Altitude of a Triangle: Definition; Formula; Properties and Solved Examples

The altitude of a triangle is a fundamental concept in geometry, defined as the perpendicular distance from a vertex to the opposite side (or its extension). The altitude of a triangle is also known as the height of the triangle. Knowing how to determine the altitude of equilateral, isosceles, or scalene triangles helps in applying area formulas and analyzing geometric relationships. In this guide, you will learn what the altitude of a triangle is, how to calculate it using formulas, and how it differs across various types of triangles. 

Table of Contents

• What is the Altitude of a Triangle?
• Altitude of a Triangle Formula
• Properties of the Altitude of a Triangle
• Altitude of Different Types of Triangles
• Differences Between Altitude and Median of a Triangle
• Solved Example on Altitude of a Triangle
• Practice Questions on Altitude of a Triangle

What is the Altitude of a Triangle?

An altitude is a perpendicular drawn from a vertex to its opposite side. In a triangle, if we draw a line segment from a vertex to any point on the opposite side such that it makes a right angle, then the line segment is called the altitude of the triangle. 

Altitude is also the height of the triangle when a side is considered its base. Every triangle has three altitudes, one from each vertex. These three altitudes always meet at a single point called the orthocentre.

In ΔPQR, QS is the altitude that joins vertex Q and point S on side PR, and QS ⊥ PR.

Similarly, PT and RU are the other two altitudes of ΔPQR.

The point of intersection of altitudes is called the orthocentre of the triangle. Here, H is the orthocentre of ΔPQR.

Altitude of a Triangle Formula

The formula for the altitude of a triangle can be derived from the formula for the area of a triangle. Area = (1/2) × base × height, where the height is actually the altitude of the triangle. By rearranging this formula, we can easily calculate the altitude.

Altitude = (2 × Area) / base

Properties of the Altitude of a Triangle

Here are the most important properties of the altitude of a triangle:

1.  Always perpendicular: An altitude is always perpendicular to the base it is drawn to. i.e., it always forms a 90° angle at the foot of the altitude.

2. Every triangle has three altitudes: Since a triangle has three sides and three vertices, there are three altitudes, one from each vertex to the opposite side.

3. The three altitudes are concurrent: All three altitudes of any triangle always meet at a single point called the orthocentre.

4. The Orthocenter Changes Position by Triangle Type

1. In an acute triangle, the orthocentre lies inside the triangle.

2. In a right triangle, the orthocentre lies at the right-angle vertex itself.

3. In an obtuse triangle, the orthocentre lies outside the triangle.

5. Altitude can lie outside the triangle: In an obtuse triangle, the altitude from an acute vertex falls outside the triangle. To draw it, you extend the base and then draw the perpendicular from the opposite vertex to that extension.

6. Altitude = Median = Perpendicular Bisector in equilateral triangles: In an equilateral triangle, the altitude, median, angle bisector, and perpendicular bisector from any vertex all coincide; they are the same line segment.

7. Used to calculate area: Area of triangle = (1/2) × Base × Altitude

Altitude of Different Types of Triangles

Altitude of an Equilateral Triangle

An equilateral triangle has all three sides equal (s). The altitude from any vertex bisects the base exactly in half and also bisects the angle at the top.

Derivation:

Using the Pythagorean theorem on the right triangle formed:

h² + (a/2)² = a²

h² = a² − a²/4 = 3a²/4

h = (√3 / 2) × a

Example: If the side of an equilateral triangle is 6 cm:

h = (√3 / 2) × 6 = 3√3 cm ≈ 5.196 cm

Altitude of an Isosceles Triangle

An isosceles triangle has two equal sides (each of length a) and one unequal base (b). The altitude drawn from the apex (the vertex between the two equal sides) acts as the perpendicular bisector of the base.

Derivation:

Using the Pythagorean theorem:

h² = a² − (b/2)²

h = √(a² − b²/4)

Example: An isosceles triangle has equal sides of 10 cm each and a base of 12 cm. Find its altitude.

h = √(10² − 6²) = √(100 − 36) = √64 = 8 cm

Altitude of a Right-Angled Triangle

A right-angled triangle has one 90° angle. 

There are two cases:

Case 1: Altitude to one of the legs:

The altitude to a leg is the other leg. For example, in a right triangle with legs p and q, the altitude to q is p, and vice versa.

Case 2: Altitude to the hypotenuse:

When a perpendicular is drawn from the right-angle vertex to the hypotenuse, two smaller similar triangles are formed.

If the hypotenuse is c and is divided into segments x and y by the foot of the altitude:  h² = x · y ⇒ h = √(x · y)

There's also a direct formula in terms of the legs p and q:

h = (p × q) / c (where c = √(p² + q²))

Example: A right triangle has legs 6 cm and 8 cm. Find the altitude to the hypotenuse.

Hypotenuse c = √(36 + 64) = √100 = 10 cm

h = (6 × 8) / 10 = 48/10 = 4.8 cm

Altitude of a Scalene Triangle

A scalene triangle has all three sides different from each other. 

To find the altitude, we first calculate the area using Heron's formula, then use the following formula:

h = (2 × Area) / base

Heron's Formula:

s = (a + b + c) / 2 (semi-perimeter)

Area = √[s(s − a)(s − b)(s − c)]

Example: A triangle has sides a = 5 cm, b = 7 cm, and c = 8 cm. Find the altitude to side b.

s = (5 + 7 + 8)/2 = 10

Area = √[10 × 5 × 3 × 2] = √300 = 10√3 cm²

h_{b} = (2 × 10√3) / 7 = 20√3 / 7 ≈ 4.95 cm

Altitude of an Obtuse Triangle

In an obtuse triangle (one angle greater than 90°), the altitudes from the two acute vertices fall outside the triangle. You have to extend the base before dropping the perpendicular.

The formula remains the same:

h = (2 × Area) / base

Formula Summary Table

Where:

• a = side of an equilateral triangle

• a = equal leg of isosceles, b = base of isosceles

• p, q = legs of right triangle; c = hypotenuse

• x, y = segments of hypotenuse

• a, b, c = sides of general triangle; s = semi-perimeter
  
  

Differences Between Altitude and Median of a Triangle

An altitude of a triangle and a median of a triangle are both drawn from a vertex, but they are very different in nature.

Solved Example on Altitude of a Triangle

Example 1: Find the altitude of an equilateral triangle with a side of 10 cm.

Solution: Given, a = 10 cm.

The length of altitude of an equilateral triangle = h = (√3 / 2) × a 

h = (√3 / 2) × 10 = 5√3 cm ≈ 8.66 cm

The altitude of an equilateral triangle with a side of 10 cm is 8.66 cm.

Example 2: An isosceles triangle has equal sides of 13 cm and a base of 10 cm. Find the altitude from the apex to the base.

Solution: G, a = 13 cm and b = 10 cm

The length of altitude of an equilateral triangle = h = √(a² − b²/4)

Where a = equal leg of the isosceles and b = base of the isosceles

h = √(13² − 5²) = √(169 − 25) = √144 = 12 cm

The altitude of an isosceles triangle with equal sides of 13 cm and a base of 10 cm is 12 cm in length. 

Example 3: In a right triangle, the two legs are 8 cm and 15 cm. Find the altitude drawn to the hypotenuse.

Solution: Given, p = 8 cm, q = 15cm

c = √(p² + q²)) 

The length of the altitude of a right-angled triangle = h = (p × q) / c

c = √(64 + 225) = √289 = 17 cm

h = (8 × 15) / 17 = 120/17 = 7.06 cm (approx.)

Example 4: The sides of a triangle are 6 cm, 8 cm, and 10 cm. Find the altitude to the side of length 8 cm.

Solution: Given, a =  6 cm, b = 8 cm, and c = 10 cm

s = (6 + 8 + 10)/2 = 12

Area = √[12 × 6 × 4 × 2] = √576 = 24 cm²

h = (2 × 24) / 8 = 6 cm

Example 5: A triangle has a base of 14 cm and an area of 84 cm². Find its altitude.

Solution: Given, b =  14 cm and A = 84 cm²

Area = (1/2) × base × height

84 = (1/2) × 14 × h

84 = 7h

h = 12 cm

Practice Questions on Altitude of a Triangle

1. The point of intersection of altitudes is called ______________.

2. A triangle has sides a = 3 cm, b = 4 cm, c = 5 cm. Find all three altitudes.

3. Find the altitude of an equilateral triangle whose side is 12 cm.

4. The altitude of a triangle is 9 cm and the base is 16 cm. Find its area.

5. A triangle has a base of 20 cm and an area of 150 cm². What is its altitude?

6. Three sides of a triangle are 7 cm, 8 cm, and 9 cm. Find the altitude of the triangle to the side of length 8 cm.

7. The altitude of an equilateral triangle is 9√3 cm. Find the side of the triangle.

8. In triangle ABC, AB = 10 cm, BC = 10 cm, and AC = 12 cm. Find the altitude from vertex B to AC.

9. In a right-angled triangle, the legs are 5 cm and 12 cm. Find the altitude to the hypotenuse.

10. A triangular park has sides 50 m, 60 m, and 70 m. Find the length of the altitude drawn to the longest side. (Use Heron's formula)