Please enter at least three relevant properties (sides and/or angles) to begin.
Understanding Triangle Calculations
Triangles are fundamental geometric shapes with three sides and three angles. Calculating their properties like area, perimeter, and missing sides or angles is crucial in various fields, including construction, navigation, engineering, and even art. This calculator helps you determine various properties of a triangle based on the information you provide.
Key Triangle Properties and Formulas:
Perimeter: The total length of all three sides.
Perimeter = sideA + sideB + sideC
Area (using two sides and the included angle – SAS): This is useful when you know two sides and the angle between them.
Area = 0.5 * sideA * sideB * sin(angleC)
(and its permutations for other side/angle pairs)
Area (using Heron's Formula – SSS): This is used when you know the lengths of all three sides.
First, calculate the semi-perimeter (s):
s = (sideA + sideB + sideC) / 2
Then, calculate the area:
Area = sqrt(s * (s – sideA) * (s – sideB) * (s – sideC))
Law of Sines: Relates the lengths of the sides of a triangle to the sines of its opposite angles. It's useful for finding missing sides or angles when you have certain combinations of known values.
sideA / sin(angleA) = sideB / sin(angleB) = sideC / sin(angleC)
Law of Cosines: Relates the lengths of the sides of a triangle to the cosine of one of its angles. It's useful for finding a missing side when you know two sides and the included angle (SAS), or for finding a missing angle when you know all three sides (SSS).
sideC² = sideA² + sideB² – 2 * sideA * sideB * cos(angleC)
(and its permutations)
How to Use This Calculator:
Enter at least three known properties of the triangle (sides and/or angles). The calculator can then attempt to find other missing properties.
To calculate the Area using SAS, provide two sides and the angle *between* them.
To calculate the Area using SSS, provide all three sides.
To calculate All Properties, provide enough information to uniquely define the triangle (e.g., three sides; two sides and an angle; one side and two angles; two angles and a side). The calculator will attempt to solve for the rest.
Note: Ensure the input values form a valid triangle (e.g., the sum of any two sides must be greater than the third side, and the sum of angles must be 180 degrees). This calculator performs basic validation but may not catch all geometric impossibilities. Angles should be entered in degrees.
// Helper function to convert degrees to radians
function degreesToRadians(degrees) {
return degrees * Math.PI / 180;
}
// Helper function to check if a number is valid
function isValidNumber(num) {
return typeof num === 'number' && isFinite(num) && num >= 0;
}
// Helper function to check triangle inequality
function checkTriangleInequality(a, b, c) {
return a + b > c && a + c > b && b + c > a;
}
// Helper function to check angle sum
function checkAngleSum(a, b, c) {
// Allow for small floating point errors
return Math.abs(a + b + c – 180) < 0.001;
}
function calculateTriangleProperties() {
var sideA = parseFloat(document.getElementById('sideA').value);
var sideB = parseFloat(document.getElementById('sideB').value);
var sideC = parseFloat(document.getElementById('sideC').value);
var angleA = parseFloat(document.getElementById('angleA').value);
var angleB = parseFloat(document.getElementById('angleB').value);
var angleC = parseFloat(document.getElementById('angleC').value);
var resultDiv = document.getElementById('result');
var message = "";
var calculatedProperties = {};
// Convert angles to radians for calculations
var angleARad = isValidNumber(angleA) ? degreesToRadians(angleA) : null;
var angleBRad = isValidNumber(angleB) ? degreesToRadians(angleB) : null;
var angleCRad = isValidNumber(angleC) ? degreesToRadians(angleC) : null;
var inputsCount = 0;
if (isValidNumber(sideA)) inputsCount++;
if (isValidNumber(sideB)) inputsCount++;
if (isValidNumber(sideC)) inputsCount++;
if (isValidNumber(angleA)) inputsCount++;
if (isValidNumber(angleB)) inputsCount++;
if (isValidNumber(angleC)) inputsCount++;
// Basic validation: Need at least 3 properties to solve most triangles
if (inputsCount < 3) {
resultDiv.innerHTML = "Please enter at least 3 properties (sides or angles).";
return;
}
// — Attempt to solve for missing properties —
// Case 1: SSS (3 sides known)
if (isValidNumber(sideA) && isValidNumber(sideB) && isValidNumber(sideC)) {
if (!checkTriangleInequality(sideA, sideB, sideC)) {
resultDiv.innerHTML = "Error: These side lengths do not form a valid triangle.";
return;
}
// Calculate angles using Law of Cosines
var cosAngleA = (sideB * sideB + sideC * sideC – sideA * sideA) / (2 * sideB * sideC);
var cosAngleB = (sideA * sideA + sideC * sideC – sideB * sideB) / (2 * sideA * sideC);
var cosAngleC = (sideA * sideA + sideB * sideB – sideC * sideC) / (2 * sideA * sideB);
if (Math.abs(cosAngleA) <= 1) angleA = Math.acos(cosAngleA) * 180 / Math.PI; else angleA = NaN;
if (Math.abs(cosAngleB) <= 1) angleB = Math.acos(cosAngleB) * 180 / Math.PI; else angleB = NaN;
if (Math.abs(cosAngleC) 0.001) {
// Try to correct the largest angle based on the sum, or simply state the issue
if (Math.abs(currentSum – 180) < 1) { // Small correction tolerance
if (angleA === Math.max(angleA, angleB, angleC)) angleA += (180 – currentSum);
else if (angleB === Math.max(angleA, angleB, angleC)) angleB += (180 – currentSum);
else angleC += (180 – currentSum);
} else {
// This might indicate an issue with the input or a degenerate triangle
message += "Warning: Calculated angles do not sum precisely to 180 degrees. ";
}
}
}
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6; // All are now known or calculated
}
// Case 2: ASA or AAS (1 side, 2 angles known)
else if (isValidNumber(sideA) && isValidNumber(angleB) && isValidNumber(angleC)) {
angleA = 180 – angleB – angleC;
if (angleA <= 0) { resultDiv.innerHTML = "Error: Invalid angles provided (sum must be 180)."; return; }
angleARad = degreesToRadians(angleA);
angleBRad = degreesToRadians(angleB);
angleCRad = degreesToRadians(angleC);
sideB = sideA * Math.sin(angleBRad) / Math.sin(angleARad);
sideC = sideA * Math.sin(angleCRad) / Math.sin(angleARad);
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6;
} else if (isValidNumber(sideB) && isValidNumber(angleA) && isValidNumber(angleC)) {
angleB = 180 – angleA – angleC;
if (angleB <= 0) { resultDiv.innerHTML = "Error: Invalid angles provided (sum must be 180)."; return; }
angleARad = degreesToRadians(angleA);
angleBRad = degreesToRadians(angleB);
angleCRad = degreesToRadians(angleC);
sideA = sideB * Math.sin(angleARad) / Math.sin(angleBRad);
sideC = sideB * Math.sin(angleCRad) / Math.sin(angleBRad);
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6;
} else if (isValidNumber(sideC) && isValidNumber(angleA) && isValidNumber(angleB)) {
angleC = 180 – angleA – angleB;
if (angleC <= 0) { resultDiv.innerHTML = "Error: Invalid angles provided (sum must be 180)."; return; }
angleARad = degreesToRadians(angleA);
angleBRad = degreesToRadians(angleB);
angleCRad = degreesToRadians(angleC);
sideA = sideC * Math.sin(angleARad) / Math.sin(angleCRad);
sideB = sideC * Math.sin(angleBRad) / Math.sin(angleCRad);
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6;
}
// Case 3: SAS (2 sides, included angle known)
else if (isValidNumber(sideA) && isValidNumber(sideB) && isValidNumber(angleC)) {
angleCRad = degreesToRadians(angleC);
sideC = Math.sqrt(sideA * sideA + sideB * sideB – 2 * sideA * sideB * Math.cos(angleCRad));
// Now we have SSS, calculate angles
var cosAngleA = (sideB * sideB + sideC * sideC – sideA * sideA) / (2 * sideB * sideC);
var cosAngleB = (sideA * sideA + sideC * sideC – sideB * sideB) / (2 * sideA * sideC);
if (Math.abs(cosAngleA) <= 1) angleA = Math.acos(cosAngleA) * 180 / Math.PI; else angleA = NaN;
if (Math.abs(cosAngleB) <= 1) angleB = Math.acos(cosAngleB) * 180 / Math.PI; else angleB = NaN;
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6;
} else if (isValidNumber(sideB) && isValidNumber(sideC) && isValidNumber(angleA)) {
angleARad = degreesToRadians(angleA);
sideA = Math.sqrt(sideB * sideB + sideC * sideC – 2 * sideB * sideC * Math.cos(angleARad));
var cosAngleB = (sideA * sideA + sideC * sideC – sideB * sideB) / (2 * sideA * sideC);
var cosAngleC = (sideA * sideA + sideB * sideB – sideC * sideC) / (2 * sideA * sideB);
if (Math.abs(cosAngleB) <= 1) angleB = Math.acos(cosAngleB) * 180 / Math.PI; else angleB = NaN;
if (Math.abs(cosAngleC) <= 1) angleC = Math.acos(cosAngleC) * 180 / Math.PI; else angleC = NaN;
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6;
} else if (isValidNumber(sideA) && isValidNumber(sideC) && isValidNumber(angleB)) {
angleBRad = degreesToRadians(angleB);
sideB = Math.sqrt(sideA * sideA + sideC * sideC – 2 * sideA * sideC * Math.cos(angleBRad));
var cosAngleA = (sideB * sideB + sideC * sideC – sideA * sideA) / (2 * sideB * sideC);
var cosAngleC = (sideA * sideA + sideB * sideB – sideC * sideC) / (2 * sideA * sideB);
if (Math.abs(cosAngleA) <= 1) angleA = Math.acos(cosAngleA) * 180 / Math.PI; else angleA = NaN;
if (Math.abs(cosAngleC) 1) { resultDiv.innerHTML = "Error: No triangle possible with these SSA inputs."; return; }
angleB = Math.asin(sinAngleB) * 180 / Math.PI;
// Check for the ambiguous case (two possible triangles)
var angleB2 = 180 – angleB;
var angleC = 180 – angleA – angleB;
var angleC2 = 180 – angleA – angleB2;
// We'll just calculate one solution for simplicity in this generic calculator
if (angleC > 0) {
angleC = 180 – angleA – angleB;
angleCRad = degreesToRadians(angleC);
sideC = sideA * Math.sin(angleCRad) / Math.sin(angleARad);
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6;
message += "Note: SSA case can sometimes yield two solutions. This calculation shows one possible solution. ";
} else {
resultDiv.innerHTML = "Error: Invalid angles resulting from SSA input."; return;
}
} // Add permutations for SSA… e.g. sideA, sideC, angleA etc.
else if (isValidNumber(sideA) && isValidNumber(sideC) && isValidNumber(angleA)) {
angleARad = degreesToRadians(angleA);
var sinAngleC = (sideC * Math.sin(angleARad)) / sideA;
if (Math.abs(sinAngleC) > 1) { resultDiv.innerHTML = "Error: No triangle possible with these SSA inputs."; return; }
angleC = Math.asin(sinAngleC) * 180 / Math.PI;
var angleC2 = 180 – angleC;
var angleB = 180 – angleA – angleC;
var angleB2 = 180 – angleA – angleC2;
if (angleB > 0) {
angleB = 180 – angleA – angleC;
angleBRad = degreesToRadians(angleB);
sideB = sideA * Math.sin(angleBRad) / Math.sin(angleARad);
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6;
message += "Note: SSA case can sometimes yield two solutions. This calculation shows one possible solution. ";
} else {
resultDiv.innerHTML = "Error: Invalid angles resulting from SSA input."; return;
}
} else if (isValidNumber(sideB) && isValidNumber(sideA) && isValidNumber(angleB)) {
angleBRad = degreesToRadians(angleB);
var sinAngleA = (sideA * Math.sin(angleBRad)) / sideB;
if (Math.abs(sinAngleA) > 1) { resultDiv.innerHTML = "Error: No triangle possible with these SSA inputs."; return; }
angleA = Math.asin(sinAngleA) * 180 / Math.PI;
var angleA2 = 180 – angleA;
var angleC = 180 – angleA – angleB;
var angleC2 = 180 – angleA2 – angleB;
if (angleC > 0) {
angleC = 180 – angleA – angleB;
angleCRad = degreesToRadians(angleC);
sideC = sideB * Math.sin(angleCRad) / Math.sin(angleBRad);
calculatedProperties = { sideA: sideA, sideB: sideB, sideC: sideC, angleA: angleA, angleB: angleB, angleC: angleC };
inputsCount = 6;
message += "Note: SSA case can sometimes yield two solutions. This calculation shows one possible solution. ";
} else {
resultDiv.innerHTML = "Error: Invalid angles resulting from SSA input."; return;
}
} // … and so on for other SSA permutations
// If we successfully calculated all properties
if (inputsCount === 6 && Object.keys(calculatedProperties).length === 6) {
var finalSideA = calculatedProperties.sideA;
var finalSideB = calculatedProperties.sideB;
var finalSideC = calculatedProperties.sideC;
var finalAngleA = calculatedProperties.angleA;
var finalAngleB = calculatedProperties.angleB;
var finalAngleC = calculatedProperties.angleC;
var perimeter = finalSideA + finalSideB + finalSideC;
var areaSAS = NaN; // Calculated if possible
var areaSSS = NaN; // Calculated if possible
// Try calculating area using SAS if applicable
if (isValidNumber(finalSideA) && isValidNumber(finalSideB) && isValidNumber(finalAngleC)) {
areaSAS = 0.5 * finalSideA * finalSideB * Math.sin(degreesToRadians(finalAngleC));
} else if (isValidNumber(finalSideB) && isValidNumber(finalSideC) && isValidNumber(finalAngleA)) {
areaSAS = 0.5 * finalSideB * finalSideC * Math.sin(degreesToRadians(finalAngleA));
} else if (isValidNumber(finalSideA) && isValidNumber(finalSideC) && isValidNumber(finalAngleB)) {
areaSAS = 0.5 * finalSideA * finalSideC * Math.sin(degreesToRadians(finalAngleB));
}
// Try calculating area using SSS (Heron's Formula) if applicable
if (isValidNumber(finalSideA) && isValidNumber(finalSideB) && isValidNumber(finalSideC)) {
var s = perimeter / 2;
var areaValue = s * (s – finalSideA) * (s – finalSideB) * (s – finalSideC);
if (areaValue >= 0) { // Ensure no negative under sqrt due to floating point errors
areaSSS = Math.sqrt(areaValue);
}
}
// Display results, prioritizing SSS area if calculated, otherwise SAS
var displayArea = areaSSS;
if (isNaN(displayArea) && !isNaN(areaSAS)) {
displayArea = areaSAS;
} else if (isNaN(displayArea) && isNaN(areaSAS)) {
displayArea = "N/A (Insufficient data)";
}
// Format numbers to a reasonable precision
var formatNum = function(num) {
if (typeof num === 'number' && isFinite(num)) {
return num.toFixed(3);
}
return "N/A";
};
// Check angle sum validity again after all calculations
var angleSumValid = checkAngleSum(finalAngleA, finalAngleB, finalAngleC);
if (!angleSumValid) {
message += "Warning: Final calculated angles do not sum precisely to 180 degrees. ";
}
resultDiv.innerHTML = message +
"
Calculated Triangle Properties:
" +
"Perimeter: " + formatNum(perimeter) + " units" +
"Area: " + formatNum(displayArea) + " square units" +
"Side A: " + formatNum(finalSideA) + " | Side B: " + formatNum(finalSideB) + " | Side C: " + formatNum(finalSideC) + "" +
"Angle A: " + formatNum(finalAngleA) + "° | Angle B: " + formatNum(finalAngleB) + "° | Angle C: " + formatNum(finalAngleC) + "°";
} else {
// Not enough info to calculate all, or an error occurred earlier
if (message.indexOf("Error:") === -1 && message.indexOf("Warning:") === -1) {
resultDiv.innerHTML = "Insufficient data to calculate all properties. Please provide more information.";
} else {
// Display error or warning if already set
resultDiv.innerHTML = message;
}
}
}
function calculateAreaSAS() {
var sideA = parseFloat(document.getElementById('sideA').value);
var sideB = parseFloat(document.getElementById('sideB').value);
var sideC = parseFloat(document.getElementById('sideC').value);
var angleA = parseFloat(document.getElementById('angleA').value);
var angleB = parseFloat(document.getElementById('angleB').value);
var angleC = parseFloat(document.getElementById('angleC').value);
var resultDiv = document.getElementById('result');
var area = NaN;
var message = "";
if (isValidNumber(sideA) && isValidNumber(sideB) && isValidNumber(angleC)) {
area = 0.5 * sideA * sideB * Math.sin(degreesToRadians(angleC));
message = "Area (SAS): " + area.toFixed(3) + " square units";
} else if (isValidNumber(sideB) && isValidNumber(sideC) && isValidNumber(angleA)) {
area = 0.5 * sideB * sideC * Math.sin(degreesToRadians(angleA));
message = "Area (SAS): " + area.toFixed(3) + " square units";
} else if (isValidNumber(sideA) && isValidNumber(sideC) && isValidNumber(angleB)) {
area = 0.5 * sideA * sideC * Math.sin(degreesToRadians(angleB));
message = "Area (SAS): " + area.toFixed(3) + " square units";
} else {
message = "Please provide two sides and the angle between them (SAS) to calculate area.";
}
resultDiv.innerHTML = message;
}
function calculateAreaSSS() {
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');
var message = "";
if (isValidNumber(sideA) && isValidNumber(sideB) && isValidNumber(sideC)) {
if (!checkTriangleInequality(sideA, sideB, sideC)) {
resultDiv.innerHTML = "Error: These side lengths do not form a valid triangle.";
return;
}
var s = (sideA + sideB + sideC) / 2;
var areaValue = s * (s – sideA) * (s – sideB) * (s – sideC);
if (areaValue >= 0) {
var area = Math.sqrt(areaValue);
message = "Area (SSS): " + area.toFixed(3) + " square units";
} else {
message = "Error calculating area. Check side lengths (Heron's formula requires non-negative value under sqrt).";
}
} else {
message = "Please provide all three side lengths (SSS) to calculate area.";
}
resultDiv.innerHTML = message;
}