What is an acceptable length for the third side of a triangle with sides 3 and 4

Let x be the third side

The sum of 5 and 6 is 11.

11 must be greater than the third side so we can say it's a triangle.

So the answer should be

0<x<=11

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Updated April 24, 2017

By Sreela Datta

Euclidean geometry, the basic geometry taught in school, requires certain relationships between the lengths of the sides of a triangle. One cannot simply take three random line segments and form a triangle. The line segments have to satisfy the triangle inequality theorems. Other theorems that define relationships between the sides of a triangle are the Pythagorean theorem and the law of cosines.

According to the first triangle inequality theorem, the lengths of any two sides of a triangle must add up to more than the length of the third side. This means that you cannot draw a triangle that has side lengths 2, 7 and 12, for instance, since 2 + 7 is less than 12. To get an intuitive feel for this, imagine first drawing a line segment 12 cm long. Now think of two other line segments 2 cm and 7 cm long attached to the two ends of the 12 cm segment. It is clear that it would not be possible to make the two end segments meet. They would have to add up at least to 12 cm.

The longest side in a triangle is across from the largest angle. This is another triangle inequality theorem and it makes intuitive sense. You can draw various conclusions from it. For example, in an obtuse triangle, the longest side has to be the one across from the obtuse angle. The converse of this is true as well. The largest angle in a triangle is the one that is across from the longest side.

The Pythagorean theorem states that, in a right triangle, the square of the length of the hypotenuse (the side across from the right angle) is equal to the sum of the squares of the other two sides. So if the length of the hypotenuse is c and the lengths of the other two sides are a and b, then c^2 = a^2 + b^2. This is an ancient theorem that has been known for thousands of years and has been used by builders and mathematicians through the ages.

The law of cosines is a generalized version of the Pythagorean theorem that applies to all triangles, not just the ones with right angles. According to this law, if a triangle had sides of length a, b and c, and the angle across from the side of length c is C, then c^2 = a^2 + b^2 - 2abcosC. You can see that when C is 90 degrees, cosC = 0 and the law of cosines is reduced to the Pythagorean theorem.

There are several different ways you can compute the length of the third side of a triangle. Depending on whether you need to know how to find the third side of a triangle on an isosceles triangle or a right triangle, or if you have two sides or two known angles, this article will review the formulas that you need to know.

Pythagorean Theorem for the Third Side of a Right Angle Triangle

The Pythagorean Theorem is used for finding the length of the hypotenuse of a right triangle.

Pythagoras was a Greek mathematician who discovered that on a triangle abc, with side c being the hypotenuse of a right triangle (the opposite side to the right angle), that:

So, as long as you are given two lengths, you can use algebra and square roots to find the length of the missing side.

Formula for the Base of an Isosceles Triangle

If you know the side length and height of an isosceles triangle, you can find the base of the triangle using this formula:

where a is the length of one of the two known, equivalent sides of the isosceles.

Find the Third Side of Any Triangle

Now that we've reviewed the two basic cases, let’s look at how to find the third unknown side for any triangle.

There are two additional concepts that you must be familiar with in trigonometry: the law of cosines and the law of sines. Both of them allow you to find the third length of a triangle.

The law of sines is the simpler one. It states that the ratio between the length of a side and its opposite angle is the same for all sides of a triangle:

Here, A, B, and C are angles, and the lengths of the sides are a, b, and c.

Because we know angle A and side a, we can use that to find side c.

The length of side c is 4.38.

The law of cosines is slightly longer and looks similar to the Pythagorean Theorem. It states that:

Here, angle C is the third angle opposite to the third side you are trying to find.

Because we know the lengths of side a and side b, as well as angle C, we can determine the missing third side:

Different Ways to Find the Third Side of a Triangle

There are a few answers to how to find the length of the third side of a triangle. To choose a formula, first assess the triangle type and any known sides or angles.

For a right triangle, use the Pythagorean Theorem. For an isosceles triangle, use the area formula for an isosceles. If you know some of the angles and other side lengths, use the law of cosines or the law of sines.

You’ll be on your way to knowing the third side in no time.

More Math Homework Help

Go back to Calculators page

To use the right angle calculator simply enter the lengths of any two sides of a right triangle into the top boxes. The calculator will then determine the length of the remaining side, the area and perimeter of the triangle, and all the angles of the triangle.

How to Find the Area and Sides of a Right Triangle

Do it yourself

If we know just two sides of a right triangle, we can use that information to find the third side, the area and perimeter of the triangle, and all the angles of the triangle. Amazing, right? Let’s review how we would find each of those parts.

How to find the Missing Side of a Right Triangle

To find the missing side of a right triangle we use the famous Pythagorean Theorem.

We need to be a little careful that we know which side we’re finding. Right triangles have two legs and a hypotenuse, which is the longest side and is always across from the right angle. When we’re trying to find the hypotenuse we substitute our two known sides for a and b. It doesn’t matter which leg is a and which is b. Then we solve for c by adding the squared values of a and b and taking the square root of both sides. 

When we’re trying to find one of the legs we enter the known leg for a and the known hypotenuse for c. Then we solve for b using simple algebra (subtract the value of a squared from both sides, then take the square root of both sides). 

How to find the Area of a Right Triangle

To find the area of a right triangle we only need to know the length of the two legs. We don’t need the hypotenuse at all. That’s because the legs determine the base and the height of the triangle in every right triangle. So we use the general triangle area formula (A = base • height/2) and substitute a and b for base and height. So our new formula for right triangle area is A = ab/2. 

How to find the Perimeter of a Right Triangle

To find the perimeter, or distance around, our triangle we simply need to add all three sides together. If we only know two of the sides we need to use the Pythagorean Theorem first to find the third side. 

How to find the Angles of a Right Triangle

To find the angles of a right triangle we use trigonometry. It’s not as difficult as it sounds. We just need to find one special button on our handheld calculators. To start we’ll need to know all the side lengths, so if we don’t know them already we’ll use the Pythagorean Theorem to find them first. 

Once we have all the sides we determine which angle we’re going to find. Then we take the side opposite that angle and divide it by the length of the hypotenuse, which is side c. That will give us a value between 0 and 1. Now we just need to find the ARCSIN button on our calculator, which is often labeled as SIN-1.  Finding the ARCSIN of our decimal value gives us our angle. Be sure that the calculator is set for angle mode rather than radian mode. 

We can repeat this process to find the other unknown angle in the triangle by once again dividing its opposite side by the hypotenuse and then taking the ARCSIN. 

Or we could show off even more triangle knowledge by using subtraction to find it since we know the interior angles of a triangle have to add up to 180°. Subtracting the angle we just found from 180° and then subtracting our known right angle (90°) will give us the third angle too. 

This calculator is great for getting all this information from just two sides of a right triangle, but it’s a fun challenge to try to find the sides, angles, area and perimeter on our own without it. Then you can use it to check our answers.