Hypotenuse Calculator

Determine the longest side of a right triangle opposite the 90-degree angle using the Pythagorean theorem or trigonometric functions.

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Calculation Examples

Calculation Case Result
Legs 3 cm and 4 cm Hypotenuse 5 cm
Leg 5 inches, angle 60° Hypotenuse ≈ 5.77 inches
Area 6, leg 3 Hypotenuse 5
12 ft and 5 ft legs Hypotenuse 13 ft

How to Find the Hypotenuse of a Triangle?

To find the length of the hypotenuse, you must first identify the known dimensions of your right triangle. This tool supports three primary input methods based on Euclidean geometry: two legs ($a$ and $b$), one leg and an acute angle ($\alpha$ or $\beta$), or the area and one side.

Enter your values into the corresponding fields and select the correct units (cm, m, inches, or feet). The calculator applies the Pythagorean theorem or SOH CAH TOA ratios to solve for the missing side. For example, entering legs of 3 and 4 will yield a hypotenuse of 5, a classic Pythagorean triple. The system also calculates the remaining angles and the second leg if only one is provided. This is essential for construction framing, verifying roof pitches, or completing geometry exercises where precision is mandatory.

Geometric Formulas and Principles

The calculation logic depends on the specific parameters provided:

1. Pythagorean Theorem: If both legs ($a$ and $b$) are known, the formula is $c = \sqrt{a^2 + b^2}$. This is the standard method for finding the longest side when the perpendicular sides are measured.

2. Trigonometry (SOH CAH TOA): If you know one leg and an acute angle, the tool uses Sine or Cosine. For an angle $\alpha$ and its opposite side, the formula is $c = a / \sin(\alpha)$. For the adjacent side, it uses $c = b / \cos(\alpha)$.

3. Area Method: When the area and one leg are known, the second leg is derived via $b = (2 \times Area) / a$, followed by the Pythagorean theorem to find the hypotenuse ($c$).

Note: These formulas only apply to triangles containing a 90-degree angle. For non-right triangles, the Law of Cosines would be required.

Pythagorean Theorem explained

Useful Tips 💡

  • The hypotenuse is always the longest side; if your result is shorter than either leg, verify your inputs.
  • Use consistent units: do not mix inches and centimeters in the same calculation.
  • For 30-60-90 triangles, the hypotenuse is always twice the length of the shorter leg.

📋Steps to Calculate

  1. Select the available data: two legs, leg + angle, or area + leg.

  2. Input the numerical values and verify the measurement units.

  3. Review the calculated hypotenuse, missing leg, and complementary angles.

Mistakes to Avoid ⚠️

  1. Applying right-triangle formulas to acute or obtuse triangles.
  2. Forgetting to take the square root of the sum when calculating manually.
  3. Confusing the opposite side with the adjacent side when using trigonometric angles.
  4. Entering the hypotenuse value into a leg field, resulting in an impossible geometry.

Practical Applications in Construction and Design📊

  1. Calculating ladder length: Determining the minimum length for an 8 ft ladder to reach a specific height safely.

  2. Architecture: Verifying the squareness of foundations using the 3-4-5 rule.

  3. Display Technology: Measuring the diagonal screen size of monitors and TVs based on height and width.

  4. Carpentry: Solving for rafter lengths and stair stringers in home building.

Questions and Answers

What is the hypotenuse in a right triangle?

The hypotenuse is the longest side of a right-angled triangle and is always located directly opposite the 90-degree (right) angle. In geometric terms, it connects the two shorter sides, known as the legs or catheti. Because it spans the widest angle in the triangle, its length is mathematically guaranteed to be greater than either individual leg.

How do you find the hypotenuse without the Pythagorean theorem?

If the lengths of the legs are unknown, you can find the hypotenuse using trigonometry provided you have one acute angle and the length of one side. Depending on which side you know, you use the Sine or Cosine ratios (SOH CAH TOA): divide the opposite side by the sine of the angle ($c = a / \sin\alpha$) or divide the adjacent side by the cosine of the angle ($c = b / \cos\alpha$).

What is the formula for the hypotenuse?

The most common formula is derived from the Pythagorean theorem: $c = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the legs. In professional drafting and engineering, trigonometric formulas are also used: $c = a / \sin(\alpha)$ for the opposite leg or $c = b / \cos(\alpha)$ for the adjacent leg. Our calculator automatically selects the correct formula based on your inputs.

Can a hypotenuse be shorter than a leg?

No, a hypotenuse cannot be shorter than a leg in Euclidean geometry. According to the Pythagorean theorem ($a^2 + b^2 = c^2$), the square of the hypotenuse is the sum of the squares of the other two sides. Therefore, the hypotenuse must always be the longest side. If your calculation shows otherwise, it usually indicates that the triangle is not a right triangle or the units were mixed.

What is a 3-4-5 triangle and why is it useful?

A 3-4-5 triangle is a specific right triangle where the sides are in a perfect ratio of 3:4:5. This is the most famous Pythagorean triple ($3^2 + 4^2 = 5^2$). Builders and carpenters frequently use the "3-4-5 rule" to ensure corners are perfectly square (90 degrees) by measuring 3 and 4 units on the legs and verifying the hypotenuse is exactly 5 units.

How do I calculate the hypotenuse for a 30-60-90 triangle?

In a 30-60-90 special right triangle, the sides follow a consistent ratio: $1 : \sqrt{3} : 2$. This means the hypotenuse is always exactly twice the length of the shorter leg (the side opposite the 30° angle). If the shortest leg is 5 cm, the hypotenuse is 10 cm, and the longer leg is $5\sqrt{3}$ cm.

How is the hypotenuse used to measure a TV or monitor size?

The advertised size of a TV or computer monitor (e.g., 55 inches) refers to its diagonal length, which is the hypotenuse of the rectangular screen. By treating the width and height of the screen as the legs of a right triangle, you can use the Pythagorean theorem to find the diagonal. For a standard 16:9 widescreen, the diagonal is approximately 1.15 times the width.

How do you find the hypotenuse of an isosceles right triangle?

In an isosceles right triangle (also known as a 45-45-90 triangle), both legs are of equal length. The formula to find the hypotenuse is simplified to $c = a\sqrt{2}$, where $a$ is the length of one leg. For example, if both legs are 10 cm, the hypotenuse is $10\sqrt{2} \approx 14.14$ cm. This is commonly used to find the diagonal of a square.

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Disclaimer: This calculator is designed to provide helpful estimates for informational purposes. While we strive for accuracy, financial (or medical) results can vary based on local laws and individual circumstances. We recommend consulting with a professional advisor for critical decisions.