Triangle Altitudes Calculator

About the Triangle Altitudes Calculator

An altitude of a triangle is a perpendicular line segment from a vertex to the line containing the opposite side (the base). Every triangle has three altitudes, and they meet at a single point called the orthocenter. The altitude is one of the most fundamental measurements in triangle geometry, directly connected to the area: Area = ½ · base · height.

The altitude to side a is h_a = 2 · Area / a. Since the area is the same regardless of which base you choose, the three altitudes are inversely proportional to their corresponding sides — the shortest side has the longest altitude and vice versa.

The orthocenter's position depends on the triangle type. For acute triangles, the orthocenter lies inside. For right triangles, it falls exactly on the vertex of the right angle. For obtuse triangles, it lies outside the triangle, on the extension of the altitude from the obtuse angle.

This calculator takes three side lengths and computes all three altitudes, the orthocenter coordinates, the triangle type (acute, right, or obtuse), the area, and visual comparison bars. A reference table shows common triangle altitudes and orthocenter locations. Presets cover right, equilateral, isosceles, and obtuse triangles for quick demonstration.

When This Page Helps

Triangle altitudes are more informative than a single height value because each side of the triangle has its own perpendicular distance and all three meet at the orthocenter. This calculator is useful when you want to compare those heights, classify the triangle, and understand where the orthocenter lands without rebuilding the geometry from scratch. It is particularly helpful in coordinate geometry, proof-based coursework, and any setting where the area and triangle type have to agree with one another.

How to Use the Inputs

1. Enter all three side lengths of the triangle (a, b, c).
2. Click a preset to load a classic triangle type.
3. View all three altitude lengths (h_a, h_b, h_c).
4. Check the triangle classification (acute, right, or obtuse).
5. Find the orthocenter coordinates and see where it lies relative to the triangle.
6. Compare altitude lengths visually in the bar chart.
7. Explore the reference table for common triangles and their orthocenter behavior.

Example Calculation

Tips & Best Practices

• The altitude to the longest side is always the shortest altitude.
• Area = ½ · base · height works for ANY base-altitude pair — all three give the same area.
• In a right triangle, two of the altitudes are the legs themselves, and the orthocenter is the right-angle vertex.
• For obtuse triangles, the orthocenter is outside the triangle — be aware when plotting.
• In an equilateral triangle, all three altitudes are equal and the orthocenter coincides with the centroid.

Area Connects All Three Altitudes

Every altitude of a triangle is tied to the same area. Once the side lengths are known, Heron's formula gives the area, and each altitude follows from h = 2A / base. This explains an important pattern: the longest side always has the shortest altitude, while the shortest side has the longest altitude. The three heights are different views of the same area rather than unrelated measurements.

What The Orthocenter Tells You

The orthocenter is the intersection point of the three altitudes, and its location reveals the triangle type immediately. In an acute triangle it lies inside the figure, in a right triangle it lands exactly on the right-angle vertex, and in an obtuse triangle it moves outside the triangle. That makes altitude calculations useful for more than just area work; they also help you interpret the shape and behavior of the whole triangle.

Using Altitudes To Check Triangle Data

Altitudes are a strong consistency check in geometry problems. If you compute h_a, h_b, and h_c from the same side lengths, each base-height pair should reproduce the same area. When one pair does not match, the mistake is usually a side-length entry error, an invalid triangle, or a rounding issue. This calculator makes that comparison immediate, which is useful in homework verification, proof writing, and coordinate-geometry setups where the orthocenter location matters.

Frequently Asked Questions

• What is the orthocenter of a triangle?

  The orthocenter is the point where all three altitudes (or their extensions) intersect. Its location depends on the triangle type: inside for acute, on the right-angle vertex for right, and outside for obtuse triangles.

• How do you calculate the altitude of a triangle?

  First find the area using Heron's formula, then h_a = 2·Area/a. This works for any triangle given three sides.

• Where is the orthocenter of an obtuse triangle?

  The orthocenter lies outside the triangle, beyond the side opposite the obtuse angle. Two altitudes must be extended beyond the triangle to meet.

• Can an altitude lie outside the triangle?

  Yes. In an obtuse triangle, the altitudes from the two acute-angle vertices extend beyond the opposite sides, so the foot of the altitude falls outside the triangle.

• What is the relationship between altitudes and area?

  Area = ½·a·h_a = ½·b·h_b = ½·c·h_c. This means the altitude to any side can compute the area, and altitudes are inversely proportional to their base lengths.

• Do the altitudes, medians, and angle bisectors ever coincide?

  Only in an equilateral triangle. In that case, each altitude is also a median, angle bisector, and perpendicular bisector.