Triangle Height 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. The altitude is always perpendicular to the base it corresponds to.

Use the formula h = 2A/b, where A is the area and b is the base. This is derived from the area formula A = (1/2) × b × h, solved for h. For area 50 and base 10: h = 100/10 = 10.

First find the area using Heron's formula: s = (a+b+c)/2, A = √(s(s-a)(s-b)(s-c)). Then compute each altitude: h_a = 2A/a, h_b = 2A/b, h_c = 2A/c.

Area A = (1/2) × a × b × sin(C), where C is the included angle. Then h_c = 2A/c, where c is found using the law of cosines. Alternatively, h = a × sin(B) for the altitude from vertex A.

Yes! In an obtuse triangle, two of the three altitudes fall outside the triangle. The altitude from an acute angle vertex to the opposite side lands on the extension of that side. The length is still the perpendicular distance.

The three altitudes of any triangle intersect at a single point called the orthocenter. The product of each altitude and its corresponding base equals twice the area: h_a × a = h_b × b = h_c × c = 2A.

For a right triangle with legs a and b and hypotenuse c: the altitude to leg a is b, the altitude to leg b is a, and the altitude to the hypotenuse is h = ab/c. The legs serve as heights to each other.

For an equilateral triangle with side a: h = (√3/2) × a ≈ 0.866a. For side 10: h ≈ 8.660. This is derived from the Pythagorean theorem on the half-triangle created by the altitude.

For an isosceles triangle with base b and legs l: h = √(l² - (b/2)²). The altitude from the apex to the base bisects the base, creating two right triangles.

In a right triangle, the altitude from the right angle to the hypotenuse has length h = ab/c, where a and b are the legs and c is the hypotenuse. This altitude divides the triangle into two smaller similar triangles.

Find the third side: c = perimeter - a - b. Then use Heron's formula to find the area. Finally, compute the altitude to any side using h = 2A/side.

The orthocenter is the point where all three altitudes of a triangle intersect. In an acute triangle, it is inside. In an obtuse triangle, it is outside. In a right triangle, it is at the vertex of the right angle.

Only for a degenerate triangle (three collinear points with zero area). For any valid triangle with positive area, all three heights are positive.

Area = (1/2) × base × height. The height is the perpendicular component that, combined with the base, determines the triangle's area. For a fixed base, doubling the height doubles the area.

For vertices (x₁,y₁), (x₂,y₂), (x₃,y₃), find the area using the coordinate formula: A = |x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| / 2. Then h = 2A/base, where base is the distance between two vertices.

The altitude is perpendicular to the base; the median connects a vertex to the midpoint of the opposite side. They are the same only in isosceles and equilateral triangles. In scalene triangles, they are different lines.

The shortest altitude corresponds to the longest side: h_min = 2A/s_max. This is because altitude is inversely proportional to the base length for a fixed area.

Builders use triangle heights when calculating roof pitch, structural truss dimensions, and load distribution. The height determines how steep a roof is and how much material is needed.

The pedal triangle is formed by connecting the feet of the three altitudes. Its properties are related to the orthocenter and have applications in advanced geometry and crystallography.

Using the formula h_a = bc/(2R), where R is the circumradius. This relates the altitude to the product of the other two sides and the circumradius. The circumradius is found from R = abc/(4A).

The Triangle Height Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Triangle Height Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.