How to Calculate Height of a Triangle

Triangle Height Calculator

Use the sections below to calculate the height of a triangle based on the information you have available.

Method 1: From Base and Area

If you know the base length and the total area of the triangle.

Method 2: From Side and Angle

If you know one side length (not the base) and the angle adjacent to the base.

function calculateHeightFromBaseArea() { var baseLength = parseFloat(document.getElementById('baseLength1').value); var triangleArea = parseFloat(document.getElementById('triangleArea1').value); var resultDiv = document.getElementById('resultHeight1'); if (isNaN(baseLength) || isNaN(triangleArea) || baseLength <= 0 || triangleArea <= 0) { resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Please enter valid positive numbers for Base Length and Area.'; return; } var height = (2 * triangleArea) / baseLength; resultDiv.style.backgroundColor = '#e9f7ef'; resultDiv.style.borderColor = '#d4edda'; resultDiv.style.color = '#155724'; resultDiv.innerHTML = 'The height of the triangle is: ' + height.toFixed(4) + ' units'; } function calculateHeightFromSideAngle() { var sideLength = parseFloat(document.getElementById('sideLength2').value); var angleDegrees = parseFloat(document.getElementById('angleDegrees2').value); var resultDiv = document.getElementById('resultHeight2'); if (isNaN(sideLength) || isNaN(angleDegrees) || sideLength <= 0 || angleDegrees = 180) { resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Please enter valid positive numbers for Side Length and an Angle between 0 and 180 degrees.'; return; } var angleRadians = angleDegrees * (Math.PI / 180); var height = sideLength * Math.sin(angleRadians); resultDiv.style.backgroundColor = '#e9f7ef'; resultDiv.style.borderColor = '#d4edda'; resultDiv.style.color = '#155724'; resultDiv.innerHTML = 'The height of the triangle is: ' + height.toFixed(4) + ' units'; }

The height of a triangle, also known as its altitude, is a fundamental measurement in geometry. It represents the perpendicular distance from a vertex to the opposite side (or its extension), which is called the base. Every triangle has three heights, one for each side chosen as the base. Understanding how to calculate the height is crucial for various geometric problems, especially for finding the area of a triangle.

What is Triangle Height (Altitude)?

Imagine a triangle standing on one of its sides. The height is the straight line drawn from the topmost point (vertex) directly down to that base, forming a 90-degree angle with the base. This line might fall inside the triangle, on one of its vertices (in a right-angled triangle), or even outside the triangle (in an obtuse triangle, where the base needs to be extended).

Methods to Calculate Triangle Height

Method 1: Using Base and Area

This is perhaps the most common and straightforward method if you already know the triangle's area and the length of its base. The formula for the area of a triangle is:

Area = (1/2) × Base × Height

From this, we can rearrange the formula to solve for the height:

Height = (2 × Area) / Base

Example 1:

Suppose a triangle has an area of 25 square units and a base length of 10 units.

• Area = 25
• Base = 10
• Height = (2 × 25) / 10 = 50 / 10 = 5 units

You can use the "Method 1: From Base and Area" section of the calculator above to verify this.

Method 2: Using a Side and an Adjacent Angle

If you know the length of one side (let's call it 'a') and the measure of an angle (let's call it 'C') that is adjacent to the base you're considering, you can use trigonometry to find the height. For a triangle ABC, if you want the height relative to base 'b' (side AC), and you know side 'a' (BC) and angle 'C', the height (h) can be calculated as:

Height = Side × sin(Angle)

Where 'Side' is the length of the side adjacent to the base, and 'Angle' is the angle between that side and the base.

Example 2:

Consider a triangle where one side (not the base) is 8 units long, and the angle between this side and the base is 30 degrees.

• Side Length = 8
• Angle = 30 degrees
• Height = 8 × sin(30°) = 8 × 0.5 = 4 units

Use the "Method 2: From Side and Angle" section of the calculator above to confirm this result.

Method 3: Using All Three Sides (Heron's Formula)

If you only know the lengths of all three sides (a, b, c) of a triangle, you can first calculate its area using Heron's formula, and then use Method 1 to find the height.

Steps:

1. Calculate the semi-perimeter (s):

   s = (a + b + c) / 2

2. Calculate the Area using Heron's Formula:

   Area = √(s × (s – a) × (s – b) × (s – c))

3. Calculate the Height: Once you have the area, choose one of the sides as the base (e.g., side 'a') and apply the formula from Method 1:

   Height_a = (2 × Area) / a

   (Similarly for Height_b and Height_c)

Example 3:

Consider a triangle with sides a=3, b=4, and c=5 (a right-angled triangle).

• Semi-perimeter (s) = (3 + 4 + 5) / 2 = 12 / 2 = 6
• Area = √(6 × (6 – 3) × (6 – 4) × (6 – 5)) = √(6 × 3 × 2 × 1) = √36 = 6 square units
• Now, let's find the height relative to base 'a' (length 3): Height_a = (2 × 6) / 3 = 12 / 3 = 4 units
• Height relative to base 'b' (length 4): Height_b = (2 × 6) / 4 = 12 / 4 = 3 units
• Height relative to base 'c' (length 5): Height_c = (2 × 6) / 5 = 12 / 5 = 2.4 units

Special Cases

• Right-Angled Triangle: If you choose one of the legs as the base, the other leg is its corresponding height. For example, if the legs are 3 and 4, and the hypotenuse is 5, then if you take the base as 3, the height is 4. If you take the base as 4, the height is 3. If you take the hypotenuse (5) as the base, you'd use Method 3 (as shown in Example 3) or Method 2 if you know an angle.
• Equilateral Triangle: All sides are equal (s). The height (h) can be calculated as:

  h = (s × √3) / 2

• Isosceles Triangle: Two sides are equal. The height drawn to the unequal base bisects that base and the angle at the apex.

Conclusion

Calculating the height of a triangle is a fundamental skill in geometry. Whether you have the area and base, a side and an angle, or all three sides, there's a method to determine this crucial dimension. The calculator above provides a quick way to find the height using the most common input scenarios.