Triangle Height Calculator
Our free triangle calculator solves triangle height problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Triangle Height Calculator
Calculator
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Formula: h = 2 x Area / base | Area = sqrt(s(s-a)(s-b)(s-c))
Worked example — h_a = 9.9216 | h_b = 7.9373 | h_c = 6.6144 | Area = 39.6863
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 Area = sqrt(15 x 7 x 5 x 3) = sqrt(1575) = 39.6863 Height to side a: h_a = 2(39.6863) / 8 = 9.9216 Height to side b: h_b = 2(39.6863) / 10 = 7.9373 Height to side c: h_c = 2(39.6863) / 12 = 6.6144 Verification: 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 750 = (1/2) x 50 x height 750 = 25 x height height = 750 / 25 = 30 meters Verification: 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.
How do you find the foot of an altitude?
The foot of an altitude is the point where the altitude meets the base (or its extension). For a triangle with vertices at known coordinates, you can find the foot by projecting the vertex onto the opposite side using the dot product formula. For a triangle with sides a, b, c and the altitude from vertex A to side a, the foot divides side a at a distance of c times cos(B) from vertex B, or equivalently b times cos(C) from vertex C. Using the law of cosines, cos(B) = (a^2 + c^2 - b^2)/(2ac). For an acute triangle, the foot always lies between the two base vertices. For an obtuse triangle, the foot of the altitude from the obtuse angle vertex lies between the base vertices, but the feet of the other two altitudes lie on the extensions of their respective bases.
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.
What is the altitude of an equilateral triangle?
For an equilateral triangle with side length s, all three altitudes are equal and each has length h = s times sqrt(3) divided by 2, which is approximately 0.866 times s. This can be derived by splitting the equilateral triangle into two 30-60-90 right triangles, where the altitude is the long leg. For example, an equilateral triangle with side 10 has an altitude of 10 times sqrt(3)/2 = 8.6603. The area of an equilateral triangle is s squared times sqrt(3)/4. The altitude of an equilateral triangle also equals the distance from any vertex to the midpoint of the opposite side (median), the angle bisector length, and the perpendicular bisector length, since all four coincide in an equilateral triangle.
How does the height change when a triangle is scaled?
When a triangle is scaled by a factor k (all sides multiplied by k), every altitude is also scaled by the same factor k. This is because scaling preserves angles and multiplies all linear measurements by k. If the original triangle has altitude h to base b, the scaled triangle has altitude kh to base kb. The area, being a two-dimensional measurement, scales by k squared. For example, doubling all sides of a triangle doubles each altitude and quadruples the area. This scaling property is directly related to similar triangles: corresponding altitudes of similar triangles are in the same ratio as corresponding sides. This is why the altitude ratio can be used to determine the scale factor between similar triangles.
What are practical applications of calculating triangle heights?
Triangle height calculations have extensive practical applications across many fields. In construction and architecture, heights determine roof pitch, gable dimensions, and structural load distribution in triangular trusses. In surveying, calculating the height of a triangular plot helps determine area for property assessment and land valuation. In navigation and aviation, triangle heights are used to compute shortest distances between a point and a line (closest approach calculations). Civil engineers use triangle heights when designing road gradients and drainage slopes. In computer graphics, triangle heights are essential for rendering algorithms, collision detection, and mesh quality assessment. Structural engineers use altitude calculations to determine the stability of triangular support structures and to calculate moment arms in force analysis.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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