Find Missing Side of Triangle Calculator

Solve for Hypotenuse, Legs, or any side using Law of Cosines

1. Right Triangle (Pythagorean Theorem)

Enter any two sides to find the third. (a² + b² = c²)

2. General Triangle (SAS Method)

Find the third side using two sides and the angle between them.

Understanding How to Find the Missing Side of a Triangle

Calculating the length of a triangle’s side depends on the information you have available. There are two primary methods used in geometry: the Pythagorean Theorem for right-angled triangles and the Law of Cosines for all other triangles.

1. Using the Pythagorean Theorem

If you are working with a right triangle (a triangle where one angle is exactly 90 degrees), you can use the famous formula:

a² + b² = c²

• c is the hypotenuse (the longest side opposite the right angle).
• a and b are the other two legs.

To find the hypotenuse, calculate the square root of (a² + b²). To find a leg, calculate the square root of (c² – other leg²).

2. Using the Law of Cosines

If the triangle is not a right triangle, or if you only know two sides and the angle between them (Side-Angle-Side or SAS), the Law of Cosines is the tool to use:

c² = a² + b² – 2ab cos(γ)

This formula allows you to find the third side (c) when you know sides a and b and the angle γ (gamma) located between them.

Practical Example: Building a Ramp

Imagine you are building a wooden ramp. You know the height of the step is 3 feet (Side A) and the distance along the ground is 4 feet (Side B). Since the ground and the step meet at a 90-degree angle, we can find the ramp surface length (Side C):

3² + 4² = c²
9 + 16 = 25
√25 = 5 feet

function calculatePythagorean() {
var a = parseFloat(document.getElementById(‘sideA’).value);
var b = parseFloat(document.getElementById(‘sideB’).value);
var c = parseFloat(document.getElementById(‘sideC’).value);
var resDiv = document.getElementById(‘pythagResult’);
resDiv.style.display = ‘block’;
resDiv.style.color = ‘#2c3e50’;
if (!isNaN(a) && !isNaN(b) && isNaN(c)) {
// Find C
var result = Math.sqrt(Math.pow(a, 2) + Math.pow(b, 2));
resDiv.innerHTML = ‘Hypotenuse (c) = ‘ + result.toFixed(4);
} else if (!isNaN(a) && !isNaN(c) && isNaN(b)) {
// Find B
if (c <= a) {
resDiv.innerHTML = 'Error: Hypotenuse must be longer than Side A';
resDiv.style.color = 'red';
} else {
var result = Math.sqrt(Math.pow(c, 2) – Math.pow(a, 2));
resDiv.innerHTML = 'Side (b) = ' + result.toFixed(4);
}
} else if (!isNaN(b) && !isNaN(c) && isNaN(a)) {
// Find A
if (c <= b) {
resDiv.innerHTML = 'Error: Hypotenuse must be longer than Side B';
resDiv.style.color = 'red';
} else {
var result = Math.sqrt(Math.pow(c, 2) – Math.pow(b, 2));
resDiv.innerHTML = 'Side (a) = ' + result.toFixed(4);
}
} else {
resDiv.innerHTML = 'Please enter exactly two values to find the third.';
resDiv.style.color = 'red';
}
}
function calculateSAS() {
var a = parseFloat(document.getElementById('sasA').value);
var b = parseFloat(document.getElementById('sasB').value);
var angleDeg = parseFloat(document.getElementById('sasAngle').value);
var resDiv = document.getElementById('sasResult');
resDiv.style.display = 'block';
resDiv.style.color = '#2c3e50';
if (isNaN(a) || isNaN(b) || isNaN(angleDeg)) {
resDiv.innerHTML = 'Please enter both sides and the angle.';
resDiv.style.color = 'red';
return;
}
if (angleDeg = 180) {
resDiv.innerHTML = ‘Angle must be between 0 and 180 degrees.’;
resDiv.style.color = ‘red’;
return;
}
// Convert degrees to radians
var angleRad = angleDeg * (Math.PI / 180);
// c^2 = a^2 + b^2 – 2ab * cos(angle)
var cSquared = Math.pow(a, 2) + Math.pow(b, 2) – (2 * a * b * Math.cos(angleRad));
var result = Math.sqrt(cSquared);
resDiv.innerHTML = ‘Missing Side (c) = ‘ + result.toFixed(4);
}