Two sides of a triangle have length 6 and 8. Which of the following are possible areas of the triangle?
I. 2
II. 12
III. 24
Possible Triangle Areas
Correct
Very Hard
0:02
1:23
First, keep in mind that the formula for the area of a triangle is:
Let’s begin by assuming that we have a right triangle, where the base is 8, and the height is 6.
The area of this triangle would be:
Notice that 24 is one of our answer choices! So we know that this answer choice works.
The remaining two answer choices (2 and 12) are both less than 24. So let’s see if we can make the area smaller. To do this, we can tilt the red side of the triangle a bit to the right:
Here we can see that height has gotten smaller: it is now less than 6. The base of the triangle has not changed (it’s still 8). Therefore, we know that the new area of the triangle is less than 24.
Let’s see what happens as we tilt the red side even further to the right:
As we tilt the red side further to the right, the height of the triangle gets smaller and smaller. In fact, we can keep tilting the red side further and further until the height of the triangle gets very close to zero. The height of the triangle will never be exactly zero, but it can get infinitesimally close.
Let’s think about what would happen if the height actually did get so small that it became zero. The area of the triangle would be:
Since the height never actually decreases all the way to zero, we know the area cannot be exactly zero. But we know that the area can get infinitesimally close to zero. (To see how this works, try calculating the area of the triangle when the height is 0.0001).
Let’s recap what we’ve done so far. We’ve found that the area of the triangle can be 24, and we know that the area can gradually get smaller, until it gets almost all the way to 0.
This means that as we are gradually tilting the red side of the triangle to the right, the area of the triangle will at some point be 12, and at some point it will be 2. In other words, the area of the triangle can be any number between 24 and 0.
Therefore, all three options (2, 12, and 24) are possible, and the correct answer is E.
Now, you might be wondering whether it’s possible for the area of the triangle to be greater than 24. If you are, please read the first question in the FAQ! :)
Frequently Asked Questions
Q: What if there’s an answer choice that is greater than 24? Can the area of the triangle be bigger than 24?
A: If you are given two sides lengths of a triangle, the area of the triangle is maximized when the triangle is a right triangle. Take another look at this setup:
As we discussed above, if the red side is tilted to the right, the height of the triangle gets smaller, and therefore the area of the triangle gets smaller as well. As the red side gets tilted to the right, the angle between the red and blue side decreases.
Well, what if we increase the angle instead of decreasing it? We can do that by tilting the red side to the left.
However, doing that would make the height of the triangle smaller. With the same base as before (8), and a smaller height, the area of the triangle will be smaller. Thus, we can see that the greatest possible area is when the triangle is a right triangle. And the greatest possible area is 24.
Q: How can the area of the triangle be 2? As per the third side rule, the unknown side must be less than the sum of the other two sides and greater than the difference, so the third side must be 2 < x < 14.
A: For any triangle, if you are given two sides, the third side must be greater than the positive difference of those two sides, but less than the sum of those sides. For this triangle, we are given sides of 6 and 8. We know the third side must be less than 6 + 8 = 14 and greater than 2. Even though our third side must be greater than 2, we can still make the area of the triangle infinitely close to zero, as the fourth diagram of the text explanation demonstrates.
Note: the third side doesn't have to be of integer length. So it could be as small as 2.0000000000001 (or smaller) or as large as 13.9999999999999... (or larger) as long as it's greater than 2 exactly and less than 14 exactly.
When you free yourself of the restriction of integer measures, you can make a triangle with an area of two and still follow the geometric rules. :)
Q: If the shortest side of the triangle must be greater than 2, how is it possible to have a height that’s so close to zero?
A: Imagine you have a stretchy string that is placed on top of a line on the ground that measures exactly 8. You pinch the string so that your hand divides the string and on one side is exactly 6 units of length and on the other side is exactly 2 units. It's not a triangle yet, but now pull that stretchy string up an infinitely small amount. You have created a triangle with a base of 8 and a height infinitely close to zero. Note that the height can be much smaller than the shortest side of the triangle! You’ll also notice that the third side is greater than 2—albeit by an infinitely small amount, but greater than 2 nonetheless.
Since we can make an area infinitely close to zero, we know we can make a triangle of any area greater than 0 and less than or equal to the maximum area, which is
If you want to look at real numbers, consider this:
Side 1 = 2.01
Side 2 = 6.00
Side 3 = 8.00
Side 1 and side 2 have a total length of 8.01 (2.01 + 6.00). So the two sides combined are just slightly longer than side 3. If you try to draw a triangle like this, you'll notice that the height has to be very very small because the sides 1 and 2 are only 0.01 longer than side 3. Now imagine a triangle that has sides (2.000001, 6, and 8). The height would be even smaller. Ultimately, you can keep shrinking the 3rd side until the height is almost 0.
Another way to visualize this is by taking another look at the diagram:
Imagine moving the higher part of the red side down towards the blue side, until the red side is almost completely on top of the blue side. There’s just a tiny tiny distance between the right end of the red side, and the blue line. That distance is the height of the triangle. The area of this triangle would be tiny! In fact, the area would be almost zero.
Q: Why did we choose 8 for the base, and not 6?
A: Which side of the triangle we choose for the base is arbitrary: it doesn’t change the area of the triangle. We could follow the same process with 6 as the base, and we would get the same result for both the minimum and maximum areas of the triangle. For instance, as long as we know that a right triangle would give us the maximum area, we find the same answer: and
are both 24, and that's as big as the triangle can get.
Q: Why do we choose a right triangle?
A: Good question! The area of a triangle is maximized when the triangle is a right triangle. If we have an acute or an obtuse triangle, that non-90 degree angle changes the shape of the triangle and we end up losing some height in the process.
Watch the lessons below for more detailed explanations of the concepts tested in this question. And don't worry, you'll be able to return to this answer from the lesson page.