Enter two sides and one angle, or two angles, to calculate the remaining angles and sides of a triangle.
Understanding Triangle Angles and Sides
Triangles are fundamental geometric shapes, defined by three sides and three interior angles. The sum of the interior angles in any Euclidean triangle is always 180 degrees. This principle is crucial for solving various geometry problems, from basic constructions to advanced engineering and physics calculations.
Key Principles:
Sum of Angles: Angle A + Angle B + Angle C = 180°.
Law of Sines: This law relates the lengths of the sides of a triangle to the sines of its opposite angles. It states that for any triangle with sides a, b, c and opposite angles A, B, C respectively:
a / sin(A) = b / sin(B) = c / sin(C)
Law of Cosines: This law relates the lengths of the sides of a triangle to the cosine of one of its angles. It's useful for finding sides or angles when the Law of Sines isn't directly applicable.
a² = b² + c² - 2bc * cos(A)b² = a² + c² - 2ac * cos(B)c² = a² + b² - 2ab * cos(C)
How This Calculator Works:
This calculator uses the fundamental properties of triangles and the Law of Sines/Cosines to determine unknown angles and sides. You can input known values (sides and/or angles) to solve for the unknowns. The calculator attempts to find a valid solution based on the trigonometric principles. Note that not all combinations of side and angle inputs will result in a valid triangle (e.g., violating the triangle inequality theorem or angle sum property).
Common Scenarios:
Angle-Angle-Side (AAS): If you know two angles and a non-included side, you can find the third angle and the other two sides.
Angle-Side-Angle (ASA): If you know two angles and the included side, you can find the third angle and the other two sides.
Side-Side-Side (SSS): If you know all three sides, you can find all three angles using the Law of Cosines.
Side-Side-Angle (SSA): This case can sometimes lead to two possible triangles (ambiguous case), which this calculator might not fully resolve without additional context.
Side-Angle-Side (SAS): If you know two sides and the included angle, you can find the third side using the Law of Cosines and then use the Law of Sines or Cosines to find the remaining angles.
Important Note: Ensure your inputs represent a geometrically possible triangle. The calculator includes basic validation to prevent nonsensical results, but complex cases or invalid inputs may yield errors or ambiguous results.
function calculateTriangleAngles() {
var sideA = parseFloat(document.getElementById('sideA').value);
var sideB = parseFloat(document.getElementById('sideB').value);
var sideC = parseFloat(document.getElementById('sideC').value);
var angleA_deg = parseFloat(document.getElementById('angleA').value);
var angleB_deg = parseFloat(document.getElementById('angleB').value);
var angleC_deg = parseFloat(document.getElementById('angleC').value);
var resultDiv = document.getElementById('result');
resultDiv.innerHTML = "; // Clear previous results
// Helper function to convert degrees to radians
function degToRad(degrees) {
return degrees * Math.PI / 180;
}
// Helper function to convert radians to degrees
function radToDeg(radians) {
return radians * 180 / Math.PI;
}
// — Input Validation —
var inputs = [sideA, sideB, sideC, angleA_deg, angleB_deg, angleC_deg];
var validInputsCount = 0;
for (var i = 0; i 0) {
validInputsCount++;
}
}
var angleInputsCount = 0;
if (!isNaN(angleA_deg)) angleInputsCount++;
if (!isNaN(angleB_deg)) angleInputsCount++;
if (!isNaN(angleC_deg)) angleInputsCount++;
var sideInputsCount = 0;
if (!isNaN(sideA)) sideInputsCount++;
if (!isNaN(sideB)) sideInputsCount++;
if (!isNaN(sideC)) sideInputsCount++;
// Basic check: Need at least 3 known pieces of information,
// with at least one side and a combination that allows solving.
// A common requirement is 2 angles + 1 side, or 1 angle + 2 sides.
// SSS (3 sides) and AAA (3 angles) are degenerate cases for this calculator's primary function.
if (validInputsCount = 2 && !isNaN(angleA_deg) && !isNaN(angleB_deg) && isNaN(angleC_deg)) {
calculatedAngles.c = 180 – angleA_deg – angleB_deg;
if (calculatedAngles.c > 0) {
resultDiv.innerHTML += 'Calculated Angle C: ' + radToDeg(calculatedAngles.c).toFixed(2) + '°';
calculated = true;
} else {
resultDiv.innerHTML += 'Error: Sum of known angles exceeds 180°.';
}
} else if (angleInputsCount >= 2 && !isNaN(angleA_deg) && !isNaN(angleC_deg) && isNaN(angleB_deg)) {
calculatedAngles.b = 180 – angleA_deg – angleC_deg;
if (calculatedAngles.b > 0) {
resultDiv.innerHTML += 'Calculated Angle B: ' + radToDeg(calculatedAngles.b).toFixed(2) + '°';
calculated = true;
} else {
resultDiv.innerHTML += 'Error: Sum of known angles exceeds 180°.';
}
} else if (angleInputsCount >= 2 && !isNaN(angleB_deg) && !isNaN(angleC_deg) && isNaN(angleA_deg)) {
calculatedAngles.a = 180 – angleB_deg – angleC_deg;
if (calculatedAngles.a > 0) {
resultDiv.innerHTML += 'Calculated Angle A: ' + radToDeg(calculatedAngles.a).toFixed(2) + '°';
calculated = true;
} else {
resultDiv.innerHTML += 'Error: Sum of known angles exceeds 180°.';
}
}
// Update the input fields with calculated angles for consistency
if (!isNaN(calculatedAngles.a)) document.getElementById('angleA').value = calculatedAngles.a.toFixed(2);
if (!isNaN(calculatedAngles.b)) document.getElementById('angleB').value = calculatedAngles.b.toFixed(2);
if (!isNaN(calculatedAngles.c)) document.getElementById('angleC').value = calculatedAngles.c.toFixed(2);
// Case 2: Two sides and an angle are known
// Subcase 2a: SAS (Side-Angle-Side) – Angle is between the two sides
if (sideInputsCount >= 2 && angleInputsCount >= 1) {
var knownAngle = 0;
var oppositeSide = 0;
var adjSide1 = 0;
var adjSide2 = 0;
if (!isNaN(angleA_deg) && !isNaN(sideB) && !isNaN(sideC) && isNaN(sideA)) { // Angle A known, find side a
knownAngle = degToRad(angleA_deg); adjSide1 = sideB; adjSide2 = sideC;
var a_squared = sideB*sideB + sideC*sideC – 2*sideB*sideC * Math.cos(knownAngle);
if (a_squared > 0) {
calculatedSides.a = Math.sqrt(a_squared);
resultDiv.innerHTML += 'Calculated Side A: ' + calculatedSides.a.toFixed(2) + ";
calculated = true;
} else {
resultDiv.innerHTML += 'Error: Cannot form a triangle with these inputs (SAS).';
}
} else if (!isNaN(angleB_deg) && !isNaN(sideA) && !isNaN(sideC) && isNaN(sideB)) { // Angle B known, find side b
knownAngle = degToRad(angleB_deg); adjSide1 = sideA; adjSide2 = sideC;
var b_squared = sideA*sideA + sideC*sideC – 2*sideA*sideC * Math.cos(knownAngle);
if (b_squared > 0) {
calculatedSides.b = Math.sqrt(b_squared);
resultDiv.innerHTML += 'Calculated Side B: ' + calculatedSides.b.toFixed(2) + ";
calculated = true;
} else {
resultDiv.innerHTML += 'Error: Cannot form a triangle with these inputs (SAS).';
}
} else if (!isNaN(angleC_deg) && !isNaN(sideA) && !isNaN(sideB) && isNaN(sideC)) { // Angle C known, find side c
knownAngle = degToRad(angleC_deg); adjSide1 = sideA; adjSide2 = sideB;
var c_squared = sideA*sideA + sideB*sideB – 2*sideA*sideB * Math.cos(knownAngle);
if (c_squared > 0) {
calculatedSides.c = Math.sqrt(c_squared);
resultDiv.innerHTML += 'Calculated Side C: ' + calculatedSides.c.toFixed(2) + ";
calculated = true;
} else {
resultDiv.innerHTML += 'Error: Cannot form a triangle with these inputs (SAS).';
}
}
// If SAS calculation succeeded, we might now have enough info for angles via Law of Sines/Cosines
if (calculated && sideInputsCount === 3 && angleInputsCount === 1) {
// If we calculated one side, and all 3 sides are now known, we can find angles
// Update input fields for consistency if a side was calculated
if (!isNaN(calculatedSides.a)) document.getElementById('sideA').value = calculatedSides.a.toFixed(2);
if (!isNaN(calculatedSides.b)) document.getElementById('sideB').value = calculatedSides.b.toFixed(2);
if (!isNaN(calculatedSides.c)) document.getElementById('sideC').value = calculatedSides.c.toFixed(2);
}
}
// Case 3: Two sides and an angle are known
// Subcase 3a: SSA (Side-Side-Angle) – Angle is NOT between the two sides
if (sideInputsCount >= 2 && angleInputsCount === 1) {
var angleRad, oppositeSide, adjSide;
var angleField, oppositeSideField, adjSideField;
if (!isNaN(angleA_deg) && !isNaN(sideB) && !isNaN(sideC) && isNaN(angleB_deg) && isNaN(angleC_deg)) { // Angle A is known (opposite side unknown)
angleRad = degToRad(angleA_deg);
oppositeSide = sideA; // This is the side OPPOSITE the known angle
adjSide = sideB; // One of the adjacent sides
angleField = 'angleA';
oppositeSideField = 'sideA';
adjSideField = 'sideB'; // We need to find angle B
} else if (!isNaN(angleB_deg) && !isNaN(sideA) && !isNaN(sideC) && isNaN(angleA_deg) && isNaN(angleC_deg)) { // Angle B is known
angleRad = degToRad(angleB_deg);
oppositeSide = sideB; // Opposite side to B
adjSide = sideA; // Adjacent side to B
angleField = 'angleB';
oppositeSideField = 'sideB';
adjSideField = 'sideA'; // We need to find angle A
} else if (!isNaN(angleC_deg) && !isNaN(sideA) && !isNaN(sideB) && isNaN(angleA_deg) && isNaN(angleB_deg)) { // Angle C is known
angleRad = degToRad(angleC_deg);
oppositeSide = sideC; // Opposite side to C
adjSide = sideA; // Adjacent side to C
angleField = 'angleC';
oppositeSideField = 'sideC';
adjSideField = 'sideA'; // We need to find angle A
}
if (angleRad !== undefined && oppositeSide !== undefined && adjSide !== undefined) {
var sinOppositeOverAdj = (oppositeSide * Math.sin(angleRad)) / adjSide;
if (sinOppositeOverAdj > 1) {
resultDiv.innerHTML += 'Error: No triangle possible (sine value > 1).';
} else if (sinOppositeOverAdj === 1) {
// One solution (right triangle)
var unknownAngle_deg = 90 – parseFloat(document.getElementById(angleField).value);
if (unknownAngle_deg > 0) {
var unknownAngleField = (angleField === 'angleA') ? 'angleB' : (angleField === 'angleB' ? 'angleA' : 'angleA');
if (unknownAngleField === 'angleB') calculatedAngles.b = unknownAngle_deg;
else if (unknownAngleField === 'angleA') calculatedAngles.a = unknownAngle_deg;
else calculatedAngles.a = unknownAngle_deg; // Default
resultDiv.innerHTML += 'Calculated ' + unknownAngleField.replace('angle', 'Angle ').toUpperCase() + ': ' + unknownAngle_deg.toFixed(2) + '°';
calculated = true;
}
} else {
// Potentially two solutions (ambiguous case)
var angle1_rad = Math.asin(sinOppositeOverAdj);
var angle1_deg = radToDeg(angle1_rad);
var angle2_deg = 180 – angle1_deg;
// We need to determine which angle is being calculated
var calculatedAngleName = ";
if (angleField === 'angleA') { // Angle A known, find B or C
if (adjSideField === 'sideB') { // Finding angle B
calculatedAngleName = 'Angle B';
// First solution
var thirdAngle1 = 180 – parseFloat(document.getElementById(angleField).value) – angle1_deg;
if (thirdAngle1 > 0) {
resultDiv.innerHTML += 'Possible Triangle 1: ' + calculatedAngleName + ': ' + angle1_deg.toFixed(2) + '°, Angle C: ' + thirdAngle1.toFixed(2) + '°';
calculated = true;
}
// Second solution
var thirdAngle2 = 180 – parseFloat(document.getElementById(angleField).value) – angle2_deg;
if (thirdAngle2 > 0) {
resultDiv.innerHTML += 'Possible Triangle 2: ' + calculatedAngleName + ': ' + angle2_deg.toFixed(2) + '°, Angle C: ' + thirdAngle2.toFixed(2) + '°';
calculated = true; // Indicate that at least one solution was found
}
}
} else if (angleField === 'angleB') { // Angle B known, find A or C
if (adjSideField === 'sideA') { // Finding angle A
calculatedAngleName = 'Angle A';
// First solution
var thirdAngle1 = 180 – parseFloat(document.getElementById(angleField).value) – angle1_deg;
if (thirdAngle1 > 0) {
resultDiv.innerHTML += 'Possible Triangle 1: ' + calculatedAngleName + ': ' + angle1_deg.toFixed(2) + '°, Angle C: ' + thirdAngle1.toFixed(2) + '°';
calculated = true;
}
// Second solution
var thirdAngle2 = 180 – parseFloat(document.getElementById(angleField).value) – angle2_deg;
if (thirdAngle2 > 0) {
resultDiv.innerHTML += 'Possible Triangle 2: ' + calculatedAngleName + ': ' + angle2_deg.toFixed(2) + '°, Angle C: ' + thirdAngle2.toFixed(2) + '°';
calculated = true;
}
}
} else if (angleField === 'angleC') { // Angle C known, find A or B
if (adjSideField === 'sideA') { // Finding angle A
calculatedAngleName = 'Angle A';
// First solution
var thirdAngle1 = 180 – parseFloat(document.getElementById(angleField).value) – angle1_deg;
if (thirdAngle1 > 0) {
resultDiv.innerHTML += 'Possible Triangle 1: ' + calculatedAngleName + ': ' + angle1_deg.toFixed(2) + '°, Angle B: ' + thirdAngle1.toFixed(2) + '°';
calculated = true;
}
// Second solution
var thirdAngle2 = 180 – parseFloat(document.getElementById(angleField).value) – angle2_deg;
if (thirdAngle2 > 0) {
resultDiv.innerHTML += 'Possible Triangle 2: ' + calculatedAngleName + ': ' + angle2_deg.toFixed(2) + '°, Angle B: ' + thirdAngle2.toFixed(2) + '°';
calculated = true;
}
}
}
}
}
}
// Case 4: Three sides are known (SSS) – calculate all angles
if (sideInputsCount === 3 && angleInputsCount === 0) {
if (!isNaN(sideA) && !isNaN(sideB) && !isNaN(sideC)) {
// Check triangle inequality theorem
if ((sideA + sideB > sideC) && (sideA + sideC > sideB) && (sideB + sideC > sideA)) {
var cosA = (sideB*sideB + sideC*sideC – sideA*sideA) / (2*sideB*sideC);
var cosB = (sideA*sideA + sideC*sideC – sideB*sideB) / (2*sideA*sideC);
var cosC = (sideA*sideA + sideB*sideB – sideC*sideC) / (2*sideA*sideB);
// Ensure values are within arccos domain [-1, 1] due to potential floating point inaccuracies
cosA = Math.max(-1, Math.min(1, cosA));
cosB = Math.max(-1, Math.min(1, cosB));
cosC = Math.max(-1, Math.min(1, cosC));
var angleA_calc_deg = radToDeg(Math.acos(cosA));
var angleB_calc_deg = radToDeg(Math.acos(cosB));
var angleC_calc_deg = radToDeg(Math.acos(cosC));
// Recalculate one angle to ensure sum is 180 due to potential floating point errors
angleC_calc_deg = 180 – angleA_calc_deg – angleB_calc_deg;
calculatedAngles.a = angleA_calc_deg;
calculatedAngles.b = angleB_calc_deg;
calculatedAngles.c = angleC_calc_deg;
resultDiv.innerHTML += 'Calculated Angle A: ' + calculatedAngles.a.toFixed(2) + '°';
resultDiv.innerHTML += 'Calculated Angle B: ' + calculatedAngles.b.toFixed(2) + '°';
resultDiv.innerHTML += 'Calculated Angle C: ' + calculatedAngles.c.toFixed(2) + '°';
calculated = true;
// Update input fields for consistency
if (!isNaN(calculatedAngles.a)) document.getElementById('angleA').value = calculatedAngles.a.toFixed(2);
if (!isNaN(calculatedAngles.b)) document.getElementById('angleB').value = calculatedAngles.b.toFixed(2);
if (!isNaN(calculatedAngles.c)) document.getElementById('angleC').value = calculatedAngles.c.toFixed(2);
} else {
resultDiv.innerHTML += 'Error: The given side lengths do not form a valid triangle (Triangle Inequality Theorem violated).';
}
}
}
// Final checks if no specific case matched, but we have some info
if (!calculated && validInputsCount >= 3) {
var totalKnownAngleSum = 0;
if (!isNaN(angleA_deg)) totalKnownAngleSum += angleA_deg;
if (!isNaN(angleB_deg)) totalKnownAngleSum += angleB_deg;
if (!isNaN(angleC_deg)) totalKnownAngleSum += angleC_deg;
if (totalKnownAngleSum >= 180) {
resultDiv.innerHTML += 'Error: Sum of known angles is 180° or more.';
}
// Add more checks if needed, e.g., if we have 2 sides and 1 angle but it wasn't SAS or SSA solvable.
}
if (!calculated && resultDiv.innerHTML === ") {
resultDiv.innerHTML = 'Could not determine a unique triangle. Please ensure valid inputs.';
}
}