Altitude of a Triangle: Formulas & Properties
Altitude of a Triangle: Formulas & Properties
TL;DR
The altitude of a triangle is the perpendicular segment from a vertex straight down to the line containing the opposite side, and its length is the height used in the area formula. This article covers the definition, the formulas for scalene, isosceles, equilateral, and right triangles, the orthocentre, six worked examples, and the mistakes students make most.
What Is the Altitude of a Triangle?
The altitude of a triangle is a line segment drawn from a vertex perpendicular to the line containing the opposite side. The point where it meets that side is the foot of the altitude, and the length of the segment from vertex to foot is the triangle's height.
Every triangle has three altitudes, one from each vertex, because any of the three sides can be treated as the base. The altitude always meets its base at a right angle (90°). Depending on the triangle, an altitude can fall inside the triangle, lie along a side, or fall outside it, which is a point we come back to for obtuse triangles.
Properties of the Altitude of a Triangle
The perpendicular-from-a-vertex definition forces a clear set of properties:
- Three altitudes per triangle, one to each side.
- Each altitude is perpendicular to its base, meeting it at exactly 90°.
- The three altitudes meet at one point, the orthocentre.
- An altitude need not bisect the base — this is the key difference from a median, which always does.
- Position depends on the triangle's type. In an acute triangle all three altitudes are inside; in a right triangle two of them are the legs themselves; in an obtuse triangle two altitudes fall outside the triangle.
The General Altitude Formula
Because an altitude is the height in the area formula, the two are tied together directly. The area of a triangle is half the base times the height:
Area = ( \frac{1}{2} \times \text{base} \times \text{height} )
Rearranging for the height gives the general formula:
( h = \frac{2 \times \text{Area}}{\text{base}} )
So if you know a triangle's area and the side you want the altitude to, one division gives the altitude.
Altitude Formulas by Triangle Type
Different triangle types give the altitude a cleaner closed form. Each formula below is just the general formula with the area worked out for that shape.
| Triangle type | Altitude formula | What the variables mean |
|---|---|---|
| Scalene (any) | ( h = \frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{b} ) | a,b,c are the sides; s=( \frac{a+b+c}{2} ) is the semi-perimeter; altitude is to side b. |
| Isosceles | ( h = \sqrt{a^2 - \frac{b^2}{4}} ) | a is each equal side; b is the base; altitude is from the apex |
| Equilateral | ( h = \frac{a \sqrt{3}}{2} ) | a is the side length |
| Right (to a leg) | the other leg | the two legs are altitudes of each other |
| Right (to hypotenuse) | ( h = \sqrt{xy} ) | x,y are the two segments the foot makes on the hypotenuse |
What Is the Orthocentre?
A natural question once you have three altitudes: do they meet anywhere special? They do. The three altitudes of any triangle always pass through a single common point called the orthocentre (often written H). Where that point sits tells you about the triangle:
- Acute triangle — the orthocentre lies inside the triangle.
- Right triangle — the orthocentre sits exactly at the right-angle vertex.
- Obtuse triangle — the orthocentre falls outside the triangle.
Altitude vs Median: What Is the Difference?
Both run from a vertex to the opposite side, but they are built on different rules. An altitude is defined by an angle (it must be perpendicular); a median is defined by a point (it must hit the midpoint).
| Feature | Altitude | Median |
|---|---|---|
| Goes from a vertex to | the opposite side, at 90° | the midpoint of the opposite side |
| Always perpendicular? | Yes | No, not usually |
| Always bisects the base? | No | Yes, by definition |
| Stays inside the triangle? | No (outside for obtuse) | Yes, always |
| Three of them meet at | the orthocentre | the centroid |
Examples of Altitude of a Triangle
Example 1: A triangle has an area of 24 cm² and a base of 6 cm. Find the altitude to that base
Use the general formula: ( h = \frac{2A}{b} = \frac{2 \times 24}{6} = 8 \text{ cm} )
Final answer: 8 cm.
Example 2: Find the altitude to the base of an isosceles triangle with equal sides 10 cm and base 12 cm
Done correctly with the isosceles formula: ( h = \sqrt{10^2 - \frac{12^2}{4}} = 8 \text{ cm} )
Example 3: Find the altitude of an equilateral triangle with side 6 cm
( h = \frac{6\sqrt{3}}{2} = 3\sqrt{3} \approx 5.20 \text{ cm} )
Example 4: A scalene triangle has sides 7 cm, 8 cm, and 9 cm. Find the altitude to the 8 cm side
First, the semi-perimeter and area:
( s = \frac{7 + 8 + 9}{2} = 12 )
( A = \sqrt{12(12-7)(12-8)(12-9)} \approx 26.83\text{ cm}^2 )
Then the altitude to the 8 cm base: ( h = \frac{2A}{b} \approx 6.71 \text{ cm} )
Example 5: In a right triangle, the altitude from the right angle meets the hypotenuse and splits it into segments of 4 cm and 9 cm. Find the altitude
( h = \sqrt{4 \times 9} = 6 \text{ cm} )
Example 6: A triangle has sides 5 cm, 12 cm, and 13 cm. Find the altitude to the 13 cm side
( h = \frac{2 \times 30}{13} \approx 4.62 \text{ cm} )
Why the Altitude of a Triangle Matters
The altitude connects area, the Pythagorean theorem, and triangle centres, making it essential for understanding geometry.
Where Students Trip Up on Altitudes
Mistake 1: Using a slant side as the height
Mistake 2: Confusing the altitude with the median
Mistake 3: Expecting every altitude to stay inside the triangle
Key Takeaways
- The altitude of a triangle is the perpendicular segment from a vertex to the opposite side.
- The general formula is ( h = \frac{2A}{b} );
- Every triangle has three altitudes that meet at the orthocentre.
Practice These Problems to Solidify Your Understanding
- A triangle has area 45 cm² and base 9 cm.
- Find the altitude of an equilateral triangle with side 10 cm.
- An isosceles triangle has equal sides 13 cm and base 10 cm.