Math
Triangle Height Calculator
The height of a triangle is the perpendicular distance from a vertex to the line of the opposite side, and each triangle has three of them.
Enter three sides to get all three heights through Heron's formula, or enter a base with its area or the two sides that meet at an angle to get the height to that base.
Try an example
Result
Height to the chosen base
—
- Exact height
- —
- Area
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- All three heights
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- Where the altitude lands
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Study path
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What the triangle height calculator solves
Every triangle has three heights (altitudes), one to each side. The height to a side is the perpendicular segment from the opposite The turning point of a parabola, or a corner point of a shape. to that side's line. All three pair with their base to give the same area: Area = ½ · base · height.
| Triangle | Heights h_a, h_b, h_c |
|---|---|
| 3, 4, 5 (right) | 4, 3, 2.4 |
| 13, 14, 15 | 12.92, 12, 11.2 |
| 10, 10, 10 (equilateral) | 8.66, 8.66, 8.66 |
| 5, 5, 8 (isosceles) | 4.8, 4.8, 3 |
How to use it
- Choose the mode: three sides, base and area, or base + side + included angle.
- For three sides, pick which side is the base. The calculator still lists all three heights.
- Enter the lengths (same unit) and any angle in degrees.
- Read the height, the exact A root expression written with the √ symbol, such as a square root. when one exists, and the area used to get it.
How to read the answer
The height is in the same unit as the sides. Multiplying half the base by the height should return the area — that is the built-in check. The largest height always belongs to the shortest side, because base × height is constant.
Common mistakes and edge cases
- Using a slanted side as the height. Unless the triangle is right-angled at the base, the side is longer than the height.
- Forgetting the ½: from base and area, h = 2A / b, not A / b.
- Pairing a height with the wrong base. h_c goes with side c.
- Sides that fail the triangle inequality have no height because they form no triangle.
- In the angle mode, the angle must be the one between the base and the other side, not the angle opposite the base.
Worked examples
Sides 13, 14, 15, height to c
Area 84, h_c = 168/15
Height to the chosen base
11.2
Sides 3, 4, 5, height to a
Height is the other leg
Height to the chosen base
4
Base 10, area 35
h = 70 / 10
Height to the chosen base
7
Base 10, side 8, angle 50°
h = 8 · sin 50°
Height to the chosen base
6.1284
Obtuse: base 4, side 6, angle 120°
Altitude lands outside
Height to the chosen base
5.1962
Impossible sides 1, 2, 3
Height to the chosen base
—
Zero area
Height to the chosen base
—
Frequently asked questions
How do you find the height of a triangle without the area?+
Use the three sides: get the area with Heron's formula, then divide twice the area by the base. Or, with two sides and the angle between them, height = side · sin(angle).
Does a triangle have more than one height?+
Yes, three — one to each side. They are different unless the triangle is equilateral, but each one times its base gives the same area.
Can the height be outside the triangle?+
Yes. In an obtuse triangle the altitudes to the two shorter sides fall outside, on the extended side lines. The length is still valid.
What is the height of an equilateral triangle?+
side · √3 / 2. For side 10 that is 5√3 ≈ 8.66.
How do I find the height of a right triangle?+
The two legs are heights to each other. The height to the hypotenuse is (leg × leg) / hypotenuse.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026