Hypotenuse
The hypotenuse is the largest side of a right triangle. It is a side opposite to the right angle in a right triangle. The Pythagoras theorem defines the relationship between the hypotenuse and the other two sides of the right triangle, the base, and the perpendicular side. The square of the hypotenuse is equal to the sum of the square of the base and the perpendicular side of the right triangle.
The Pythagoras theorem has given the Pythagorean triplets and the largest value in Pythagorean triplets is the hypotenuse. Let us learn more about the hypotenuse in this article.
1.  What is a Hypotenuse? 
2.  Formula of Hypotenuse 
3.  Hypotenuse using Pythagoras Theorem 
4.  FAQs on Hypotenuse 
What is a Hypotenuse?
Hypotenuse is the longest side of a rightangled triangle. It is represented by the side opposite to the right angle. The hypotenuse is related to the other sides of the right triangle by the Pythagoras Theorem. The square of the measure of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle. The hypotenuse can be easily recognized in a right triangle as the largest side. Let us look at the below realworld examples of a hypotenuse in right triangleshaped objects.
Hypotenuse Definition: In a rightangled triangle, the longest side or the side opposite to the right angle is termed hypotenuse. The hypotenuse is related to the base and the altitude of the triangle, by the formula: Hypotenuse^{2} = Base^{2} + Altitude^{2}. The base and the altitude are the arms of the right angle, and the hypotenuse is the side opposite to the right angle, in a rightangled triangle.
Formula of Hypotenuse
To derive a formula of the hypotenuse, years ago there was an interesting fact revealed about triangles. Hypotenuse formula: The fact states that with a rightangled triangle or a triangle with a 90º angle, squares can be framed using each of the three sides of the triangle. After putting squares against each side, it was observed that the biggest square has the exact same area as the other two squares. To simplify the whole observation, it was later put in a short equation that can also be called a hypotenuse formula.
So, Hypotenuse formula = a^{2} + b^{2} = c^{2}, where c is the length of the hypotenuse and a and b are the other two sides of the rightangled triangle.
Now, look at the image given below to understand the derivation of the above formula. Here we have a = Perpendicular, b= Base, c = Hypotenuse.
Important Notes
The following important points help for a better understanding of the hypotenuse and its relation with the other two sides of the right triangle.
 The Pythagoras theorem states that in a rightangled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (base and perpendicular).
 This is represented as: Hypotenuse^{2} = Base^{2} + Perpendicular^{2}
 Hypotenuse equation is a^{2} + b^{2} = c^{2}. Here, a and b are opposite and adjacent sides respectively.
 The hypotenuse leg theorem states that two triangles are congruent if the hypotenuse and one leg of one right triangle are congruent to the other right triangle's hypotenuse and leg side.
Hypotenuse using Pythagoras Theorem
The hypotenuse can be related to the other two sides of the rightangled triangle by the Pythagoras theorem. The Pythagoras theorem states that the square of the hypotenuse is equal to the sum of the square of the base of the triangle, and the square of the altitude of the triangle. Among the three sides of the right triangle, the hypotenuse is the largest side, and Hypotenuse^{2} = Base^{2} + Altitude^{2}. The lengths of the hypotenuse, altitude, and base of the triangle, are together defined as a set called the Pythagorean triplets. A few examples of Pythagorean triplets are (5, 4, 3), (10, 8, 6), (25, 24, 7).
Let us consider the following example to find the hypotenuse of this triangle. The longest side of the triangle is the hypotenuse and the other two sides of the right triangle, are the perpendicular side with a measure of 8 inches, and the base with a measure of 6 inches.
The following formula is helpful to calculate the measure of the hypotenuse. (Hypotenuse)^{2} = (Base)^{2} + (Perpendicular)^{2 }= 6^{2} + 8^{2} = 36 + 64 = 100. Hypotenuse = √100= 10 inches. Also, any of the other two sides, the base or the perpendicular side can be easily calculated for the given value of the hypotenuse.
Challenging Questions
Having understood the concepts relating to the hypotenuse, now try out these two challenging questions.
 A 5 m in ladder stands on horizontal ground and reaches 3 m up a vertical wall. How far is the foot of the ladder the wall?
 Town B is 9 km north and 16 km west of town A. How far are the two towns apart?
Given below is the list of topics that are closely connected to the hypotenuse. These topics will also give you a glimpse of how such concepts are covered in Cuemath.
Solved Examples on Hypotenuse

Example 1: Find the value of the longest side of the given bread slice that is in the shape of a rightangle triangle with a given perpendicular height of 12 inches and the base of 5 inches.
Solution:
Given dimensions are perpendicular(P)= 12 inch, base(B)= 5 inch, hypotenuse(H)= ?. Putting the given dimensions in the formula H^{2} = B^{2} + P^{2}, we get H^{2} = 5^{2} + 12^{2}, H = √{25+144}= √169, H= 13 inches. Therefore the longest side of the bread slice is 13 inches.

Example 2: In a right triangle the hypotenuse is 5 units, and the perpendicular is 4 units. Find the measure of the base of the triangle.
Solution:
Given dimensions are perpendicular(P)= 4 units, hypotenuse(H)= 5 units,and base(B)= ? We know that (H)^{2} = (B)^{2} + (P)^{2}; (B)^{2} = (H)^{2}  (P)^{2}. Putting the given dimensions in the formula B^{2}= (5)^{2}  (4)^{2}, B =
√{2516}; B =√9= 3 units. Therefore the base is 3 units.
Practice Questions on Hypotenuse
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
FAQs on Hypotenuse
How do you Find the Hypotenuse of a Triangle?
By using the Pythagorean theorem (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}, we can calculate the hypotenuse. If the values of the other two sides are known, the hypotenuse can be easily calculated with this formula.
How Do you Find the Longest Side of a Triangle?
The hypotenuse is termed as the longest side of a rightangled triangle. To find the longest side we use the Pythagoras theorem, (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}. For example, A bread slice is in the shape of a rightangled triangle. If the base is 4 inches and the perpendicular height is 3 inches. find the longest side.Ans: Given dimensions are Perpendicular= 3 inch, Base = 4 inch. Putting the given dimensions in the formula. (Hypotenuse)^{2} = (Base)^{2} + (Perpendicular)^{2}. (H)^{2}= (4)^{2} + (3)^{2}, H = √ {16+9}, H =√25 =5 inch.
How Do You Find the Pythagoras Theorem?
The Pythagoras theorem is for a right triangle and it is represented by the formula (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}. The largest side of the right triangle is the hypotenuse and the other two sides are the base and the altitude of the triangle.
What are the Pythagorean Triplets?
The Pythagorean triplets are the set of three numbers representing the side lengths of a right triangle, and which follow the Pythagoras rule  (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}. Some of the examples of Pythagorean triplets are (3, 4, 5 ), (6, 8, 10), (12, 5, 13).
What is the Difference Between the Hypotenuse and Other Sides of a Triangle?
The hypotenuse is the largest side of the triangle and is opposite to the other two sides. The other two sides of the triangle are the base and the altitude of the right triangle. These are related to each other with the formula (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}.
How is the Hypotenuse Related to the Right Angle?
The hypotenuse is a side opposite to the right angle. The hypotenuse is the largest side of a right triangle and is drawn opposite to the largest angle, which is the right angle.
Can a Hypotenuse be Drawn for Any Triangle?
The hypotenuse can be drawn for only a right triangle, and not for any other triangle. The side opposite to the 90º angle is the hypotenuse. And hence a right angle is there in a right triangle, and it has a hypotenuse.
How to Calculate Hypotenuse?
The formula to calculate the hypotenuse is (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}. The largest side of the right triangle is the hypotenuse, and it can be calculated if the other two values are known.