Using Two Sides and Angle Between (SAS – Area)
Using Three Sides (SSS – Law of Cosines)
Using Two Sides of a Right Triangle
For SAS (Area Calculation)
For SSS (Law of Cosines)
For Right Triangle (Trigonometry)
Hypotenuse
Opposite
Adjacent
Opposite
Adjacent
Hypotenuse
Result
—
Degrees
Understanding Angle Calculations
Angles are fundamental to geometry, trigonometry, physics, engineering, and many other fields. They represent the amount of rotation between two intersecting lines or rays. Calculating angles accurately is crucial for tasks ranging from designing structures and navigation to analyzing motion and understanding spatial relationships.
Methods for Calculating Angles
There are several ways to calculate an angle, depending on the information you have about the geometric figure or scenario:
1. Using Two Sides and the Included Angle (SAS – for Area Calculation)
While this calculator primarily focuses on finding angles, it includes a common related calculation: finding the Area of a Triangle given Side-Angle-Side (SAS). The formula is:
Area = 0.5 * a * b * sin(C)
where 'a' and 'b' are the lengths of two sides, and 'C' is the angle between them. If you have three sides and want to find an angle, you use the Law of Cosines.
2. Using Three Sides (SSS – Law of Cosines)
If you know the lengths of all three sides of a triangle (a, b, c), you can find any of its angles using the Law of Cosines. This is an extension of the Pythagorean theorem. To find angle A, the formula is:
a² = b² + c² - 2bc * cos(A)
Rearranging to solve for cos(A):
cos(A) = (b² + c² - a²) / (2bc)
Then, you take the arccosine (inverse cosine) to find the angle A:
A = arccos((b² + c² - a²) / (2bc))
Similarly, for angle B:
cos(B) = (a² + c² - b²) / (2ac)B = arccos((a² + c² - b²) / (2ac))
And for angle C:
cos(C) = (a² + b² - c²) / (2ab)C = arccos((a² + b² - c²) / (2ab))
The result will be in radians, which this calculator converts to degrees.
3. Using Two Sides of a Right Triangle (Trigonometry)
In a right-angled triangle, the relationships between the angles and the lengths of the sides are defined by trigonometric functions: sine (sin), cosine (cos), and tangent (tan).
Sine (sin θ) = Opposite / Hypotenuse
Cosine (cos θ) = Adjacent / Hypotenuse
Tangent (tan θ) = Opposite / Adjacent
If you know two sides, you can determine which trigonometric function relates them to the angle you want to find. Then, you use the inverse trigonometric function (arcsin, arccos, arctan) to calculate the angle.
If you know Opposite and Hypotenuse: θ = arcsin(Opposite / Hypotenuse)
If you know Adjacent and Hypotenuse: θ = arccos(Adjacent / Hypotenuse)
If you know Opposite and Adjacent: θ = arctan(Opposite / Adjacent)
The calculator uses these principles to find the unknown acute angle in a right triangle.
Use Cases for Angle Calculations
Construction and Architecture: Ensuring correct slopes, roof pitches, and structural stability.
Navigation: Determining directions, courses, and positions using bearings and headings.
Engineering: Designing mechanical systems, analyzing forces, and calculating trajectories.
Computer Graphics: Rendering 3D scenes, applying rotations, and creating visual effects.
Surveying: Measuring land boundaries and elevations.
Physics: Studying projectile motion, wave propagation, and rotational dynamics.
This Angle Calculator provides a convenient way to perform these essential calculations, helping you solve problems in various technical and scientific domains.
function toggleInputs() {
var method = document.getElementById("inputMethod").value;
document.getElementById("sasInputs").style.display = "none";
document.getElementById("sssInputs").style.display = "none";
document.getElementById("rightTriangleInputs").style.display = "none";
if (method === "sides") {
document.getElementById("sasInputs").style.display = "block";
document.getElementById("result-unit").innerText = "Square Units (Area)";
} else if (method === "sides-angle") {
document.getElementById("sssInputs").style.display = "block";
document.getElementById("result-unit").innerText = "Degrees";
} else if (method === "right-triangle") {
document.getElementById("rightTriangleInputs").style.display = "block";
document.getElementById("result-unit").innerText = "Degrees";
}
}
function calculateAngle() {
var method = document.getElementById("inputMethod").value;
var resultValue = "–";
var resultUnit = "Degrees"; // Default unit
if (method === "sides") {
// SAS – Area Calculation
var sideA = parseFloat(document.getElementById("sideA").value);
var sideB = parseFloat(document.getElementById("sideB").value);
var angleC_deg = parseFloat(document.getElementById("angleC").value);
if (!isNaN(sideA) && !isNaN(sideB) && !isNaN(angleC_deg) && sideA > 0 && sideB > 0 && angleC_deg > 0 && angleC_deg 0 && sideB_sss > 0 && sideC_sss > 0;
if (validSides) {
// Triangle Inequality Theorem check
if (!((sideA_sss + sideB_sss > sideC_sss) && (sideA_sss + sideC_sss > sideB_sss) && (sideB_sss + sideC_sss > sideA_sss))) {
resultValue = "Invalid Triangle Sides";
} else {
var cosValue;
if (angleToFind_sss === "A") {
cosValue = (Math.pow(sideB_sss, 2) + Math.pow(sideC_sss, 2) – Math.pow(sideA_sss, 2)) / (2 * sideB_sss * sideC_sss);
} else if (angleToFind_sss === "B") {
cosValue = (Math.pow(sideA_sss, 2) + Math.pow(sideC_sss, 2) – Math.pow(sideB_sss, 2)) / (2 * sideA_sss * sideC_sss);
} else if (angleToFind_sss === "C") {
cosValue = (Math.pow(sideA_sss, 2) + Math.pow(sideB_sss, 2) – Math.pow(sideC_sss, 2)) / (2 * sideA_sss * sideB_sss);
} else {
resultValue = "Invalid Angle Identifier";
}
if (cosValue !== undefined && cosValue >= -1 && cosValue 0 && otherSideValue > 0;
var angleToFind_rt = "A"; // Conventionally find the angle opposite side 'a'
if (validInputs) {
var ratio;
var trigFunction; // 'sin', 'cos', 'tan'
// Determine which sides are opposite, adjacent, hypotenuse relative to the angle we might find
// Assume we're finding an angle (let's call it theta) in the triangle.
// We need to identify which of the *given* sides are opposite, adjacent, or hypotenuse relative to *this* unknown angle.
// Simplified approach: Identify the ratio based on the types of sides provided.
// The calculator needs to calculate an angle. The unknown angle is NOT the 90 degree one.
// Let's assume the sides provided are such that we can find ONE of the acute angles.
// Scenario 1: Opposite and Hypotenuse are given
if ((knownSideType === "opposite" && otherSideType === "hypotenuse") || (knownSideType === "hypotenuse" && otherSideType === "opposite")) {
ratio = otherSideValue / knownSideValue; // opposite / hypotenuse
trigFunction = "sin";
}
// Scenario 2: Adjacent and Hypotenuse are given
else if ((knownSideType === "adjacent" && otherSideType === "hypotenuse") || (knownSideType === "hypotenuse" && otherSideType === "adjacent")) {
ratio = otherSideValue / knownSideValue; // adjacent / hypotenuse
trigFunction = "cos";
}
// Scenario 3: Opposite and Adjacent are given
else if ((knownSideType === "opposite" && otherSideType === "adjacent") || (knownSideType === "adjacent" && otherSideType === "opposite")) {
ratio = otherSideValue / knownSideValue; // opposite / adjacent
trigFunction = "tan";
}
else {
resultValue = "Invalid Side Combination for Calculation";
}
if (ratio !== undefined && trigFunction !== undefined && ratio >= 0 && ratio <= 1) {
var angle_rad;
if (trigFunction === "sin") {
angle_rad = Math.asin(ratio);
} else if (trigFunction === "cos") {
angle_rad = Math.acos(ratio);
} else if (trigFunction === "tan") {
angle_rad = Math.atan(ratio);
}
if (angle_rad !== undefined) {
var angle_deg = angle_rad * (180 / Math.PI);
resultValue = angle_deg.toFixed(2);
resultUnit = "Degrees";
}
} else if (ratio !== undefined) {
resultValue = "Invalid Ratio for Trig Function";
}
} else {
resultValue = "Invalid Input";
}
}
document.getElementById("result-value").innerText = resultValue;
document.getElementById("result-unit").innerText = resultUnit;
}
// Initialize the correct input fields on page load
window.onload = toggleInputs;