Right Triangle Altitude From Sides Calculator

How This Right Triangle Altitude From Sides Calculator Works

Enter leg a, leg b, and hypotenuse c to find the altitude drawn from the right-angle vertex to the hypotenuse. The calculator shows the formula, substitution, and result so you can follow every step.

This calculator is designed for the case where you know leg a, leg b, and hypotenuse c and need the perpendicular distance from the right-angle vertex to the hypotenuse. No angles or projections required.

Right Triangle Altitude From Sides Formula

The altitude to the hypotenuse connects the right-angle vertex to the hypotenuse at a perfect 90° angle. Its length comes from a simple relationship between the triangle’s area and its sides.

The area of a right triangle can be written two ways. First, using the legs: Area = (a × b) / 2. Second, using the hypotenuse as the base and the altitude as the height: Area = (c × h) / 2. Setting these equal and solving for h gives the formula.

Because both expressions describe the same area, they must be equal: (a × b) / 2 = (c × h) / 2. Cancel the 2 from both sides, then divide by c to isolate h. The result is h = (a × b) / c.

How to Use This Calculator

1. Identify the two legs of your right triangle. These are the sides that form the 90° angle. Label them a and b.
2. Identify the hypotenuse c. This is the longest side, opposite the right angle.
3. Make sure all three measurements are in the same unit. Convert if needed.
4. Enter leg a into the first field.
5. Enter leg b into the second field.
6. Enter hypotenuse c into the third field.
7. Click Calculate. The tool multiplies a and b, divides by c, and shows the altitude h with step-by-step work.

Worked Example: Altitude of a 6-8-10 Right Triangle

Suppose you have a right triangle with leg a = 6, leg b = 8, and hypotenuse c = 10.

The altitude to the hypotenuse is 4.8 units. You can verify this: Area = (6 × 8) / 2 = 24, and also Area = (10 × 4.8) / 2 = 24. Both area calculations agree, confirming the altitude is correct.

What the Result Means

The altitude h is the shortest distance from the right-angle vertex to the hypotenuse. Think of it as the “height” of the triangle when the hypotenuse is laid flat as the base.

This value is always positive, always shorter than either leg, and always shorter than the hypotenuse. If your answer is larger than one of the legs, double-check your inputs—something is likely entered incorrectly.

When to Use This Calculator

This method is ideal whenever you already have all three side lengths and need the altitude. It avoids trigonometry and square roots entirely.

Common situations:

• A geometry problem gives all three sides and asks for the altitude to the hypotenuse.
• You need the altitude to compute the area using Area = (c × h) / 2 as a cross-check.
• A construction plan shows the span and rise of a triangular brace, and you need the perpendicular height to the longest member.
• You are proving the geometric mean relationships and need the altitude value first.
• You want to verify that a measured altitude matches the calculated value for quality control.

Why This Formula Works

The formula h = (a × b) / c is a direct consequence of the equal-area principle. Every triangle has exactly one area, but you can compute it using different base-height pairs.

Using the legs as base and height gives Area = (a × b) / 2. Using the hypotenuse as the base gives Area = (c × h) / 2. Since both equal the same area, you can set them equal and solve for h. The factor of 2 cancels from both sides, leaving h = (a × b) / c.

This relationship holds for every right triangle, regardless of size or proportions. It is one of the most useful identities in elementary geometry.

Common Mistakes

The formula is simple, but entering the wrong values is surprisingly common. Watch out for these.

• Using the hypotenuse c as a leg: If you accidentally put c in the numerator and a leg in the denominator, the result will be too large and incorrect.
• Entering a hypotenuse shorter than a leg: The hypotenuse must be the longest side. If c < a or c < b, your side labels are wrong or the triangle is not a right triangle.
• Forgetting to divide by c: The product a × b alone is not the altitude. You must divide by the hypotenuse.
• Mixing different units: If leg a is in centimeters and leg b is in meters, the result is meaningless. Convert to the same unit first.
• Confusing altitude h with leg a or leg b: The altitude is a separate segment inside the triangle, not one of the original sides. It is always shorter than either leg.

Additional Example: The 5-12-13 Triangle

The 5-12-13 right triangle is another common Pythagorean triple. Let’s find its altitude to the hypotenuse.

• h = (a × b) / c
• h = (5 × 12) / 13
• h = 60 / 13
• h ≈ 4.615 units
• Cross-check: Area = (5 × 12) / 2 = 30. Also Area = (13 × 4.615) / 2 ≈ 30 ✓