Triangle Height Calculator
Find the altitude (height) of any triangle from area and base, three side lengths, or two sides and an included angle.
Pick a calculation method, enter the measurements you know, and get the perpendicular height of your triangle instantly.
Triangle Height Calculator
Find the altitude (height) of any triangle from area and base, three side lengths, or two sides and an included angle.
About the Triangle Height Calculator
The height of a triangle, also called its altitude, is the perpendicular distance from a vertex to the line containing the opposite side. Every triangle has three altitudes — one from each vertex — and their lengths differ unless the triangle is equilateral. The altitude is a fundamental measurement because it directly links a triangle's linear dimensions to its area: Area = ½ × base × height, which means height = (2 × Area) / base.
The three altitudes of any triangle are concurrent: they all pass through a single point called the orthocenter. In an acute triangle the orthocenter lies inside the triangle. In a right triangle it coincides with the vertex of the right angle. In an obtuse triangle it falls outside the triangle, which means two of the three altitudes must be extended beyond the sides to reach the orthocenter. This geometric property is important in advanced geometry and trigonometry.
This calculator offers three methods to find the height depending on what data you have. The Area & Base method is the most direct: if you already know the triangle's area and one side (the base), the height relative to that side is simply h = (2 × Area) / base. This is especially handy after you have computed the area separately.
The Three Sides method works when you know all three side lengths but neither the area nor any height directly. The calculator first applies Heron's formula to find the area, then uses the h = 2A / side formula three times to return the height to each of the three sides simultaneously. This is particularly useful in surveying and construction, where side lengths are measured directly but heights are not.
The SAS method (Two Sides & Angle) uses trigonometry. If sides a and b are known along with the included angle C, then the height relative to side b is h_b = a × sin(C). This derivation comes from expressing the perpendicular component of side a using the sine of the included angle. It is widely used in physics and engineering where force components, vectors, and included angles arise naturally.
Common mistakes when working with triangle heights include using a slant side length instead of the perpendicular height, or applying the wrong formula for the available data. The altitude is always perpendicular to its corresponding base — this right-angle requirement distinguishes it from the median (which connects a vertex to the midpoint of the opposite side) and from any other line from a vertex. The calculator handles all three methods reliably so you can focus on applying the results rather than worrying about the algebra.
Triangle Height Examples
Worked examples for each calculation method with realistic numbers.
| Input | Height | Method & Notes |
|---|---|---|
| Area = 24, Base = 8 | h = 6 | Area & Base: h = (2 × 24) / 8 = 6. The most direct method when area is already known. |
| Sides a = 5, b = 12, c = 13 | h_a = 12, h_b = 5, h_c ≈ 4.62 | Three Sides: Area = 30 (right triangle); h_a = 60/5 = 12, h_b = 60/12 = 5, h_c = 60/13 ≈ 4.62. |
| Side A = 6, Side B = 8, Angle C = 45° | h_b ≈ 4.24 | SAS: h_b = a × sin(C) = 6 × sin(45°) = 6 × 0.7071 ≈ 4.24. |
| Area = 50, Base = 10 | h = 10 | Area & Base: h = (2 × 50) / 10 = 10. Equal base and height for this triangle. |
How to Use the Triangle Height Calculator
- Select the calculation method based on the data you have: Area & Base, Three Sides, or Two Sides & Angle (SAS).
- Enter the required values. For Area & Base, provide the triangle area and the base length. For Three Sides, enter all three side lengths. For SAS, enter two sides and the included angle in degrees.
- Click "Calculate Height" to compute the altitude. For the Three Sides method, all three heights are shown at once.
- Review the formula displayed with the result to confirm you used the right inputs.
- Click "Reset" to clear the fields and start a new calculation with different values or a different method.
Triangle Height FAQ
What is the altitude of a triangle?
The altitude (or height) of a triangle is the perpendicular line segment from a vertex to the line containing the opposite side. Every triangle has three altitudes, each corresponding to a different vertex-side pair. They all meet at a single point called the orthocenter.
Is the height the same as the slant side?
No. The height is strictly perpendicular to the base; it forms a 90° angle with the base line. The slant side is the actual edge of the triangle connecting two vertices. Confusing the two is the most common mistake when computing triangle areas and heights.
Can the altitude fall outside the triangle?
Yes, in an obtuse triangle two of the three altitudes lie outside the triangle. The altitude is drawn from a vertex perpendicular to the extension of the opposite side. Only in acute triangles do all three altitudes fall inside the triangle.
What is the difference between altitude and median?
An altitude is a perpendicular segment from a vertex to the opposite side (or its extension). A median connects a vertex to the midpoint of the opposite side — it bisects the opposite side but is not necessarily perpendicular. They coincide only in equilateral triangles or for the special vertex of an isosceles triangle.
How do I find the height if I only know the three sides?
Use the Three Sides method. The calculator applies Heron's formula to find the area from the side lengths, then divides twice the area by each side to get the corresponding altitude. All three heights are returned simultaneously.
Why does the SAS formula use sine?
In the SAS configuration, if side a is one edge and C is the angle it makes with side b, then the perpendicular component of a relative to b equals a × sin(C). This perpendicular component is exactly the height of the triangle with base b. Sine measures the perpendicular (opposite) component of a vector or line segment.