How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one  angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the  angle and they are labeled  and . The side of the triangle that is opposite of the  angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides. 

Simplify.

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract  from both sides of the equation.

Take the square root of both sides.

Simplify.

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract  from both sides of the equation.

Take the square root of both sides.

Simplify.

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

Recall how to find the perimeter of a triangle:

The given triangle has  of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

Since we are finding the length of the hypotenuse, , rewrite the equation.

Plug in the values of  and .

Now, plug in all three values into the equation to find the perimeter. Use a calculator and round to  decimal places.

How do you find the perimeter of a right triangle?

There are three primary methods used to find the perimeter of a right triangle.

1. When side lengths are given, add them together.
2. Solve for a missing side using the Pythagorean theorem.
3. If we know side-angle-side information, solve for the missing side using the Law of Cosines.

Method 1:

This method will show you how to calculate the perimeter of a triangle when all sides lengths are known. Consider the following figure:

If we know the lengths of sides , , and , then we can simply add them together to find the perimeter of the triangle. It is important to note several things. First, we need to make sure that all the units given match one another. Second, when all the side lengths are known, then the perimeter formula may be used on all types of triangles (e.g. right, acute, obtuse, equilateral, isosceles, and scalene). The perimeter formula is written formally in the following format:

Method 2:

In right triangles, we can calculate the perimeter of a triangle when we are provided only two sides. We can do this by using the Pythagorean theorem. Let's first discuss right triangles in a general sense. A right triangle is a triangle that has one  angle. It is a special triangle and needs to be labeled accordingly. The legs of the triangle form the  angle and they are labeled  and . The side of the triangle that is opposite of the  angle and connects the two legs is known as the hypotenuse. The hypotenuse is the longest side of the triangle and is labeled as .

If a triangle appears in this format, then we can use the Pythagorean theorem to solve for any missing side. This formula is written in the following manner:

We can rearrange it in a number of ways to solve for each of the sides of the triangle. Let's rearrange it to solve for the hypotenuse, .

Rearrange and take the square root of both sides. 

Simplify.

Now, let's use the Pythagorean theorem to solve for one of the legs, .

Subtract  from both sides of the equation.

Take the square root of both sides.

Simplify.

Last, let's use the Pythagorean theorem to solve for the adjacent leg, .

Subtract  from both sides of the equation.

Take the square root of both sides.

Simplify.

It is important to note that we can only use the following formulas to solve for the missing side of a right triangle when two other sides are known:

After we find the missing side, we can use the perimeter formula to calculate the triangle's perimeter.  

Method 3:

This method is the most complicated method and can only be used when we know two side lengths of a triangle as well as the measure of the angle that is between them. When we know side-angle-side (SAS) information, we can use the Law of Cosines to find the missing side. In order for this formula to accurately calculate the missing side we need to label the triangle in the following manner:

When the triangle is labeled in this way each side directly corresponds to the angle directly opposite of it. If we label our triangle carefully, then we can use the following formulas to find missing sides in any triangle given SAS information:

After, we calculate the right side of the equation, we need to take the square root of both sides in order to obtain the final side length of the missing side. Last, we need to use the perimeter formula to obtain the distance of the side lengths of the polygon.

Solution:

Now, that we have discussed the three methods used to calculate the perimeter of a triangle, we can use this information to solve the problem.

Recall how to find the perimeter of a triangle:

The given triangle has  of the three sides needed. Use the Pythagorean theorem to find the length of the third side.

Recall the Pythagorean theorem:

Since we are finding the length of the hypotenuse, , rewrite the equation.

Plug in the values of  and .

Now, plug in all three values into the equation to find the perimeter. Use a calculator and round to  decimal places.

The area of a triangle is given by the equation:

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

The area of a triangle is given by the equation:

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

The area of a triangle is given by the equation:

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

The area of a triangle is given by the equation:

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

The area of a triangle is given by the equation:

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

The area of a triangle is given by the equation:

Since the base leg of the given triangle is 4 cm, while the height is 3 cm, this gives:

The formula for the area of a triangle is

where  is the base of the triangle and  is the height.

For the triangle shown, side  is the base and side  is the height.

Therefore, the area is equal to

or, based on the units given, 42 square centimeters

An equilateral triangle has three congruent sides. The area of a triangle is given by where is the base and is the height.

The equilateral triangle can be broken into two right triangles, where the legs are and and the hypotenuses is .

Using the Pythagorean Theorem we get or and the area is