Calculate the angles of a triangle using side lengths or two sides and an angle.
Or, for non-right triangles, you can use two sides and the included angle (in degrees):
Understanding Triangle Angle Calculation
Triangles are fundamental geometric shapes with three sides and three interior angles. The sum of the interior angles of any triangle is always 180 degrees. Calculating these angles is a common task in geometry, trigonometry, engineering, and surveying.
Methods for Calculating Angles:
Depending on the information you have about a triangle, different methods can be used to find its angles:
1. Using All Three Side Lengths (SSS – Side-Side-Side):
If you know the lengths of all three sides (let's call them a, b, and c), you can use the Law of Cosines to find each angle. The Law of Cosines states:
c² = a² + b² - 2ab * cos(C)
b² = a² + c² - 2ac * cos(B)
a² = b² + c² - 2bc * cos(A)
To find an angle (e.g., angle A), you can rearrange the formula:
cos(A) = (b² + c² - a²) / (2bc)
Then, use the inverse cosine function (arccos or cos⁻¹) to find the angle in degrees: A = arccos((b² + c² - a²) / (2bc)).
This calculator implements this by taking side lengths a, b, and c. Note that for a valid triangle, the sum of any two sides must be greater than the third side (triangle inequality theorem).
2. Using Two Sides and the Included Angle (SAS – Side-Angle-Side):
If you know two sides (e.g., a and b) and the angle between them (angle C), you can first find the length of the third side (c) using the Law of Cosines:
c² = a² + b² - 2ab * cos(C)
Once you have all three sides, you can proceed as in the SSS method to find the remaining angles using the Law of Cosines again, or you can use the Law of Sines to find one of the other angles:
a / sin(A) = b / sin(B) = c / sin(C)
This calculator uses the SAS input to first calculate the third side (c) using the Law of Cosines, and then uses the SSS approach to find all angles.
Use Cases:
Navigation & Surveying: Determining distances and positions by triangulation.
Engineering & Architecture: Designing structures, calculating forces, and ensuring stability.
Computer Graphics: Rendering 3D models and calculating object orientations.
Physics: Analyzing projectile motion and wave interference.
Mathematics Education: Practicing trigonometric principles and problem-solving.
function calculateAngles() {
var sideA = parseFloat(document.getElementById("sideA").value);
var sideB = parseFloat(document.getElementById("sideB").value);
var sideC = parseFloat(document.getElementById("sideC").value);
var sideA_SAS = parseFloat(document.getElementById("sideA_SAS").value);
var sideB_SAS = parseFloat(document.getElementById("sideB_SAS").value);
var angleC_SAS_deg = parseFloat(document.getElementById("angleC_SAS").value);
var resultDiv = document.getElementById("result");
resultDiv.style.display = "none"; // Hide previous results
var angleA, angleB, angleC;
var message = "";
// — SSS Calculation —
if (!isNaN(sideA) && !isNaN(sideB) && !isNaN(sideC) && sideA > 0 && sideB > 0 && sideC > 0) {
// 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 the valid range for arccos [-1, 1] due to floating point errors
cosA = Math.max(-1, Math.min(1, cosA));
cosB = Math.max(-1, Math.min(1, cosB));
cosC = Math.max(-1, Math.min(1, cosC));
angleA = Math.acos(cosA) * (180 / Math.PI);
angleB = Math.acos(cosB) * (180 / Math.PI);
angleC = Math.acos(cosC) * (180 / Math.PI);
message = "
Results (SSS):
" +
"Angle A: " + angleA.toFixed(2) + "°" +
"Angle B: " + angleB.toFixed(2) + "°" +
"Angle C: " + angleC.toFixed(2) + "°" +
"Sum of Angles: " + (angleA + angleB + angleC).toFixed(2) + "°";
resultDiv.innerHTML = message;
resultDiv.style.display = "block";
} else {
resultDiv.innerHTML = "Error: The given side lengths do not form a valid triangle.";
resultDiv.style.display = "block";
}
}
// — SAS Calculation —
else if (!isNaN(sideA_SAS) && !isNaN(sideB_SAS) && !isNaN(angleC_SAS_deg) && sideA_SAS > 0 && sideB_SAS > 0 && angleC_SAS_deg > 0 && angleC_SAS_deg < 180) {
var angleC_SAS_rad = angleC_SAS_deg * (Math.PI / 180);
var sideC_SAS = Math.sqrt(sideA_SAS * sideA_SAS + sideB_SAS * sideB_SAS – 2 * sideA_SAS * sideB_SAS * Math.cos(angleC_SAS_rad));
// Now we have SSS case with sideA_SAS, sideB_SAS, and calculated sideC_SAS
var cosA = (sideB_SAS * sideB_SAS + sideC_SAS * sideC_SAS – sideA_SAS * sideA_SAS) / (2 * sideB_SAS * sideC_SAS);
var cosB = (sideA_SAS * sideA_SAS + sideC_SAS * sideC_SAS – sideB_SAS * sideB_SAS) / (2 * sideA_SAS * sideC_SAS);
cosA = Math.max(-1, Math.min(1, cosA));
cosB = Math.max(-1, Math.min(1, cosB));
angleA = Math.acos(cosA) * (180 / Math.PI);
angleB = Math.acos(cosB) * (180 / Math.PI);
angleC = angleC_SAS_deg; // Already known
message = "
Results (SAS):
" +
"Calculated Side C: " + sideC_SAS.toFixed(2) + "" +
"Angle A: " + angleA.toFixed(2) + "°" +
"Angle B: " + angleB.toFixed(2) + "°" +
"Angle C: " + angleC.toFixed(2) + "°" +
"Sum of Angles: " + (angleA + angleB + angleC).toFixed(2) + "°";
resultDiv.innerHTML = message;
resultDiv.style.display = "block";
}
else {
resultDiv.innerHTML = "Please enter valid inputs for either SSS (all three sides) or SAS (two sides and the included angle).";
resultDiv.style.display = "block";
}
}