Find the Area of a Triangle Calculator

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Triangle Area Calculator

Base and Height Three Sides (Heron's Formula) Side-Angle-Side (SAS)
Resulting Area:
0

How to Find the Area of a Triangle

Finding the area of a triangle is a fundamental skill in geometry. Depending on the information you have available (base and height, all three sides, or two sides and an angle), different formulas are applied to reach the same conclusion.

1. Using Base and Height

This is the most common method. If you know the length of the base and the perpendicular height (altitude) of the triangle, use the formula:

Area = ½ × Base × Height

Example: If a triangle has a base of 10 cm and a height of 5 cm, the area is 0.5 × 10 × 5 = 25 cm².

2. Heron's Formula (Three Sides)

When you know the lengths of all three sides (a, b, and c) but not the height, you use Heron's Formula. First, calculate the semi-perimeter (s):

s = (a + b + c) / 2

Then, apply the area formula:

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

3. Side-Angle-Side (SAS) Method

If you have two sides and the measure of the angle between them, trigonometry makes the calculation easy:

Area = ½ × a × b × sin(C)

This is particularly useful for scalene triangles where measuring the vertical height is difficult.

Triangle Inequality Theorem

Remember that for a triangle to exist, the sum of any two sides must be strictly greater than the length of the third side. If your measurements don't satisfy this rule (e.g., sides of 2, 2, and 10), it is geometrically impossible to form a triangle.

function switchMethod() { var method = document.getElementById("calcMethod").value; document.getElementById("method-bh").style.display = (method === "bh") ? "block" : "none"; document.getElementById("method-heron").style.display = (method === "heron") ? "block" : "none"; document.getElementById("method-sas").style.display = (method === "sas") ? "block" : "none"; document.getElementById("tri-result").style.display = "none"; document.getElementById("error-msg").style.display = "none"; } function calculateTriangleArea() { var method = document.getElementById("calcMethod").value; var resultDiv = document.getElementById("tri-result"); var areaOutput = document.getElementById("area-output"); var perimeterOutput = document.getElementById("perimeter-output"); var errorDiv = document.getElementById("error-msg"); var area = 0; var perimeter = ""; errorDiv.style.display = "none"; resultDiv.style.display = "none"; if (method === "bh") { var b = parseFloat(document.getElementById("base").value); var h = parseFloat(document.getElementById("height").value); if (isNaN(b) || isNaN(h) || b <= 0 || h <= 0) { showError("Please enter valid positive numbers for base and height."); return; } area = 0.5 * b * h; perimeter = "Perimeter cannot be calculated with only base and height."; } else if (method === "heron") { var a = parseFloat(document.getElementById("sideA").value); var b = parseFloat(document.getElementById("sideB").value); var c = parseFloat(document.getElementById("sideC").value); if (isNaN(a) || isNaN(b) || isNaN(c) || a <= 0 || b <= 0 || c <= 0) { showError("Please enter valid positive numbers for all sides."); return; } if ((a + b <= c) || (a + c <= b) || (b + c <= a)) { showError("Triangle Inequality Violation: The sum of any two sides must be greater than the third side."); return; } var s = (a + b + c) / 2; area = Math.sqrt(s * (s – a) * (s – b) * (s – c)); perimeter = "Perimeter: " + (a + b + c).toFixed(2); } else if (method === "sas") { var s1 = parseFloat(document.getElementById("sasSideA").value); var s2 = parseFloat(document.getElementById("sasSideB").value); var ang = parseFloat(document.getElementById("angle").value); if (isNaN(s1) || isNaN(s2) || isNaN(ang) || s1 <= 0 || s2 <= 0 || ang = 180) { showError("Please enter valid dimensions. Angle must be between 0 and 180 degrees."); return; } var radians = ang * (Math.PI / 180); area = 0.5 * s1 * s2 * Math.sin(radians); // Law of Cosines for third side to get perimeter var s3 = Math.sqrt(Math.pow(s1, 2) + Math.pow(s2, 2) – (2 * s1 * s2 * Math.cos(radians))); perimeter = "Perimeter: " + (s1 + s2 + s3).toFixed(2); } areaOutput.innerHTML = area.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}) + " units²"; perimeterOutput.innerHTML = perimeter; resultDiv.style.display = "block"; } function showError(msg) { var errorDiv = document.getElementById("error-msg"); errorDiv.innerHTML = msg; errorDiv.style.display = "block"; }

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