Enter three values, including at least one side length, to solve the triangle. Angles should be in degrees.
Calculation Results:
Understanding Triangle Trigonometry
Trigonometry is the branch of mathematics that explores the relationships between the side lengths and angles of triangles. Whether you are dealing with a right-angled triangle or an oblique triangle, certain fundamental laws allow you to solve for missing dimensions.
Key Trigonometric Principles
The Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($a² + b² = c²$).
SOH CAH TOA: These ratios (Sine, Cosine, Tangent) define the relationships between angles and side ratios in right triangles.
Law of Sines: $a / \sin(A) = b / \sin(B) = c / \sin(C)$. This is used when you know an angle and its opposite side.
Law of Cosines: $c² = a² + b² – 2ab \cos(C)$. This is essential for Solving SAS (Side-Angle-Side) or SSS (Side-Side-Side) triangles.
Triangle Solving Examples
Example 1: The 3-4-5 Triangle (SSS)
If you enter Side a = 3, Side b = 4, and Side c = 5, the calculator uses the Law of Cosines to determine that Angle C is exactly 90°, Angle A is approximately 36.87°, and Angle B is 53.13°.
Example 2: Side-Angle-Side (SAS)
Suppose you know Side a = 10, Side b = 12, and the included Angle C = 45°. The calculator finds Side c using the Law of Cosines and then determines the remaining angles using the Law of Sines.
Important Tips for Accuracy
To ensure valid results, remember that the sum of all internal angles in a triangle must always equal 180°. Additionally, the sum of any two sides must be strictly greater than the length of the third side (Triangle Inequality Theorem). If your inputs violate these rules, the calculator will notify you that the triangle is impossible.
function calculateTriangle() {
var a = parseFloat(document.getElementById('side_a').value);
var b = parseFloat(document.getElementById('side_b').value);
var c = parseFloat(document.getElementById('side_c').value);
var A = parseFloat(document.getElementById('angle_A').value);
var B = parseFloat(document.getElementById('angle_B').value);
var C = parseFloat(document.getElementById('angle_C').value);
var resDiv = document.getElementById('trig-results');
var content = document.getElementById('results-content');
var warn = document.getElementById('warning-msg');
resDiv.style.display = 'block';
content.innerHTML = ";
warn.innerHTML = ";
var rad = Math.PI / 180;
var deg = 180 / Math.PI;
var inputs = [a, b, c, A, B, C].filter(function(x) { return !isNaN(x); }).length;
var sideCount = [a, b, c].filter(function(x) { return !isNaN(x); }).length;
if (inputs < 3 || sideCount === 0) {
warn.innerHTML = "Error: Please enter at least 3 values, including at least one side.";
return;
}
// Logic Tree
try {
// Case SSS
if (!isNaN(a) && !isNaN(b) && !isNaN(c)) {
if (a + b <= c || a + c <= b || b + c <= a) throw "Invalid sides: Triangle inequality failed.";
A = Math.acos((b*b + c*c – a*a) / (2*b*c)) * deg;
B = Math.acos((a*a + c*c – b*b) / (2*a*c)) * deg;
C = 180 – A – B;
}
// Case SAS
else if (!isNaN(a) && !isNaN(b) && !isNaN(C)) {
c = Math.sqrt(a*a + b*b – 2*a*b*Math.cos(C * rad));
A = Math.acos((b*b + c*c – a*a) / (2*b*c)) * deg;
B = 180 – A – C;
}
else if (!isNaN(a) && !isNaN(c) && !isNaN(B)) {
b = Math.sqrt(a*a + c*c – 2*a*c*Math.cos(B * rad));
A = Math.acos((b*b + c*c – a*a) / (2*b*c)) * deg;
C = 180 – A – B;
}
else if (!isNaN(b) && !isNaN(c) && !isNaN(A)) {
a = Math.sqrt(b*b + c*c – 2*b*c*Math.cos(A * rad));
B = Math.acos((a*a + c*c – b*b) / (2*a*c)) * deg;
C = 180 – A – B;
}
// Case ASA / AAS
else if (!isNaN(A) && !isNaN(B) && (!isNaN(a) || !isNaN(b) || !isNaN(c))) {
C = 180 – A – B;
if (C <= 0) throw "Invalid angles: Sum exceeds 180°.";
if (!isNaN(a)) {
b = a * Math.sin(B * rad) / Math.sin(A * rad);
c = a * Math.sin(C * rad) / Math.sin(A * rad);
} else if (!isNaN(b)) {
a = b * Math.sin(A * rad) / Math.sin(B * rad);
c = b * Math.sin(C * rad) / Math.sin(B * rad);
} else {
a = c * Math.sin(A * rad) / Math.sin(C * rad);
b = c * Math.sin(B * rad) / Math.sin(C * rad);
}
}
else if (!isNaN(A) && !isNaN(C) && (!isNaN(a) || !isNaN(b) || !isNaN(c))) {
B = 180 – A – C;
if (B <= 0) throw "Invalid angles: Sum exceeds 180°.";
if (!isNaN(a)) {
b = a * Math.sin(B * rad) / Math.sin(A * rad);
c = a * Math.sin(C * rad) / Math.sin(A * rad);
} else if (!isNaN(b)) {
a = b * Math.sin(A * rad) / Math.sin(B * rad);
c = b * Math.sin(C * rad) / Math.sin(B * rad);
} else {
a = c * Math.sin(A * rad) / Math.sin(C * rad);
b = c * Math.sin(B * rad) / Math.sin(C * rad);
}
}
else if (!isNaN(B) && !isNaN(C) && (!isNaN(a) || !isNaN(b) || !isNaN(c))) {
A = 180 – B – C;
if (A 1) throw "No such triangle exists.";
B = Math.asin(sinB) * deg;
C = 180 – A – B;
c = a * Math.sin(C * rad) / Math.sin(A * rad);
warn.innerHTML = "Note: This setup (SSA) can have multiple solutions. One solution shown.";
}
else if (!isNaN(a) && !isNaN(c) && !isNaN(A)) {
var sinC = c * Math.sin(A * rad) / a;
if (sinC > 1) throw "No such triangle exists.";
C = Math.asin(sinC) * deg;
B = 180 – A – C;
b = a * Math.sin(B * rad) / Math.sin(A * rad);
}
else if (!isNaN(b) && !isNaN(a) && !isNaN(B)) {
var sinA = a * Math.sin(B * rad) / b;
if (sinA > 1) throw "No such triangle exists.";
A = Math.asin(sinA) * deg;
C = 180 – A – B;
c = b * Math.sin(C * rad) / Math.sin(B * rad);
}
else if (!isNaN(b) && !isNaN(c) && !isNaN(B)) {
var sinC = c * Math.sin(B * rad) / b;
if (sinC > 1) throw "No such triangle exists.";
C = Math.asin(sinC) * deg;
A = 180 – B – C;
a = b * Math.sin(A * rad) / Math.sin(B * rad);
}
else if (!isNaN(c) && !isNaN(a) && !isNaN(C)) {
var sinA = a * Math.sin(C * rad) / c;
if (sinA > 1) throw "No such triangle exists.";
A = Math.asin(sinA) * deg;
B = 180 – A – C;
b = c * Math.sin(B * rad) / Math.sin(C * rad);
}
else if (!isNaN(c) && !isNaN(b) && !isNaN(C)) {
var sinB = b * Math.sin(C * rad) / c;
if (sinB > 1) throw "No such triangle exists.";
B = Math.asin(sinB) * deg;
A = 180 – B – C;
a = c * Math.sin(A * rad) / Math.sin(C * rad);
}
else {
throw "Combination not supported or mathematically insufficient.";
}
if (isNaN(a) || isNaN(b) || isNaN(c) || isNaN(A) || isNaN(B) || isNaN(C)) {
throw "Unable to calculate with provided values.";
}
var perimeter = a + b + c;
var s = perimeter / 2;
var area = Math.sqrt(s * (s – a) * (s – b) * (s – c));
content.innerHTML =
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