Calculate Area of Triangle with 3 Sides

Triangle Area Calculator (Heron's Formula)

function calculateTriangleArea() { var sideA = parseFloat(document.getElementById('sideA').value); var sideB = parseFloat(document.getElementById('sideB').value); var sideC = parseFloat(document.getElementById('sideC').value); var resultDiv = document.getElementById('result'); if (isNaN(sideA) || isNaN(sideB) || isNaN(sideC) || sideA <= 0 || sideB <= 0 || sideC sideC) && (sideA + sideC > sideB) && (sideB + sideC > sideA))) { resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'Invalid triangle: The sum of any two sides must be greater than the third side.'; return; } // Heron's Formula var s = (sideA + sideB + sideC) / 2; // Semi-perimeter var areaSquared = s * (s – sideA) * (s – sideB) * (s – sideC); // Due to floating point inaccuracies, areaSquared might be slightly negative for valid triangles. // If it's very close to zero but negative, treat it as zero. if (areaSquared -1e-9) { // Check if it's a very small negative number areaSquared = 0; } if (areaSquared < 0) { // This case should ideally not happen for valid triangles, but as a safeguard resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.borderColor = '#f5c6cb'; resultDiv.style.color = '#721c24'; resultDiv.innerHTML = 'An error occurred during calculation. Please check your inputs.'; return; } var area = Math.sqrt(areaSquared); resultDiv.style.backgroundColor = '#e9f7ef'; resultDiv.style.borderColor = '#d4edda'; resultDiv.style.color = '#155724'; resultDiv.innerHTML = 'The area of the triangle is: ' + area.toFixed(4) + ' square units.'; }

Understanding the Area of a Triangle with Three Sides (Heron's Formula)

Calculating the area of a triangle is a fundamental concept in geometry. While the most common formula involves the base and height (Area = 0.5 * base * height), this isn't always practical if the height isn't readily known or easily measurable. This is where Heron's Formula becomes incredibly useful, allowing you to find the area of any triangle solely from the lengths of its three sides.

What is Heron's Formula?

Heron's Formula, named after Hero of Alexandria, is a powerful tool for determining the area of a triangle when only the lengths of its three sides (let's call them 'a', 'b', and 'c') are known. It bypasses the need to calculate angles or perpendicular heights, making it highly versatile.

The Formula Explained

Heron's Formula involves two main steps:

1. Calculate the Semi-Perimeter (s): The semi-perimeter is half the perimeter of the triangle.

   s = (a + b + c) / 2

2. Calculate the Area: Once you have the semi-perimeter, you can plug it into the main formula:

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

The Triangle Inequality Theorem

Before applying Heron's Formula, it's crucial to ensure that the given side lengths can actually form a triangle. This is governed by the Triangle Inequality Theorem, which states:

• The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In mathematical terms, for sides a, b, and c:

• a + b > c
• a + c > b
• b + c > a

If these conditions are not met, the given side lengths cannot form a valid triangle, and Heron's Formula will either yield an error or a non-real result (like the square root of a negative number).

How to Use the Calculator

Our Triangle Area Calculator simplifies this process for you:

1. Enter Side A Length: Input the length of the first side of your triangle into the "Side A Length" field.
2. Enter Side B Length: Input the length of the second side into the "Side B Length" field.
3. Enter Side C Length: Input the length of the third side into the "Side C Length" field.
4. Click "Calculate Area": The calculator will instantly apply Heron's Formula, perform the necessary checks (like the Triangle Inequality Theorem), and display the area of your triangle in square units.

Example Calculation

Let's say you have a triangle with the following side lengths:

• Side A = 7 units
• Side B = 8 units
• Side C = 9 units

Here's how Heron's Formula would be applied:

1. Check Triangle Inequality:
   • 7 + 8 > 9 (15 > 9) – True
   • 7 + 9 > 8 (16 > 8) – True
   • 8 + 9 > 7 (17 > 7) – True

   Since all conditions are met, this is a valid triangle.

2. Calculate Semi-Perimeter (s):

   s = (7 + 8 + 9) / 2 = 24 / 2 = 12

3. Calculate Area:

   Area = √(12 * (12 – 7) * (12 – 8) * (12 – 9))

   Area = √(12 * 5 * 4 * 3)

   Area = √(720)

   Area ≈ 26.8328 square units

Using the calculator with these values (7, 8, 9) will yield approximately 26.8328 square units, confirming the manual calculation.

This calculator is a handy tool for students, engineers, architects, or anyone needing to quickly determine the area of a triangle without relying on angles or heights.