Triangle Height (Altitude) Calculator

About the Triangle Height (Altitude) Calculator

The height (altitude) of a triangle is the perpendicular distance from a vertex to the opposite side (or its extension). Every triangle has three altitudes, one from each vertex, and they always intersect at a single point called the orthocenter. Finding the height is fundamental — it is needed for area calculations (A = ½ × base × height), for structural analysis, and for coordinate geometry.

This calculator supports three input methods. The simplest is the area + base method: if you already know the area and one side, the height to that side is h = 2A / base. The three-sides method uses Heron's formula to first compute the area from s = (a+b+c)/2, then derives all three altitudes. The coordinate method takes three vertex points (x, y) and computes everything from the geometry directly.

Altitudes appear in many real-world contexts: land surveying (finding the width of a river using the altitude of a triangle formed by landmarks), architecture (roof pitch calculations), and physics (decomposing forces perpendicular to a surface). The orthocenter location also depends on the triangle type — inside for acute, at the right-angle vertex for right, and outside for obtuse triangles.

This calculator outputs all three altitudes, the area, perimeter, angles, and orthocenter type. Visual altitude ratio bars, presets, and a reference table make it easy to explore different triangles.

When This Page Helps

This calculator is useful because altitude problems rarely come in one standard form. Sometimes you know area and base, sometimes three sides, and sometimes only coordinates from a diagram or map. By supporting all three approaches and reporting every altitude together with area, angles, and orthocenter behavior, the tool helps students verify derivations and helps designers or surveyors move from raw measurements to triangle geometry quickly.

How to Use the Inputs

1. Choose the input method: Area + Base, Three Sides, or Coordinates.
2. For Area + Base: enter the known area and the base side length.
3. For Three Sides: enter all three side lengths (a, b, c).
4. For Coordinates: enter the (x, y) coordinates of vertices A, B, C.
5. Or click a preset to load a common example.
6. View all three altitudes, area, angles, and orthocenter type.
7. Compare altitude lengths visually in the bar chart.

Example Calculation

Tips & Best Practices

• The altitude is always the shortest perpendicular from a vertex to the opposite side — not the slant distance.
• In a right triangle, two of the altitudes are the legs themselves, and the third is to the hypotenuse.
• The orthocenter lies inside an acute triangle, at the 90° vertex of a right triangle, and outside an obtuse triangle.
• For equilateral triangles, all three altitudes are equal: h = (√3/2) × side.
• In coordinate geometry, the "shoelace formula" gives area directly from vertex coordinates — then altitudes follow from h = 2A/base.

Choosing the Best Input Method

Triangle height problems often hide the same idea behind different givens. If area and base are known, the altitude is immediate. If only the three sides are known, Heron's formula creates the area first and then turns that into all three heights. If a problem is drawn on axes, coordinate input can be the fastest route because the side lengths and area come straight from the points.

Reading the Orthocenter from the Altitudes

Altitudes do more than support the area formula. Their intersection determines the orthocenter, and its location tells you something important about the triangle itself. Acute triangles keep the orthocenter inside, right triangles place it at the right-angle vertex, and obtuse triangles push it outside. Seeing the altitudes and orthocenter type together helps connect computation with geometric structure.

Using Heights in Real Problems

Altitudes appear whenever you need a true perpendicular distance rather than a slanted edge length. That shows up in land measurement, roof framing, force decomposition, and coordinate geometry. By comparing all three heights instead of just one, you also get a quick sense of which side acts as the longest and shortest effective base for the same triangle area.

Frequently Asked Questions

• What is the height (altitude) of a triangle?

  The height is the perpendicular distance from a vertex to the line containing the opposite side. Each triangle has three altitudes.

• How do I find the height if I know the area and base?

  Use h = 2A / base. For an area of 24 and base of 8: h = 2 × 24 / 8 = 6.

• How do I find the height from three sides?

  First compute the area using Heron's formula: s = (a+b+c)/2, A = √[s(s−a)(s−b)(s−c)]. Then h_a = 2A/a for each side.

• What is the orthocenter?

  The orthocenter is the point where all three altitudes of a triangle intersect. Its location depends on the triangle type: inside for acute, on the vertex for right, outside for obtuse.

• Can a triangle have an altitude outside the triangle?

  Yes. In an obtuse triangle, the altitude from the vertex of the obtuse angle falls outside the triangle, landing on the extension of the opposite side.

• How do I find the height from vertex coordinates?

  Compute the area using the shoelace formula: A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. Then h = 2A / base, where the base is the distance between the two other vertices.