What Is Triangle Height?
The height, or altitude, of a triangle is the perpendicular distance from a vertex to the opposite side or to the extension of that side. Every triangle has three altitudes because any of its three sides can be treated as the base.
Use the calculator above to find height from area and base, from all three sides, from a side and angle, or to find the altitude to the hypotenuse of a right triangle.
Triangle Height Formulas
Area + Base: h = 2A ÷ b
Three Sides: A = √[s(s−a)(s−b)(s−c)], then h = 2A ÷ base
Side + Angle: h = s × sin(θ)
Right Triangle Altitude to Hypotenuse: h = ab ÷ c
Equilateral Triangle: h = (√3 ÷ 2)a
Isosceles Triangle: h = √(s² − (b ÷ 2)²)
How to Find Triangle Height from Three Sides
When the area is unknown but all three side lengths are available, first use Heron’s formula. Calculate the semiperimeter \(s=(a+b+c)/2\), then the area \(A=\sqrt{s(s-a)(s-b)(s-c)}\). Once the area is known, the altitude to any chosen side is \(h=2A/base\).
Three-sides mode calculates all three altitudes automatically: hₐ, hᵦ, and h꜀. It also validates that the side lengths satisfy the triangle inequality before applying Heron’s formula.
Height from a Side and Angle
If a side adjacent to the base and the included angle are known, the perpendicular height can be found with \(h=s\sin(\theta)\). The angle must be greater than 0° and less than 180°.
Special Triangle Heights
| Triangle | Height Formula | Key Point |
|---|---|---|
| Right triangle | h = ab ÷ c | Altitude to the hypotenuse |
| Equilateral | h = (√3/2)a | All three altitudes are equal |
| Isosceles | h = √(s² − (b/2)²) | Altitude to the base bisects the base |
| Obtuse | Depends on known values | Some altitudes meet an extension of a side |
Worked Examples
Example 1: Area and Base
Area = 24 cm² and base = 6 cm.
h = 2A ÷ b
h = 2(24) ÷ 6
h = 48 ÷ 6
h = 8 cm
Example 2: Three Sides
Sides are 5, 12, and 13. Find the height to base 13.
s = (5 + 12 + 13) ÷ 2 = 15
A = √[15(10)(3)(2)] = 30
h = 2(30) ÷ 13
h ≈ 4.615 units
Example 3: Side and Angle
Adjacent side = 10 m and included angle = 30°.
h = s sin(θ)
h = 10 sin(30°)
h = 5 m
Where Does the Altitude Fall?
In an acute triangle, all three altitudes fall inside the triangle. In a right triangle, two altitudes are the legs themselves. In an obtuse triangle, some altitudes meet an extension of the opposite side outside the triangle. The three altitude lines always intersect at a single point called the orthocenter.
Common Triangle Height Mistakes
- Using the wrong base: Height is specific to the side chosen as the base.
- Mixing units: All length measurements should use the same unit.
- Assuming every altitude lies inside: Some altitudes of an obtuse triangle meet a side extension.
- Treating a side as a height: A side is a height only when it is perpendicular to the selected base.
- Skipping triangle validation: Three side lengths must satisfy the triangle inequality before Heron’s formula is used.
How to Verify Your Answer
Recalculate the triangle area using A = ½bh. The area produced by the calculated height should match the original area, allowing only a small difference caused by display rounding.
Need related geometry tools? Use the Area Calculator, Pythagorean Theorem Calculator, Perimeter Calculator, or the Distance Between Points Calculator.
Frequently Asked Questions
Yes. In triangle geometry, height and altitude refer to the perpendicular distance from a vertex to the opposite side or its extension.
Yes. Every triangle has three altitudes, one corresponding to each side used as the base.
Use Heron’s formula to calculate the area, then divide twice the area by the chosen base. Three-sides mode calculates all three altitudes automatically.
You can use all three sides with Heron’s formula, use a side and included angle with h = s sin(θ), or use the right-triangle altitude formula when appropriate.
In an obtuse triangle, the perpendicular from some vertices meets an extension of the opposite side rather than the side segment itself.
The orthocenter is the point where the three altitude lines of a triangle intersect.
How we calculate
Area-and-base mode uses h = 2A/b. Three-sides mode uses Heron’s formula to calculate the triangle area before finding the altitude to each side. Side-and-angle mode uses h = s sin(θ). Right-triangle mode uses h = ab/c for the altitude to the hypotenuse. Results are rounded only for display.
About this calculator: Maintained by Utilixea. Formula logic, triangle validation rules, and worked test cases are checked whenever this calculator is updated.
Last reviewed: July 22, 2026
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