Triangle Height Calculator
Our free triangle calculator solves triangle height problems. Get worked examples, visual aids, and downloadable results.
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The height to any side is found by dividing twice the triangle area by that side length. The area is calculated using Heron formula where s = (a+b+c)/2 is the semi-perimeter.
Last reviewed: December 2025
Worked Examples
Example 1: Finding All Three Heights from Sides
Example 2: Finding Height from Base and Area
Background & Theory
The Triangle Height Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Triangle Height Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
Formula
h = 2 x Area / base | Area = sqrt(s(s-a)(s-b)(s-c))
The height to any side is found by dividing twice the triangle area by that side length. The area is calculated using Heron formula where s = (a+b+c)/2 is the semi-perimeter.
Worked Examples
Example 1: Finding All Three Heights from Sides
Problem: A triangle has sides a = 8, b = 10, c = 12. Find all three altitudes.
Solution: Semi-perimeter s = (8 + 10 + 12) / 2 = 15\nArea = sqrt(15 x 7 x 5 x 3) = sqrt(1575) = 39.6863\n\nHeight to side a: h_a = 2(39.6863) / 8 = 9.9216\nHeight to side b: h_b = 2(39.6863) / 10 = 7.9373\nHeight to side c: h_c = 2(39.6863) / 12 = 6.6144\n\nVerification: h_a x a = h_b x b = h_c x c = 79.3726 = 2 x Area
Result: h_a = 9.9216 | h_b = 7.9373 | h_c = 6.6144 | Area = 39.6863
Example 2: Finding Height from Base and Area
Problem: A triangular plot of land has a base of 50 meters and an area of 750 square meters. What is the height?
Solution: Using Area = (1/2) x base x height\n750 = (1/2) x 50 x height\n750 = 25 x height\nheight = 750 / 25 = 30 meters\n\nVerification: Area = (1/2) x 50 x 30 = 750 sq meters
Result: Height = 30 meters
Frequently Asked Questions
What is the height (altitude) of a triangle?
The height or altitude of a triangle is the perpendicular distance from a vertex to the line containing the opposite side (called the base). Every triangle has three altitudes, one from each vertex. The altitude creates a right angle where it meets the base or the extension of the base. For acute triangles, all three altitudes fall inside the triangle. For right triangles, two of the altitudes are the legs themselves. For obtuse triangles, two altitudes fall outside the triangle and meet the extensions of the sides rather than the sides themselves. The point where all three altitudes intersect is called the orthocenter, which is one of the four classical triangle centers along with the centroid, incenter, and circumcenter.
How do you calculate the height of a triangle from three sides?
To find the height from three known sides, first calculate the area using Heron formula, then use the area-base-height relationship. Heron formula gives Area = sqrt(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. Once you have the area, the height to any side is found using h = 2 times Area divided by the base. For example, with sides 5, 6, 7: s = 9, Area = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.697. Height to side a: h_a = 2(14.697)/5 = 5.879. Height to side b: h_b = 2(14.697)/6 = 4.899. Height to side c: h_c = 2(14.697)/7 = 4.199. The height to the longest side is always the shortest altitude.
What is the relationship between area and height of a triangle?
The area of a triangle is directly related to its height through the fundamental formula: Area = (1/2) times base times height. This means that height = 2 times Area divided by base. This relationship is incredibly versatile because it works for any triangle, regardless of type. If you know the area and any base, you can find the corresponding height. Conversely, if you know the base and height, you can find the area. For equilateral triangles with side s, the height is s times sqrt(3)/2 and the area is s squared times sqrt(3)/4. This area-height relationship is preserved under shearing transformations, which is why parallelograms with the same base and height have twice the triangle area.
What is the orthocenter and how does it relate to triangle heights?
The orthocenter is the point where all three altitudes of a triangle intersect. This point always exists for any triangle, though its position varies by triangle type. For an acute triangle, the orthocenter lies inside the triangle. For a right triangle, the orthocenter is located exactly at the vertex of the right angle. For an obtuse triangle, the orthocenter lies outside the triangle, beyond the side opposite the obtuse angle. The orthocenter is denoted H and is one of the four classical triangle centers. It has an interesting property: reflecting the orthocenter over any side of the triangle places it on the circumscribed circle. The orthocenter, centroid, and circumcenter are always collinear, lying on a line called the Euler line.
What is the formula for height using trigonometry?
Using trigonometry, the height can be calculated directly from a side and an angle. The height from vertex C to side c equals a times sin(B) or equivalently b times sin(A). More generally, if you know side b and the angle A between sides b and c, the height from C to side c is h = b times sin(A). This works because in the right triangle formed by the altitude, the altitude is the side opposite to the known angle, and the known side is the hypotenuse. For example, in a triangle with b = 10 and angle A = 30 degrees, the height h = 10 times sin(30) = 10 times 0.5 = 5. This trigonometric approach is often more direct than using Heron formula when angle information is available.
How are the three altitudes of a triangle related?
The three altitudes of a triangle are inversely proportional to their corresponding bases. Since all three heights give the same area (Area = base times height / 2), we get h_a times a = h_b times b = h_c times c = 2 times Area. This means the altitude to the longest side is always the shortest, and the altitude to the shortest side is always the longest. There is also a beautiful reciprocal relationship: 1/h_a + 1/h_b + 1/h_c relates to the triangle area and its properties. The three altitudes also satisfy the constraint that they must be concurrent (meet at the orthocenter), which places restrictions on which sets of three line segments can serve as altitudes of a valid triangle.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy