Enter two sides and an angle (in degrees), or three sides, or two angles and a side to solve for the unknown properties of a triangle.
Results
Enter information to see the solution.
Understanding Triangle Solutions
A triangle is a fundamental geometric shape defined by three sides and three angles. Solving a triangle means determining the lengths of all its sides and the measures of all its angles when some of these values are known. There are specific sets of known information (called "givens") that allow for a unique triangle solution. These are:
Three Sides (SSS): If you know the lengths of all three sides (a, b, c), you can find all the angles using the Law of Cosines.
Two Sides and the Included Angle (SAS): If you know two sides (e.g., a, b) and the angle between them (C), you can find the third side using the Law of Cosines, and then find the remaining angles using the Law of Sines or Law of Cosines.
Two Angles and the Included Side (ASA): If you know two angles (e.g., A, B) and the side between them (c), you can find the third angle (since A + B + C = 180°). Then, you can find the other two sides using the Law of Sines.
Two Angles and a Non-Included Side (AAS): If you know two angles (e.g., A, B) and a side opposite one of them (e.g., a), you can find the third angle (C). Then, you can find the remaining sides using the Law of Sines.
Two Sides and a Non-Included Angle (SSA – Ambiguous Case): This case is known as the "ambiguous case" because it can result in zero, one, or two possible triangles. If you know sides a, b, and angle A, you first use the Law of Sines to find angle B. If sin(B) > 1, there's no solution. If sin(B) = 1, there's one right-angled triangle. If sin(B) < 1, there are two possible values for angle B (B and 180° – B), which can lead to two different triangles.
Key Laws Used:
Law of Sines: For any triangle with sides a, b, c and opposite angles A, B, C:
a / sin(A) = b / sin(B) = c / sin(C)
Sum of Angles: The sum of the interior angles of any triangle is always 180 degrees.
A + B + C = 180°
How to Use This Calculator:
This calculator aims to solve for unknown sides and angles of a triangle. You should provide *at least* three pieces of information (sides or angles). The calculator will attempt to apply the appropriate trigonometric laws to find the missing values.
Enter known values: Fill in the side lengths (a, b, c) and/or angle measures (A, B, C in degrees).
Click "Solve Triangle": The calculator will process your inputs.
View Results: The calculated unknown sides and angles will be displayed. If no valid triangle can be formed with the given inputs, or if there's an ambiguous case not fully resolved by the inputs, an appropriate message will be shown.
Note: Ensure angles are entered in degrees. The calculator handles basic cases (SSS, SAS, ASA, AAS) and attempts to resolve common SSA scenarios. Numerical precision may lead to very small discrepancies in angle sums.
function calculateTriangle() {
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
var inputs = [sideA, sideB, sideC, angleA_deg, angleB_deg, angleC_deg];
var knownCount = inputs.filter(val => !isNaN(val) && val > 0).length;
if (knownCount 0) {
calculatedAngleC_rad = calculatedAngleC_deg * Math.PI / 180;
calculatedSideC = sideA * Math.sin(calculatedAngleC_rad) / Math.sin(angleA_rad); // Use side A as known for calculation
// Re-calculate sides B and C using Law of Sines with the newly found side C
calculatedSideB = calculatedSideC * Math.sin(angleB_rad) / Math.sin(calculatedAngleC_rad);
calculatedSideA = calculatedSideC * Math.sin(angleA_rad) / Math.sin(calculatedAngleC_rad);
} else {
resultDiv.innerHTML = "Invalid triangle: Angles sum to 180 degrees or less.";
return;
}
} else if (!sidesKnown[1] && anglesKnown[0] && anglesKnown[2] && isNaN(angleB_deg)) {
calculatedAngleB_deg = 180 – angleA_deg – angleC_deg;
if (calculatedAngleB_deg > 0) {
calculatedAngleB_rad = calculatedAngleB_deg * Math.PI / 180;
calculatedSideB = sideA * Math.sin(calculatedAngleB_rad) / Math.sin(angleA_rad); // Use side A as known
calculatedSideC = calculatedSideB * Math.sin(angleC_rad) / Math.sin(calculatedAngleB_rad);
calculatedSideA = calculatedSideB * Math.sin(angleA_rad) / Math.sin(calculatedAngleB_rad);
} else {
resultDiv.innerHTML = "Invalid triangle: Angles sum to 180 degrees or less.";
return;
}
} else if (!sidesKnown[0] && anglesKnown[1] && anglesKnown[2] && isNaN(angleA_deg)) {
calculatedAngleA_deg = 180 – angleB_deg – angleC_deg;
if (calculatedAngleA_deg > 0) {
calculatedAngleA_rad = calculatedAngleA_deg * Math.PI / 180;
calculatedSideA = sideB * Math.sin(calculatedAngleA_rad) / Math.sin(angleB_rad); // Use side B as known
calculatedSideC = calculatedSideA * Math.sin(angleC_rad) / Math.sin(calculatedAngleA_rad);
calculatedSideB = calculatedSideA * Math.sin(angleB_rad) / Math.sin(calculatedAngleA_rad);
} else {
resultDiv.innerHTML = "Invalid triangle: Angles sum to 180 degrees or less.";
return;
}
}
// Case 4: Two Angles and Non-Included Side (AAS) – Handled by first finding third angle then using ASA logic
else if (anglesKnown[0] && anglesKnown[1] && sidesKnown[2] && isNaN(angleC_deg)) {
calculatedAngleC_deg = 180 – angleA_deg – angleB_deg;
if (calculatedAngleC_deg > 0) {
calculatedAngleC_rad = calculatedAngleC_deg * Math.PI / 180;
calculatedSideA = sideC * Math.sin(angleA_rad) / calculatedAngleC_rad;
calculatedSideB = sideC * Math.sin(angleB_rad) / calculatedAngleC_rad;
} else {
resultDiv.innerHTML = "Invalid triangle: Angles sum to 180 degrees or less.";
return;
}
} else if (anglesKnown[0] && anglesKnown[2] && sidesKnown[1] && isNaN(angleB_deg)) {
calculatedAngleB_deg = 180 – angleA_deg – angleC_deg;
if (calculatedAngleB_deg > 0) {
calculatedAngleB_rad = calculatedAngleB_deg * Math.PI / 180;
calculatedSideA = sideB * Math.sin(angleA_rad) / calculatedAngleB_rad;
calculatedSideC = sideB * Math.sin(angleC_rad) / calculatedAngleB_rad;
} else {
resultDiv.innerHTML = "Invalid triangle: Angles sum to 180 degrees or less.";
return;
}
} else if (anglesKnown[1] && anglesKnown[2] && sidesKnown[0] && isNaN(angleA_deg)) {
calculatedAngleA_deg = 180 – angleB_deg – angleC_deg;
if (calculatedAngleA_deg > 0) {
calculatedAngleA_rad = calculatedAngleA_deg * Math.PI / 180;
calculatedSideB = sideA * Math.sin(angleB_rad) / calculatedAngleA_rad;
calculatedSideC = sideA * Math.sin(angleC_rad) / calculatedAngleA_rad;
} else {
resultDiv.innerHTML = "Invalid triangle: Angles sum to 180 degrees or less.";
return;
}
}
// Case 5: Two Sides and Non-Included Angle (SSA – Ambiguous Case)
else if (sidesKnown[0] && sidesKnown[1] && anglesKnown[0] && isNaN(angleB_deg) && isNaN(angleC_deg)) {
// Find angle B using Law of Sines
var sinB = (sideB * Math.sin(angleA_rad)) / sideA;
if (sinB > 1) {
resultDiv.innerHTML = "No triangle possible (sin B > 1).";
return;
} else if (sinB === 1) { // One solution (right triangle)
calculatedAngleB_rad = Math.PI / 2;
calculatedAngleB_deg = 90;
calculatedAngleC_rad = Math.PI – angleA_rad – calculatedAngleB_rad;
calculatedAngleC_deg = calculatedAngleC_rad * 180 / Math.PI;
calculatedSideC = sideA * Math.sin(calculatedAngleC_rad) / Math.sin(angleA_rad);
} else { // Potentially two solutions
var angleB1_rad = Math.asin(sinB);
var angleB1_deg = angleB1_rad * 180 / Math.PI;
var angleC1_rad = Math.PI – angleA_rad – angleB1_rad;
var angleC1_deg = angleC1_rad * 180 / Math.PI;
var sideC1 = sideA * Math.sin(angleC1_rad) / Math.sin(angleA_rad);
var angleB2_rad = Math.PI – angleB1_rad;
var angleB2_deg = angleB2_rad * 180 / Math.PI;
var angleC2_rad = Math.PI – angleA_rad – angleB2_rad;
var angleC2_deg = angleC2_rad * 180 / Math.PI;
var sideC2 = sideA * Math.sin(angleC2_rad) / Math.sin(angleA_rad);
// Check if the second triangle is valid (angle C2 must be positive)
if (angleC2_deg > 0) {
resultDiv.innerHTML = "Ambiguous Case: Two possible triangles.";
resultDiv.innerHTML += "Triangle 1:";
resultDiv.innerHTML += ` Angle B: ${angleB1_deg.toFixed(2)}°, Angle C: ${angleC1_deg.toFixed(2)}°, Side c: ${sideC1.toFixed(2)}`;
resultDiv.innerHTML += "Triangle 2:";
resultDiv.innerHTML += ` Angle B: ${angleB2_deg.toFixed(2)}°, Angle C: ${angleC2_deg.toFixed(2)}°, Side c: ${sideC2.toFixed(2)}`;
return; // Exit after displaying ambiguous case
} else { // Only one valid triangle
calculatedAngleB_deg = angleB1_deg;
calculatedAngleB_rad = angleB1_rad;
calculatedAngleC_deg = angleC1_deg;
calculatedAngleC_rad = angleC1_rad;
calculatedSideC = sideC1;
}
}
} else if (sidesKnown[0] && sidesKnown[2] && anglesKnown[0] && isNaN(angleB_deg) && isNaN(angleC_deg)) { // SSA with side a, c, angle A
var sinC = (sideC * Math.sin(angleA_rad)) / sideA;
if (sinC > 1) {
resultDiv.innerHTML = "No triangle possible (sin C > 1).";
return;
} else if (sinC === 1) {
calculatedAngleC_rad = Math.PI / 2;
calculatedAngleC_deg = 90;
calculatedAngleB_rad = Math.PI – angleA_rad – calculatedAngleC_rad;
calculatedAngleB_deg = calculatedAngleB_rad * 180 / Math.PI;
calculatedSideB = sideA * Math.sin(calculatedAngleB_rad) / Math.sin(angleA_rad);
} else {
var angleC1_rad = Math.asin(sinC);
var angleC1_deg = angleC1_rad * 180 / Math.PI;
var angleB1_rad = Math.PI – angleA_rad – angleC1_rad;
var angleB1_deg = angleB1_rad * 180 / Math.PI;
var sideB1 = sideA * Math.sin(angleB1_rad) / Math.sin(angleA_rad);
var angleC2_rad = Math.PI – angleC1_rad;
var angleC2_deg = angleC2_rad * 180 / Math.PI;
var angleB2_rad = Math.PI – angleA_rad – angleC2_rad;
var angleB2_deg = angleB2_rad * 180 / Math.PI;
var sideB2 = sideA * Math.sin(angleB2_rad) / Math.sin(angleA_rad);
if (angleB2_deg > 0) {
resultDiv.innerHTML = "Ambiguous Case: Two possible triangles.";
resultDiv.innerHTML += "Triangle 1:";
resultDiv.innerHTML += ` Angle C: ${angleC1_deg.toFixed(2)}°, Angle B: ${angleB1_deg.toFixed(2)}°, Side b: ${sideB1.toFixed(2)}`;
resultDiv.innerHTML += "Triangle 2:";
resultDiv.innerHTML += ` Angle C: ${angleC2_deg.toFixed(2)}°, Angle B: ${angleB2_deg.toFixed(2)}°, Side b: ${sideB2.toFixed(2)}`;
return;
} else {
calculatedAngleC_deg = angleC1_deg;
calculatedAngleC_rad = angleC1_rad;
calculatedAngleB_deg = angleB1_deg;
calculatedAngleB_rad = angleB1_rad;
calculatedSideB = sideB1;
}
}
} else if (sidesKnown[1] && sidesKnown[2] && anglesKnown[0] && isNaN(angleB_deg) && isNaN(angleC_deg)) { // SSA with side b, c, angle A
// This case is not standard SSA, as angle A is opposite side a.
// If side A is unknown, and we know b, c, A, this is SAS.
// If we know a, b, A, this is SSA.
// If we know a, c, A, this is SSA.
// If we know b, c, B, this is SSA.
// If we know b, c, C, this is SSA.
// This particular combination seems to imply we know sides b and c, and angle A. This is SAS.
// Re-evaluating the condition: we have sideB, sideC, angleA, and we want angleB, angleC, sideA
if (sidesKnown[1] && sidesKnown[2] && anglesKnown[0] && isNaN(sideA) && isNaN(angleB_deg) && isNaN(angleC_deg)) { // SAS case
calculatedSideA = Math.sqrt(sideB*sideB + sideC*sideC – 2*sideB*sideC*Math.cos(angleA_rad));
calculatedAngleB_rad = Math.asin((sideB * Math.sin(angleA_rad)) / calculatedSideA);
calculatedAngleC_rad = Math.PI – angleA_rad – calculatedAngleB_rad;
calculatedAngleB_deg = calculatedAngleB_rad * 180 / Math.PI;
calculatedAngleC_deg = calculatedAngleC_rad * 180 / Math.PI;
} else {
resultDiv.innerHTML = "Invalid input combination for SSA or other cases.";
return;
}
} else {
resultDiv.innerHTML = "Could not determine triangle solution with the provided inputs. Ensure you have at least 3 values and they form a valid case (SSS, SAS, ASA, AAS, SSA).";
return;
}
// Final Check for consistency and display
var angleSum = (isNaN(calculatedAngleA_deg) ? 0 : calculatedAngleA_deg) +
(isNaN(calculatedAngleB_deg) ? 0 : calculatedAngleB_deg) +
(isNaN(calculatedAngleC_deg) ? 0 : calculatedAngleC_deg);
var output = "";
output += "Side a: " + (isNaN(calculatedSideA) ? "N/A" : calculatedSideA.toFixed(3)) + "";
output += "Side b: " + (isNaN(calculatedSideB) ? "N/A" : calculatedSideB.toFixed(3)) + "";
output += "Side c: " + (isNaN(calculatedSideC) ? "N/A" : calculatedSideC.toFixed(3)) + "";
output += "Angle A: " + (isNaN(calculatedAngleA_deg) ? "N/A" : calculatedAngleA_deg.toFixed(3)) + "°";
output += "Angle B: " + (isNaN(calculatedAngleB_deg) ? "N/A" : calculatedAngleB_deg.toFixed(3)) + "°";
output += "Angle C: " + (isNaN(calculatedAngleC_deg) ? "N/A" : calculatedAngleC_deg.toFixed(3)) + "°";
if (Math.abs(angleSum – 180) > 1e-6 && !isNaN(angleSum)) { // Check if angle sum is close to 180
output += "Note: Angle sum is approximately " + angleSum.toFixed(3) + "°, minor precision differences may occur.";
}
resultDiv.innerHTML = output;
}
// Helper function to check triangle inequality theorem
function isTriangleValid(a, b, c) {
if (isNaN(a) || isNaN(b) || isNaN(c) || a <= 0 || b <= 0 || c c) && (a + c > b) && (b + c > a);
}