How Do We Calculate the Area of a Triangle

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Triangle Area Calculator

Base and Height (Standard) Three Sides (Heron's Formula) Two Sides and Included Angle (SAS)

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How to Calculate the Area of a Triangle

Calculating the area of a triangle is a fundamental skill in geometry. Depending on the information you have available (sides, heights, or angles), there are three primary methods used to find the area.

1. Using Base and Height

This is the most common method. If you know the length of the bottom (base) and the vertical distance from the base to the opposite peak (height), the formula is:

Area = (Base × Height) / 2

Example: If a triangle has a base of 10 cm and a height of 5 cm, the area is (10 × 5) / 2 = 25 cm².

2. Using Three Sides (Heron's Formula)

If you know all three side lengths (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, the area is calculated as:

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

3. Side-Angle-Side (SAS) Method

If you know two side lengths and the angle between them, you can use trigonometry:

Area = 0.5 × a × b × sin(Angle)

This is particularly useful when the vertical height is difficult to measure directly.

Frequently Asked Questions

  • Does the orientation matter? No, any side can be the base as long as the height is measured perpendicular to that base.
  • What units should I use? Always ensure all measurements are in the same unit (e.g., all cm or all inches). The result will be in square units.
  • Can Heron's formula fail? Yes, if the sum of any two sides is not greater than the third side, a triangle cannot exist, and the calculation will result in an error.
function switchMethod() { var method = document.getElementById("calcMethod").value; document.getElementById("methodBaseHeight").style.display = (method === "baseHeight") ? "block" : "none"; document.getElementById("methodThreeSides").style.display = (method === "threeSides") ? "block" : "none"; document.getElementById("methodSAS").style.display = (method === "sas") ? "block" : "none"; document.getElementById("triangleResult").style.display = "none"; } function calculateTriangleArea() { var method = document.getElementById("calcMethod").value; var area = 0; var formulaUsed = ""; var isValid = true; if (method === "baseHeight") { var b = parseFloat(document.getElementById("baseInput").value); var h = parseFloat(document.getElementById("heightInput").value); if (isNaN(b) || isNaN(h) || b <= 0 || h <= 0) { alert("Please enter valid positive numbers for base and height."); isValid = false; } else { area = 0.5 * b * h; formulaUsed = "Formula: (Base × Height) / 2 = (" + b + " × " + h + ") / 2"; } } else if (method === "threeSides") { 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) { alert("Please enter valid positive numbers for all three sides."); isValid = false; } else if ((a + b <= c) || (a + c <= b) || (b + c <= a)) { alert("The sum of any two sides must be greater than the third side (Triangle Inequality Theorem)."); isValid = false; } else { var s = (a + b + c) / 2; area = Math.sqrt(s * (s – a) * (s – b) * (s – c)); formulaUsed = "Heron's Formula: s = " + s.toFixed(2) + "; Area = √(" + s.toFixed(2) + "(" + s.toFixed(2) + "-" + a + ")(" + s.toFixed(2) + "-" + b + ")(" + s.toFixed(2) + "-" + c + "))"; } } else if (method === "sas") { var s1 = parseFloat(document.getElementById("sasSideA").value); var s2 = parseFloat(document.getElementById("sasSideB").value); var angle = parseFloat(document.getElementById("sasAngle").value); if (isNaN(s1) || isNaN(s2) || isNaN(angle) || s1 <= 0 || s2 <= 0 || angle = 180) { alert("Please enter valid numbers. The angle must be between 0 and 180 degrees."); isValid = false; } else { var rad = angle * (Math.PI / 180); area = 0.5 * s1 * s2 * Math.sin(rad); formulaUsed = "SAS Formula: 0.5 × " + s1 + " × " + s2 + " × sin(" + angle + "°)"; } } if (isValid) { document.getElementById("resultText").innerText = "Area = " + area.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}) + " square units"; document.getElementById("resultFormula").innerText = formulaUsed; document.getElementById("triangleResult").style.display = "block"; } }

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