Right Triangle Altitude Calculator

In geometry, an altitude of a triangle is a perpendicular line segment drawn from a vertex to the opposite side (or line containing the opposite side). Because every triangle has three vertices, every triangle possesses three distinct altitudes.

For a right triangle, two of its three altitudes are identical to its perpendicular legs $a$ and $b$, because each leg is already perpendicular to the other. However, the third altitude—specifically the altitude dropped from the $90^\circ$ right-angle vertex to the hypotenuse $c$—is an interior perpendicular segment denoted by $h$.

This specific altitude $h$ is of primary interest in geometry because it establishes fundamental proportional relationships, divides the original triangle into two smaller similar right triangles, and obeys the famous Geometric Mean Theorem.

The altitude to the hypotenuse is the perpendicular segment extending from the right-angle vertex $C$ to meet the hypotenuse $c$ at point $D$ at a $90^\circ$ angle.

When altitude $h$ is drawn to hypotenuse $c$, it divides hypotenuse $c$ into two smaller segments: segment $p$ (adjacent to leg $a$) and segment $q$ (adjacent to leg $b$), such that:

The fundamental formula to calculate the altitude $h$ to the hypotenuse from side lengths is:

Where:

• $h$ = altitude to the hypotenuse
• $a$ = length of the first perpendicular leg
• $b$ = length of the second perpendicular leg
• $c$ = length of the hypotenuse ($c = \sqrt{a^2 + b^2}$)

In plain language: The altitude to the hypotenuse equals the product of the two perpendicular legs divided by the hypotenuse.

The altitude formula $h = \frac{ab}{c}$ is directly derived using the basic formula for the area of a triangle.

The area $A$ of any triangle is given by $A = \frac{1}{2} \times \text{base} \times \text{height}$. A right triangle can be evaluated in two distinct ways:

1. Using the legs: Choosing leg $b$ as the base means leg $a$ is the perpendicular height: $$\text{Area} = \frac{1}{2} a b$$
2. Using the hypotenuse: Choosing hypotenuse $c$ as the base means altitude $h$ is the corresponding perpendicular height: $$\text{Area} = \frac{1}{2} c h$$

Because both expressions calculate the exact same enclosed area of the same triangle, we set them equal to each other:

Multiply both sides of the equation by $2$:

Dividing both sides by $c$ isolates the altitude $h$:

This clean, elegant derivation proves that $h = \frac{ab}{c}$ holds true for all right triangles.

When only the two perpendicular legs $a$ and $b$ are known, first calculate the hypotenuse $c$ using the Pythagorean theorem:

Then substitute $c$ back into the primary altitude formula:

When all three sides ($a, b, c$) are known, calculating the altitude is a straightforward single-step process.

Before computing, always verify that the sides form a valid right triangle by checking that $a^2 + b^2 = c^2$.

If you know the total enclosed area $A$ of the right triangle and its hypotenuse $c$, you can solve for altitude $h$ by rearranging the area equation $A = \frac{1}{2} c h$:

Note: Area $A$ is expressed in square units (e.g., $\text{cm}^2$), while altitude $h$ and hypotenuse $c$ are linear distance measurements (e.g., $\text{cm}$).

The altitude to the hypotenuse is intrinsically connected to the triangle's area. Because $\text{Area} = \frac{1}{2} c h$, the altitude $h$ represents the exact perpendicular height required when using hypotenuse $c$ as the base.

If you double the area of a right triangle while holding hypotenuse $c$ constant, the altitude $h$ doubles proportionally.

To explore general area calculations, visit our dedicated Right Triangle Area Calculator →.

Because $h = \frac{ab}{\sqrt{a^2 + b^2}}$, altitude $h$ is sensitive to both leg lengths. If either leg increases while the other remains fixed, the altitude $h$ increases, but it remains strictly less than both legs $a$ and $b$.

In any right triangle, the altitude to the hypotenuse is always strictly shorter than both leg $a$ and leg $b$:

One of the most important theorems in geometry is the Right Triangle Altitude Theorem (or Geometric Mean Altitude Theorem).

When the altitude $h$ is drawn to hypotenuse $c$, it splits the hypotenuse into two segments $p$ and $q$. The length of altitude $h$ is the geometric mean of segments $p$ and $q$:

This relationship arises because the altitude creates two smaller right triangles that are geometrically similar to each other and to the original triangle. Comparing proportional side ratios yields $\frac{h}{p} = \frac{q}{h} \implies h^2 = pq$.

In addition to $h^2 = pq$, drawing altitude $h$ creates two additional leg-segment relationships known as the Leg Theorem:

• Leg $a$ relationship: $a^2 = c \times p \implies a = \sqrt{cp}$
• Leg $b$ relationship: $b^2 = c \times q \implies b = \sqrt{cq}$
• Segment sum: $p + q = c$
• Altitude geometric mean: $h^2 = p \times q$

Altitude calculations often yield rational fractions or radical expressions.

• Exact Fraction / Radical Form: Keeps precision without rounding (e.g., $h = \frac{60}{13}$ or $h = 5\sqrt{2}$).
• Decimal Form: Provides practical decimal approximations (e.g., $h \approx 4.6154$).

To maintain mathematical rigor, avoid rounding intermediate numbers like hypotenuse $c$ before computing altitude $h$.

Altitude is a 1-dimensional (linear) distance. Always express altitude in linear measurement units:

mm, cm, m, km, in, ft, yd

Correct: $h = 4.8 \text{ cm}$.
Incorrect: $h = 4.8 \text{ cm}^2$ (Square units designate area, not altitude).

Verify your altitude calculations using three independent mathematical checks:

1. Check 1 (Leg Product): $h = \frac{a \times b}{c}$
2. Check 2 (Area Formula): $h = \frac{2A}{c}$ (Verify $\frac{1}{2}ab = \frac{1}{2}ch$)
3. Check 3 (Geometric Mean): $h = \sqrt{p \times q}$ (where $p = \frac{a^2}{c}$ and $q = \frac{b^2}{c}$)

All three methods must yield the identical altitude result.

In common language, "altitude" and "height" are often used interchangeably. However, in right triangle geometry:

• Height: Refers generally to any vertical leg relative to a horizontal base leg ($a$ relative to $b$, or $b$ relative to $a$).
• Altitude to Hypotenuse: Refers specifically to the interior perpendicular line segment $h$ drawn from the $90^\circ$ right angle to hypotenuse $c$.

It is essential not to confuse hypotenuse $c$ with altitude $h$:

• Hypotenuse ($c$): The longest outer boundary side of the triangle, located directly opposite the $90^\circ$ angle.
• Altitude ($h$): An interior line segment extending from the $90^\circ$ vertex perpendicular to the hypotenuse.

An altitude and a median serve fundamentally different geometric roles:

• Altitude ($h$): Meets the hypotenuse at a strict $90^\circ$ perpendicular angle (Does NOT necessarily bisect the hypotenuse unless $a = b$).
• Median ($m_c$): Connects the $90^\circ$ vertex to the exact midpoint of the hypotenuse (In a right triangle, median to hypotenuse length is always exactly $m_c = \frac{c}{2}$).

Suppose an architectural truss frame forms a right triangle with horizontal base leg $b = 12 \text{ ft}$ and vertical leg $a = 5 \text{ ft}$. An interior structural support beam must be installed from the $90^\circ$ corner perpendicular to the diagonal rafter (hypotenuse).

Step 1: Calculate diagonal rafter length $c = \sqrt{5^2 + 12^2} = 13 \text{ ft}$.

Step 2: Calculate exact beam length $h = \frac{5 \times 12}{13} = \frac{60}{13} \approx 4.615 \text{ ft}$ (approx $4 \text{ ft } 7.38 \text{ in}$).