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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 squares 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.  Hypotenuse Equation 
3.  How to Find Hypotenuse? 
4.  Hypotenuse Theorem 
5.  FAQs on Hypotenuse 
What is a Hypotenuse?
The hypotenuse is the longest side of a rightangled triangle. It is represented by the side opposite to the right angle. It 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.
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}. Let us look at the below realworld examples of a hypotenuse in right triangleshaped objects.
Hypotenuse Equation
To derive an equation or a formula of the hypotenuse, years ago there was an interesting fact revealed about triangles. Hypotenuse equation: 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 equation.
So, the hypotenuse equation = 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.
Tips and Tricks on Hypotenuse:
The following points will help you to get a better understanding of the hypotenuse and its relation to 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 the legs of the right triangle and c is the hypotenuse.
 The hypotenuse leg theorem states that two triangles are congruent if the hypotenuse and one leg of one right triangle are congruent/equal to the other right triangle's hypotenuse and leg side.
How to Find Hypotenuse?
To find the length of the hypotenuse of a triangle, we will be using the above equation. For that, we should know the values of the base and perpendicular of the triangle. For example, in a right triangle, if the length of the base is 3 units, and the length of the perpendicular side is 4 units, then the length of the hypotenuse can be found by using the formula Hypotenuse^{2} = Base^{2} + Perpendicular^{2}. By substituting the values of the base and perpendicular, we get, Hypotenuse^{2} = 3^{2} + 4^{2} = 9 + 16 = 25. This implies that the length of the hypotenuse is 5 units. This is how we can easily find the length of the hypotenuse by using the hypotenuse equation.
Follow the steps given below to find the hypotenuse length in a rightangled triangle:
 Step 1: Identify the values of base and perpendicular sides.
 Step 2: Substitute the values of base and perpendicular in the formula: Hypotenuse^{2} = Base^{2} + Perpendicular^{2}.
 Step 3: Solve the equation and get the answer.
Let us consider one more example to find the hypotenuse of a 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. This implies, 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 using the same equation.
Hypotenuse 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 squares 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}. This is known as the hypotenuse theorem. 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 triples are (5, 4, 3), (10, 8, 6), and (25, 24, 7).
Challenging Questions:
Having understood the concepts related to the hypotenuse of a triangle, now try out these two challenging questions.
 A 5 meters ladder stands on horizontal ground and reaches 3 m up a vertical wall. How far is the foot of the ladder from the wall?
 Town B is 9 km north and 16 km west of town A. What is the shortest distance to go from town A to town B?
► Related Topics:
Check these articles related to the concept of the hypotenuse of a triangle.
Hypotenuse Examples

Example 1: Find the value of the longest side of a 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 inches, and base (B) = 5 inches. 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 inches
H = 13 inches.
Therefore the length of the hypotenuse (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, and hypotenuse (H) = 5 units. We know that (H)^{2} = (B)^{2} + (P)^{2} ⇒ (B)^{2} = (H)^{2}  (P)^{2}.
Putting the given dimensions in the formula, we get,
B^{2} = (5)^{2}  (4)^{2}
B = √{2516}
B = √9 = 3 units
Therefore, the length of the base is 3 units. 
Example 3: How to find the missing hypotenuse of a triangle with base = 7 units and perpendicular = 24 units?
Solution:
Given dimensions are base (B) = 7 units and perpendicular (P) = 24 units. To find the hypotenuse (H), we will use the equation: (H)^{2} = (B)^{2} + (P)^{2}.
Putting the given dimensions in the equation, we get,
H^{2} = (7)^{2} + (24)^{2}
B = √{49+576}
B = √625 = 25 units
Therefore, the length of the hypotenuse is 25 units.
FAQs on Hypotenuse
What is the Meaning of Hypotenuse?
In mathematics, the hypotenuse of a triangle is defined as the longest side of a right triangle. It is the side opposite to the 90degree angle. It is equal to the square root of the sum of the squares of the other two sides.
What is the Length of the Hypotenuse?
The length of the hypotenuse is greater than the lengths of the other two sides of a right triangle. The square of the hypotenuse length is equal to the sum of squares of the other two sides of the triangle. Mathematically, it can be expressed in the form of an equation as Hypotenuse^{2} = Base^{2} + Perpendicular^{2}.
What is the Hypotenuse Leg Theorem?
The hypotenuse leg theorem states that two right triangles are congruent if the lengths of the hypotenuse and any one of the legs of a triangle are equal to the hypotenuse and the leg of the other triangle.
How to Find the Missing Hypotenuse?
The missing hypotenuse can be easily known if we know the lengths of the other two sides by using the hypotenuse equation: Hypotenuse^{2} = Base^{2} + Perpendicular^{2}. For example, if the base and perpendicular of a right triangle measure 6 units and 8 units respectively, then the hypotenuse is equal to:
Hypotenuse^{2} = 6^{2} + 8^{2}
= 36 + 64
= 100
Therefore, hypotenuse = 10 units.
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 hypotenuse theorem, (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}. For example, a bread slice is given in the shape of a rightangled triangle. If the base is 4 inches and the height is 3 inches, then the hypotenuse is (H)^{2} = (4)^{2} + (3)^{2} = √{16+9} = √25 = 5 inches.
How to Find Hypotenuse with Angle and Side?
If an angle and a side are known, then we can calculate hypotenuse by applying the formula of trigonometric ratios. If A is the angle known, then we have,
 sin A = Perpendicular/Hypotenuse
 cos A = Base/Hypotenuse
So, if the length of the base is given, then the cos formula can be used and if height is known then the sin formula can be used to find the hypotenuse length.
What is the Difference between the Hypotenuse and Other Sides of a Triangle?
The hypotenuse is the largest side of the triangle. The other two sides 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 the 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 only for a right triangle, and not for any other triangle. The side opposite to the 90° angle is the hypotenuse. And since a right angle is there in a right triangle, 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 sides are known.
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