Triangle Height Calculator
Find the height of any triangle using whichever measurements you already have. Choose one of four methods: three sides (Heron's formula), area and base, two sides with their included angle, or the equilateral shortcut. The calculator returns all three altitudes for the general case, shows the step-by-step working, and draws the triangle to scale.
Formula
Worked example
For a triangle with sides a = 5 m, b = 6 m, c = 7 m: s = (5+6+7)/2 = 9. Area = sqrt(9 * 4 * 3 * 2) = sqrt(216) = 6*sqrt(6) approximately 14.697 m2. h_a = 2*14.697/5 = 5.879 m, h_b = 2*14.697/6 = 4.899 m, h_c = 2*14.697/7 = 4.199 m.
What is the height (altitude) of a triangle?
The height of a triangle, also called an altitude, is the perpendicular distance from a vertex to the opposite side (or to the line extending that side). Every triangle has exactly three altitudes, one from each vertex. The point where all three meet is called the orthocenter. For an acute triangle the orthocenter lies inside the triangle; for an obtuse triangle it falls outside; and for a right triangle it sits exactly at the right-angle vertex. The altitude is crucial because it connects geometry to area: Area = (1/2) x base x height, so knowing any altitude and the matching base is enough to recover the full area.
How to calculate triangle height - four methods
Method 1 - From area and base: If you already know the area A and the length of the base b, the altitude to that base is simply h = 2A/b. This is the most direct formula. Method 2 - From three sides (Heron's formula): First compute the semi-perimeter s = (a+b+c)/2, then the area using Area = sqrt(s*(s-a)*(s-b)*(s-c)). Each altitude follows from h_x = 2*Area/x where x is the matching side. Method 3 - From two sides and included angle: When you know two sides p and q and the angle between them, the area is 0.5*p*q*sin(angle). Divide by the appropriate base to get any altitude. Method 4 - Equilateral triangle shortcut: Because all three sides are equal, the altitude is h = a*sqrt(3)/2, and it is simultaneously a median, an angle bisector, and a perpendicular bisector.
Special cases: isosceles and right triangles
For an isosceles triangle with two equal legs of length a and a base of length b, the altitude from the apex to the base is h = sqrt(a^2 - (b/2)^2). This works because the altitude bisects the base exactly, creating two congruent right triangles. For a right triangle with legs a and b and hypotenuse c, the two legs themselves are altitudes to each other. The altitude from the right-angle vertex to the hypotenuse is the shortest of the three and equals h_c = a*b/c. This quantity appears in many geometry proofs and is the geometric mean of the two segments the altitude cuts on the hypotenuse.
Relationship between altitudes and area
The three altitudes of any triangle satisfy a reciprocal identity: 1/h_a + 1/h_b + 1/h_c relates to the circumradius and inradius. More practically, the product of any side and its altitude is always twice the area: a*h_a = b*h_b = c*h_c = 2*Area. This means you can freely switch base-height pairs. A longer side always corresponds to a shorter altitude, and vice versa. In a triangle with sides in ratio 3:4:5 (a right triangle), the altitudes are in the inverse ratio 1/3:1/4:1/5, or equivalently 20:15:12 after scaling.
Altitude formulas by triangle type
| Triangle type | Known inputs | Altitude formula |
|---|---|---|
| Any triangle | Area (A), base (b) | h = 2A / b |
| Any triangle (3 sides) | Sides a, b, c - Heron | h_a = 2 × sqrt(s(s-a)(s-b)(s-c)) / a |
| Any triangle | Two sides p, q; angle γ | h = p × q × sin(γ) / base |
| Equilateral | Side a | h = a√3 / 2 |
| Isosceles | Legs a, base b | h = sqrt(a² - (b/2)²) |
| Right triangle | Legs a, b; hyp. c | h_c = a × b / c |
Quick-reference formulas for the altitude h to each base.
Frequently asked questions
Can a triangle have two equal altitudes?
Yes. If two sides of a triangle are equal (an isosceles triangle), the altitudes to those two sides are also equal. If all three sides are equal (equilateral), all three altitudes are the same length. For a scalene triangle, all three altitudes are different.
What is the altitude of a right triangle to its hypotenuse?
For a right triangle with legs a and b and hypotenuse c, the altitude from the right-angle vertex to the hypotenuse is h = (a * b) / c. For example, a 3-4-5 right triangle has a hypotenuse altitude of (3*4)/5 = 2.4 units. This altitude also equals the geometric mean of the two hypotenuse segments it creates.
How do I find the height if I only know two sides and the included angle?
Use the formula: Area = 0.5 * p * q * sin(angle), then divide twice the area by whichever side you want to use as the base. If sides p = 5 m and q = 6 m with an included angle of 60 degrees, the area is 0.5 * 5 * 6 * sin(60) = 12.99 m2. The altitude to side p (using p as the base) is 2 * 12.99 / 5 = 5.196 m.
Does the altitude of a triangle always fall inside the triangle?
Only for acute triangles. In an obtuse triangle, the altitudes from the two acute vertices fall outside the triangle because the foot of the perpendicular lies on the extension of the opposite side. The altitude from the obtuse vertex always falls inside. In a right triangle, the altitude from the right-angle vertex lands exactly on the hypotenuse, and the other two altitudes coincide with the legs.
What is the formula for the height of an equilateral triangle?
For an equilateral triangle with side length a, the height is h = a * sqrt(3) / 2, which is approximately 0.866 * a. For example, an equilateral triangle with side 10 m has a height of 10 * 1.732 / 2 = 8.66 m. All three altitudes are identical and each one also bisects the opposite side.
How are the three altitudes of a triangle related to each other?
Because each altitude equals 2 * Area divided by its base, the altitudes are inversely proportional to the side lengths. A longer side has a shorter altitude, and a shorter side has a longer altitude. For any triangle, a * h_a = b * h_b = c * h_c = 2 * Area.