Sides A and B (Opposite and Adjacent)
Side A and Hypotenuse C
Side B and Hypotenuse C
Understanding Right Triangles and Angle Calculation
A right triangle is a fundamental geometric shape defined by having one angle that measures exactly 90 degrees (a right angle). The sides opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides are referred to as the legs. If we label the vertices of the triangle as A, B, and C, and the right angle is at vertex C, then side c is the hypotenuse, side a is opposite angle A, and side b is opposite angle B.
The Pythagorean Theorem
For any right triangle, the relationship between the lengths of the sides is described by the Pythagorean theorem: a² + b² = c², where a and b are the lengths of the legs, and c is the length of the hypotenuse.
Trigonometric Functions (SOH CAH TOA)
To find the angles (other than the 90-degree angle) in a right triangle when you know the side lengths, we use trigonometric functions:
Sine (sin): sin(angle) = Opposite / Hypotenuse (SOH)
To find the angle itself, we use the inverse trigonometric functions (arcsine, arccosine, arctangent):
Angle A = arcsin(Opposite / Hypotenuse)
Angle A = arccos(Adjacent / Hypotenuse)
Angle A = arctan(Opposite / Adjacent)
Similarly for Angle B:
Angle B = arcsin(Opposite / Hypotenuse)
Angle B = arccos(Adjacent / Hypotenuse)
Angle B = arctan(Opposite / Adjacent)
In our calculator, 'Side A' is often considered the side opposite to Angle B, and 'Side B' is adjacent to Angle B. Angle A is opposite Side A, and Angle B is opposite Side B. The hypotenuse is always side C.
How this Calculator Works
This calculator allows you to input the lengths of the sides of a right triangle. You can then choose which pair of sides you want to use to calculate the angles. The calculator uses the appropriate trigonometric function (or Pythagorean theorem for validation) to determine the measures of Angle A and Angle B in degrees.
Example Scenarios:
Scenario 1: Using Sides A and B
If Side A = 3 and Side B = 4, this implies the tangent of Angle B is 3/4.
tan(B) = Opposite / Adjacent = 3 / 4
Angle B = arctan(3/4) ≈ 36.87 degrees
Since the sum of angles in a triangle is 180 degrees and one angle is 90 degrees, Angle A = 180 – 90 – Angle B ≈ 180 – 90 – 36.87 = 53.13 degrees.
The calculator will prompt you to enter values for Side A and Side B, and select "Sides A and B".
Scenario 2: Using Side A and Hypotenuse C
If Side A = 3 and Hypotenuse C = 5, this implies the sine of Angle B is 3/5.
sin(B) = Opposite / Hypotenuse = 3 / 5
Angle B = arcsin(3/5) ≈ 36.87 degrees
Angle A = 180 – 90 – 36.87 ≈ 53.13 degrees.
The calculator will prompt you to enter values for Side A and Hypotenuse C, and select "Side A and Hypotenuse C".
Scenario 3: Using Side B and Hypotenuse C
If Side B = 4 and Hypotenuse C = 5, this implies the cosine of Angle B is 4/5.
cos(B) = Adjacent / Hypotenuse = 4 / 5
Angle B = arccos(4/5) ≈ 36.87 degrees
Angle A = 180 – 90 – 36.87 ≈ 53.13 degrees.
The calculator will prompt you to enter values for Side B and Hypotenuse C, and select "Side B and Hypotenuse C".
Important Note: The calculator first validates if the provided side lengths can form a right triangle using the Pythagorean theorem. If the sides do not satisfy the theorem within a small tolerance, it will indicate an invalid triangle. When calculating angles, ensure you select the correct option based on the sides you have available.
function calculateAngles() {
var sideA = parseFloat(document.getElementById("sideA").value);
var sideB = parseFloat(document.getElementById("sideB").value);
var sideC = parseFloat(document.getElementById("sideC").value);
var calculationType = document.getElementById("calculationType").value;
var resultDiv = document.getElementById("result");
var angleA_resultDiv = document.getElementById("angleA_result");
var angleB_resultDiv = document.getElementById("angleB_result");
// Clear previous results
angleA_resultDiv.innerHTML = "";
angleB_resultDiv.innerHTML = "";
resultDiv.style.backgroundColor = "var(–success-green)"; // Reset background
var isValid = true;
var errorMessage = "";
// Basic validation for positive numbers
if (isNaN(sideA) || sideA <= 0) {
if (calculationType === "a_b" || calculationType === "a_c") {
errorMessage += "Side A must be a positive number. ";
isValid = false;
}
}
if (isNaN(sideB) || sideB <= 0) {
if (calculationType === "a_b" || calculationType === "b_c") {
errorMessage += "Side B must be a positive number. ";
isValid = false;
}
}
if (isNaN(sideC) || sideC tolerance) {
errorMessage += "The provided side lengths do not form a valid right triangle according to the Pythagorean theorem. ";
isValid = false;
}
} else if (!isNaN(sideA) && !isNaN(sideB)) { // If only a and b are given, calculate c
var calculatedC = Math.sqrt(sideA * sideA + sideB * sideB);
if (Math.abs(calculatedC – sideC) > tolerance && !isNaN(sideC)) {
errorMessage += "Hypotenuse C is inconsistent with sides A and B. ";
isValid = false;
}
} else if (!isNaN(sideA) && !isNaN(sideC)) { // If only a and c are given, calculate b
if (sideA >= sideC) {
errorMessage += "Side A cannot be greater than or equal to Hypotenuse C. ";
isValid = false;
} else {
var calculatedB = Math.sqrt(sideC * sideC – sideA * sideA);
if (Math.abs(calculatedB – sideB) > tolerance && !isNaN(sideB)) {
errorMessage += "Side B is inconsistent with side A and Hypotenuse C. ";
isValid = false;
}
}
} else if (!isNaN(sideB) && !isNaN(sideC)) { // If only b and c are given, calculate a
if (sideB >= sideC) {
errorMessage += "Side B cannot be greater than or equal to Hypotenuse C. ";
isValid = false;
} else {
var calculatedA = Math.sqrt(sideC * sideC – sideB * sideB);
if (Math.abs(calculatedA – sideA) > tolerance && !isNaN(sideA)) {
errorMessage += "Side A is inconsistent with side B and Hypotenuse C. ";
isValid = false;
}
}
}
if (!isValid) {
angleA_resultDiv.innerHTML = errorMessage;
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
var angleA_rad, angleB_rad;
if (calculationType === "a_b") {
if (isNaN(sideA) || isNaN(sideB) || sideA <= 0 || sideB <= 0) {
angleA_resultDiv.innerHTML = "Please provide positive values for Side A and Side B.";
resultDiv.style.backgroundColor = "#dc3545";
return;
}
angleB_rad = Math.atan(sideA / sideB); // Angle B = arctan(Opposite/Adjacent) = arctan(a/b)
angleA_rad = Math.atan(sideB / sideA); // Angle A = arctan(Opposite/Adjacent) = arctan(b/a)
} else if (calculationType === "a_c") {
if (isNaN(sideA) || isNaN(sideC) || sideA <= 0 || sideC = sideC) {
angleA_resultDiv.innerHTML = "Please provide positive values for Side A and Hypotenuse C, with Side A < Hypotenuse C.";
resultDiv.style.backgroundColor = "#dc3545";
return;
}
angleB_rad = Math.asin(sideA / sideC); // Angle B = arcsin(Opposite/Hypotenuse) = arcsin(a/c)
angleA_rad = Math.acos(sideA / sideC); // Angle A = arccos(Adjacent/Hypotenuse) = arccos(a/c)
} else if (calculationType === "b_c") {
if (isNaN(sideB) || isNaN(sideC) || sideB <= 0 || sideC = sideC) {
angleA_resultDiv.innerHTML = "Please provide positive values for Side B and Hypotenuse C, with Side B < Hypotenuse C.";
resultDiv.style.backgroundColor = "#dc3545";
return;
}
angleB_rad = Math.acos(sideB / sideC); // Angle B = arccos(Adjacent/Hypotenuse) = arccos(b/c)
angleA_rad = Math.asin(sideB / sideC); // Angle A = arcsin(Opposite/Hypotenuse) = arcsin(b/c)
} else {
angleA_resultDiv.innerHTML = "Invalid calculation type selected.";
resultDiv.style.backgroundColor = "#dc3545";
return;
}
var angleA_deg = angleA_rad * (180 / Math.PI);
var angleB_deg = angleB_rad * (180 / Math.PI);
angleA_resultDiv.innerHTML = "Angle A: " + angleA_deg.toFixed(2) + "°";
angleB_resultDiv.innerHTML = "Angle B: " + angleB_deg.toFixed(2) + "°";
}