Area of Triangle with Three Sides Calculator

Area of Triangle with Three Sides Calculator

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.innerHTML = "These side lengths do not form a valid triangle (Triangle Inequality Theorem violated)."; 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 degenerate triangles. // Ensure it's not negative before taking square root. if (areaSquared < 0) { areaSquared = 0; // Treat as a degenerate triangle with zero area } var area = Math.sqrt(areaSquared); resultDiv.innerHTML = "The area of the triangle is: " + area.toFixed(4) + " square units."; }

Understanding the Area of a Triangle with Three Sides

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), what if you only know the lengths of the three sides and not the height? This is where Heron's Formula comes to the rescue.

What is Heron's Formula?

Heron's Formula, named after Hero of Alexandria, provides a way to calculate the area of a triangle when the lengths of all three sides are known. It's particularly useful when you cannot easily determine the height of the triangle.

The formula is as follows:

Area = √[s(s – a)(s – b)(s – c)]

Where:

• a, b, and c are the lengths of the three sides of the triangle.
• s is the semi-perimeter of the triangle, calculated as s = (a + b + c) / 2.

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:

• 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 the area would be zero (or undefined in a practical sense).

How to Use the Calculator

Our Area of Triangle with Three Sides 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 and display the area of your triangle in square units.

The calculator also includes a check for the Triangle Inequality Theorem, ensuring that the side lengths you provide can indeed form a real triangle.

Examples

Let's look at a few examples to illustrate Heron's Formula:

Example 1: A Right-Angled Triangle

Consider a triangle with sides 3, 4, and 5 units. This is a classic right-angled triangle.

• Side A = 3
• Side B = 4
• Side C = 5

Semi-perimeter (s) = (3 + 4 + 5) / 2 = 12 / 2 = 6

Area = √[6 * (6 – 3) * (6 – 4) * (6 – 5)]

Area = √[6 * 3 * 2 * 1]

Area = √[36]

Area = 6 square units

(You can test these values in the calculator above.)

Example 2: An Equilateral Triangle

Consider an equilateral triangle with all sides equal to 7 units.

• Side A = 7
• Side B = 7
• Side C = 7

Semi-perimeter (s) = (7 + 7 + 7) / 2 = 21 / 2 = 10.5

Area = √[10.5 * (10.5 – 7) * (10.5 – 7) * (10.5 – 7)]

Area = √[10.5 * 3.5 * 3.5 * 3.5]

Area = √[450.09375]

Area ≈ 21.2176 square units

Example 3: An Invalid Triangle

Consider sides 1, 2, and 5 units.

• Side A = 1
• Side B = 2
• Side C = 5

Here, 1 + 2 = 3, which is NOT greater than 5. This violates the Triangle Inequality Theorem. The calculator will correctly identify that these sides do not form a valid triangle.

Whether you're a student, an engineer, or just curious, this calculator provides a quick and accurate way to find the area of any triangle given its three side lengths.