Why does the altitude formula for a right triangle use leg1 times leg2 divided by hypotenuse?

The area of the right triangle is (leg1 times leg2) / 2. Using Area = (1/2) times base times height with base = hypotenuse: height = 2 times Area / hypotenuse = 2 times (leg1 times leg2 / 2) / hypotenuse = leg1 times leg2 / hypotenuse. This is the geometric mean result: altitude squared = product of the two hypotenuse segments.

Triangle Height Calculator

Calculate the height of any triangle from base and area, three sides, or a right triangle.

What is the Height (Altitude) of a Triangle?

The height of a triangle, also called the altitude, is the perpendicular distance from a vertex to the opposite side (or its extension). Every triangle has exactly three heights, one from each vertex. These three altitudes always intersect at a single point called the orthocenter, which lies inside the triangle for acute triangles, at the right-angle vertex for right triangles, and outside the triangle for obtuse triangles.

The most familiar use of triangle height is in calculating area: Area = (1/2) times base times height. This formula works for any triangle as long as the height is measured perpendicular to the chosen base. However, finding the height is not always straightforward - if you know the area and base, you can work backwards. If you only know the three sides, you need Heron’s formula to find the area first, then derive all three heights from it.

For a right triangle, there is a special case worth knowing: the altitude from the right-angle vertex to the hypotenuse equals (leg1 times leg2) divided by the hypotenuse. This result connects to the geometric mean: the altitude squared equals the product of the two segments it divides the hypotenuse into. This is a key result in Euclidean geometry and appears frequently in trigonometry and engineering.

The three heights of a triangle are related to each other through the area: h_a times a = h_b times b = h_c times c = 2 times Area. This means the tallest altitude is always opposite the shortest side, and the shortest altitude is always opposite the longest side. In an equilateral triangle, all three altitudes are equal, each measuring side times sqrt(3) / 2.

Understanding triangle altitudes is essential for many practical applications: calculating roof pitch and rafter length in construction, finding the depth of a triangular cross-section in structural engineering, determining the height of a triangular land parcel in surveying, and computing stress distributions in triangular finite elements in computational mechanics.

Formula and Derivation

Mode 1 - Base and Area:

h = 2A / b

h = Height (altitude to base b)

A = Area of the triangle

b = Base length

Derived from: Area = (1/2) × b × h

Mode 2 - Three Sides (Heron’s formula):

s = (a + b + c) / 2

Area = √(s(s−a)(s−b)(s−c))

h_a = 2A / a h_b = 2A / b h_c = 2A / c

s = Semi-perimeter

h_a, h_b, h_c = Altitudes to sides a, b, c respectively

Mode 3 - Right Triangle:

leg_2 = √(hyp² − leg²)

h = leg_1 × leg_2 / hyp

h = Altitude from right angle to hypotenuse

leg_1 = Known leg, hyp = Hypotenuse

How to Use This Calculator

Choose the mode that matches what you know. “Base and Area” is simplest if you already know those. “Three Sides” uses Heron’s formula to find all three altitudes. “Right Triangle” finds the altitude to the hypotenuse from one leg and the hypotenuse.
Enter the values - use any consistent unit of length. For Three Sides mode, verify the triangle inequality: the sum of any two sides must be greater than the third.
Click Calculate - in Base and Area mode you get one height. In Three Sides mode you see all three altitudes h_a, h_b, h_c. In Right Triangle mode you see the altitude to the hypotenuse.
Read the formula note - in Three Sides mode this shows the area calculated by Heron’s formula, which confirms the triangle is valid and shows the intermediate step.
Try example links - each worked example below includes a pre-fill link so you can verify the results immediately.

Example Calculations

Example 1 - Base 8, Area 24

Find the height of a triangle with base 8 cm and area 24 sq cm.

Base = 8 cm, Area = 24 cm²

h = 2 × Area / base = 2 × 24 / 8 = 48 / 8 = 6 cm

Verify: Area = (1/2) × 8 × 6 = 24 cm² ✓

Height = 6.0000 cm

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Example 2 - Right triangle with sides 3, 4, 5

Find all three altitudes of the classic 3-4-5 right triangle.

a = 3, b = 4, c = 5. s = (3+4+5)/2 = 6

Area = √(6 × 3 × 2 × 1) = √36 = 6 sq units

h_a = 2 × 6 / 3 = 4.0000 h_b = 2 × 6 / 4 = 3.0000 h_c = 2 × 6 / 5 = 2.4000

h_c is the altitude to the hypotenuse: 3 × 4 / 5 = 12/5 = 2.4 confirms this

h_a = 4.0000 | h_b = 3.0000 | h_c = 2.4000

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Example 3 - Right triangle: leg 6, hypotenuse 10

Find the altitude to the hypotenuse of a right triangle with one leg 6 and hypotenuse 10.

Leg1 = 6, hypotenuse = 10. Leg2 = √(100 − 36) = √64 = 8

Altitude to hypotenuse = leg1 × leg2 / hyp = 6 × 8 / 10 = 48 / 10 = 4.8 units

Verify: Area = (1/2) × 6 × 8 = 24 = (1/2) × 10 × 4.8 = 24 ✓

Altitude to hypotenuse = 4.8000 units

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Example 4 - Three sides 5, 12, 13

Find all three altitudes of a 5-12-13 right triangle (all three sides known).

a = 5, b = 12, c = 13. s = (5+12+13)/2 = 15

Area = √(15 × 10 × 3 × 2) = √900 = 30 sq units

h_a = 60/5 = 12.0000 h_b = 60/12 = 5.0000 h_c = 60/13 = 4.6154

Note: 5 × 12 / 13 = 60/13 = 4.6154 confirms altitude to hypotenuse

h_a = 12.0000 | h_b = 5.0000 | h_c = 4.6154

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Frequently Asked Questions

How do I find the height of a triangle given the base and area?+

Rearrange the area formula: Area = (1/2) times base times height. Solving for height: h = 2 times Area divided by base. Example: base = 8, area = 24. h = 2 times 24 / 8 = 48 / 8 = 6 units.

How do I find all three altitudes of a triangle from its three sides?+

First use Heron's formula to find the area: s = (a+b+c)/2, Area = sqrt(s(s-a)(s-b)(s-c)). Then each altitude is h_a = 2A/a, h_b = 2A/b, h_c = 2A/c. Example: sides 3, 4, 5. s = 6, Area = sqrt(6 times 3 times 2 times 1) = sqrt(36) = 6. h_a = 12/3 = 4, h_b = 12/4 = 3, h_c = 12/5 = 2.4.

What is the altitude to the hypotenuse of a right triangle?+

For a right triangle with legs a and b and hypotenuse c, the altitude h from the right angle to the hypotenuse = (a times b) / c. Example: legs 3 and 4, hypotenuse 5. h = (3 times 4) / 5 = 12/5 = 2.4. If you only know one leg and the hypotenuse, find the other leg first: b = sqrt(c squared minus a squared).

What is the difference between a height and an altitude of a triangle?+

The terms height and altitude of a triangle mean the same thing: the perpendicular distance from a vertex to the opposite side (or its extension for obtuse triangles). Every triangle has three altitudes. The one most commonly called "the height" is the altitude to the base, used in Area = (1/2) times base times height.

Can a triangle height be outside the triangle?+

Yes. For an obtuse triangle (one angle greater than 90 degrees), two of the three altitudes fall outside the triangle. The foot of the altitude lands on the extension of the opposite side beyond the triangle. Only the altitude from the obtuse vertex's opposite side lands inside. For right and acute triangles, all three altitudes land inside.

How do I find the height of an equilateral triangle?+

For an equilateral triangle with side s, the height = s times sqrt(3) / 2 approximately 0.866 times s. This can be derived from the 30-60-90 triangle formed by the altitude. Example: equilateral triangle with side 10. Height = 10 times sqrt(3) / 2 = 10 times 0.866 = 8.660 units. Alternatively use this calculator with all three sides equal to s.

What is Heron's formula and why is it used here?+

Heron's formula finds the area of any triangle from its three side lengths without needing the height. It states: Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 is the semi-perimeter. Once we have the area, we can find all three heights: h_a = 2A/a. This calculator uses Heron's formula automatically when you enter three sides.

What is the relationship between the three altitudes of a triangle?+

The three altitudes of any triangle meet at a single point called the orthocenter. For an acute triangle, the orthocenter is inside the triangle. For a right triangle, it is at the right-angle vertex. For an obtuse triangle, it is outside the triangle. Also: the product of any altitude and its base equals twice the area: h_a times a = h_b times b = h_c times c = 2A.

How do I calculate the height of a right triangle if I know one leg and the hypotenuse?+

First find the missing leg: leg2 = sqrt(hypotenuse squared minus leg1 squared). Then the altitude to the hypotenuse = leg1 times leg2 / hypotenuse. Example: leg1 = 6, hypotenuse = 10. leg2 = sqrt(100 - 36) = sqrt(64) = 8. Altitude = 6 times 8 / 10 = 48/10 = 4.8 units.

What is the altitude of a triangle with sides 5, 12, 13?+

Sides a=5, b=12, c=13. s = (5+12+13)/2 = 15. Area = sqrt(15 times 10 times 3 times 2) = sqrt(900) = 30. h_a = 2 times 30 / 5 = 12. h_b = 2 times 30 / 12 = 5. h_c = 2 times 30 / 13 = 4.615. Note that this is a right triangle (5 squared + 12 squared = 169 = 13 squared), so h_c = leg1 times leg2 / hyp = 60/13 = 4.615 confirms this.

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📌 Quick Tips

💡Every triangle has three heights (altitudes), one from each vertex to the opposite side. This calculator finds all three when you enter all three side lengths.

💡The altitude from a vertex is the perpendicular distance to the opposite side. For obtuse triangles, two of the altitudes land outside the triangle on extensions of the sides.

💡In a right triangle, the altitude from the right angle to the hypotenuse equals (leg1 times leg2) divided by the hypotenuse. This is a key result in right triangle geometry.