * - * - * - where are the three sides of the triangle. So this is going to be 10 * - 10 - 9 * - actually is 15 not 10. don't know why put 10. So, let me just make that correction. So, - is going to be 15 - 9. - is 15 - 10. And - is 15 - 11. 15 - 9 is 6. 15 - 10 is 5. 15 - 11 is 4. Now, we need to simplify this answer. So, if you don't have calculator, here's what you could do. 15, you could break it into 5 * 3. 6 is 2 * 3. And four, well, that's just perfect square, so we can leave that alone. So we have the square of 4. 5 * 5 is 25. So I'm going to write that as square of 25. 3 * 3 is 9. And then we have two left over on the inside. The square of 4 is 2. The square of 25 is 5. The of 9 is 3. 2 * 5 is 10. 10 * 3 is 30. So the area the exact answer is 30 2. So that's how you can use Hon's formula to find the area of triangle when you have all three sides. If you have triangle where the three sides are the same, it's an equilateral triangle. You could simply use this formula as we've discussed early in the video. But if the sides are different, then use Hon's formula.