What Is the Altitude of a Triangle?
An altitude is a perpendicular line segment drawn from a vertex to the line containing the opposite side. The point where the altitude meets the base is called the foot of the altitude. Each triangle has exactly three altitudes, one from each vertex.
The term altitude and height are used interchangeably in triangle geometry. Dropping an altitude means constructing this perpendicular from a chosen vertex. In some triangles, particularly obtuse triangles, an altitude may fall outside the triangle itself, intersecting an extended line of the opposite side rather than the side directly.
The altitude is crucial for calculating the area of a triangle using the formula: Area = ½ × Base × Height. Conversely, if you know the area and one base, you can rearrange this to find the corresponding height.
Height Formulas for Different Triangle Types
Different triangle types require different approaches. For a scalene triangle (all sides different), you can use Heron's formula to find the area first, then calculate heights. For special triangles like equilateral or isosceles, direct formulas exist that depend only on the known side lengths. Right triangles offer the simplest case: the two legs themselves serve as altitudes.
Scalene Triangle (given all three sides):
Area = √[s(s − a)(s − b)(s − c)]
where s = (a + b + c) ÷ 2
ha = 2 × Area ÷ a
hb = 2 × Area ÷ b
hc = 2 × Area ÷ c
Equilateral Triangle (all sides equal):
h = a × √3 ÷ 2
Isosceles Triangle (two equal legs):
hbase = √[a² − (0.5 × b)²]
hleg = (hbase × b) ÷ a
Right Triangle (one 90° angle):
hc = (leg₁ × leg₂) ÷ hypotenuse
a, b, c— Side lengths of the triangles— Semi-perimeter: (a + b + c) ÷ 2h<sub>a</sub>, h<sub>b</sub>, h<sub>c</sub>— Altitudes from vertices opposite sides a, b, cArea— Area of the triangle
Calculating Heights for Equilateral and Isosceles Triangles
Equilateral triangles have a special property: all three altitudes are identical. If the side length is a, each altitude equals a × √3 ÷ 2 (approximately 0.866 × a). This formula emerges because an altitude bisects both the opposite side and the apex angle in an equilateral triangle, creating two 30-60-90 right triangles.
Isosceles triangles contain two equal legs and a different base. The altitude from the apex (the vertex between the two equal sides) bisects the base and can be found using the Pythagorean theorem: h = √[leg² − (base ÷ 2)²]. The two altitudes from the base vertices are equal in length but different from the apex altitude. Calculate these using the area method: h = 2 × Area ÷ base.
Heights in Right Triangles and Using Area
In a right triangle, two of the three altitudes are immediately obvious: they are simply the two perpendicular legs. The third altitude—from the right angle vertex to the hypotenuse—requires calculation.
For any triangle where you know the area, computing height becomes straightforward. Rearrange the area formula: if Area = ½ × Base × Height, then Height = 2 × Area ÷ Base. This method works universally across all triangle types. To find the area itself, use Heron's formula when you have all three side lengths, or the standard formula Area = ½ × Base × Height if you already know one altitude.
Common Pitfalls When Finding Triangle Heights
Avoid these frequent mistakes when calculating altitudes:
- Confusing height with side length — The altitude is always perpendicular to the base, but it is not necessarily one of the triangle's sides unless the triangle is a right triangle. In acute and obtuse triangles, the altitude is typically a new line segment, not part of the original triangle outline.
- Forgetting about obtuse triangles — In obtuse triangles, one or two altitudes extend outside the triangle's boundary. The altitude from an acute-angle vertex to the opposite side (which is part of an obtuse angle) will meet an extension of that side, not the side itself. Always account for this when sketching or visualizing the problem.
- Using only angle measurements — You cannot determine a triangle's altitude from angles alone. Infinitely many triangles can share the same three angles but have different sizes and therefore different altitude lengths. You must know at least one side length to proceed.
- Mixing up the base in isosceles triangles — In isosceles triangles, the term 'base' typically refers to the side that differs from the two equal legs. The altitude from the apex to the base is straightforward; altitudes from base vertices involve a different calculation and are usually longer than the apex altitude.