Right Triangle Solver
Use this calculator to find the unknown sides and angles of a right triangle. Enter any two known values (sides or acute angles), and the calculator will determine the remaining properties.
.right-triangle-calculator-container {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f9f9f9;
padding: 25px;
border-radius: 10px;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1);
max-width: 700px;
margin: 30px auto;
border: 1px solid #e0e0e0;
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.right-triangle-calculator-container h2 {
text-align: center;
color: #333;
margin-bottom: 20px;
font-size: 28px;
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.right-triangle-calculator-container p {
margin-bottom: 15px;
line-height: 1.6;
color: #555;
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.calculator-form .form-group {
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align-items: center;
margin-bottom: 15px;
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.calculator-form .form-group label {
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margin-right: 15px;
color: #444;
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.calculator-form .form-group input[type="number"] {
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border: 1px solid #ccc;
border-radius: 5px;
font-size: 16px;
max-width: 150px;
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.calculator-form .form-group span {
margin-left: 10px;
color: #666;
}
.calculator-form .form-actions {
text-align: center;
margin-top: 25px;
}
.calculate-button, .clear-button {
background-color: #007bff;
color: white;
padding: 12px 25px;
border: none;
border-radius: 5px;
cursor: pointer;
font-size: 17px;
transition: background-color 0.3s ease, transform 0.2s ease;
margin: 0 10px;
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.clear-button {
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}
.calculate-button:hover {
background-color: #0056b3;
transform: translateY(-2px);
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.clear-button:hover {
background-color: #5a6268;
transform: translateY(-2px);
}
.calculator-result {
margin-top: 30px;
padding: 20px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 8px;
color: #155724;
font-size: 17px;
}
.calculator-result h3 {
color: #0f5132;
margin-top: 0;
margin-bottom: 15px;
font-size: 22px;
}
.calculator-result p {
margin-bottom: 8px;
color: #155724;
}
.calculator-result p.error {
color: #721c24;
background-color: #f8d7da;
border-color: #f5c6cb;
padding: 10px;
border-radius: 5px;
}
.calculator-result p.info {
color: #0c5460;
background-color: #d1ecf1;
border-color: #bee5eb;
padding: 10px;
border-radius: 5px;
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.right-triangle-calculator-container h3 {
color: #333;
margin-top: 25px;
margin-bottom: 15px;
font-size: 24px;
}
.right-triangle-calculator-container ul {
list-style-type: disc;
margin-left: 20px;
margin-bottom: 15px;
color: #555;
}
.right-triangle-calculator-container li {
margin-bottom: 8px;
}
function toRadians(degrees) {
return degrees * (Math.PI / 180);
}
function toDegrees(radians) {
return radians * (180 / Math.PI);
}
function calculateTriangle() {
var sideA_input = document.getElementById('sideA').value;
var sideB_input = document.getElementById('sideB').value;
var sideC_input = document.getElementById('sideC').value;
var angleA_input = document.getElementById('angleA').value;
var angleB_input = document.getElementById('angleB').value;
var a = parseFloat(sideA_input);
var b = parseFloat(sideB_input);
var c = parseFloat(sideC_input);
var A_deg = parseFloat(angleA_input);
var B_deg = parseFloat(angleB_input);
var knownValues = [];
if (!isNaN(a) && a > 0) knownValues.push({ type: 'a', value: a });
if (!isNaN(b) && b > 0) knownValues.push({ type: 'b', value: b });
if (!isNaN(c) && c > 0) knownValues.push({ type: 'c', value: c });
if (!isNaN(A_deg) && A_deg > 0 && A_deg 0 && B_deg < 90) knownValues.push({ type: 'B', value: B_deg });
if (knownValues.length 2) {
document.getElementById('triangleResult').innerHTML = 'Please enter exactly two values to solve the triangle. Clear and try again if you entered too many.';
return;
}
var res_a = NaN, res_b = NaN, res_c = NaN;
var res_A_deg = NaN, res_B_deg = NaN;
var val1 = knownValues[0];
var val2 = knownValues[1];
// Helper to get values by type
function getVal(typeChar) {
if (val1.type === typeChar) return val1.value;
if (val2.type === typeChar) return val2.value;
return NaN;
}
// Case 1: Two sides (a, b)
if ((val1.type === 'a' && val2.type === 'b') || (val1.type === 'b' && val2.type === 'a')) {
res_a = getVal('a');
res_b = getVal('b');
res_c = Math.sqrt(res_a * res_a + res_b * res_b);
res_A_deg = toDegrees(Math.atan(res_a / res_b));
res_B_deg = 90 – res_A_deg;
}
// Case 2: Side a and Hypotenuse c
else if ((val1.type === 'a' && val2.type === 'c') || (val1.type === 'c' && val2.type === 'a')) {
res_a = getVal('a');
res_c = getVal('c');
if (res_a >= res_c) {
document.getElementById('triangleResult').innerHTML = 'Side A cannot be greater than or equal to the hypotenuse C.';
return;
}
res_b = Math.sqrt(res_c * res_c – res_a * res_a);
res_A_deg = toDegrees(Math.asin(res_a / res_c));
res_B_deg = 90 – res_A_deg;
}
// Case 3: Side b and Hypotenuse c
else if ((val1.type === 'b' && val2.type === 'c') || (val1.type === 'c' && val2.type === 'b')) {
res_b = getVal('b');
res_c = getVal('c');
if (res_b >= res_c) {
document.getElementById('triangleResult').innerHTML = 'Side B cannot be greater than or equal to the hypotenuse C.';
return;
}
res_a = Math.sqrt(res_c * res_c – res_b * res_b);
res_B_deg = toDegrees(Math.asin(res_b / res_c));
res_A_deg = 90 – res_B_deg;
}
// Case 4: Side a and Angle A
else if ((val1.type === 'a' && val2.type === 'A') || (val1.type === 'A' && val2.type === 'a')) {
res_a = getVal('a');
res_A_deg = getVal('A');
var A_rad = toRadians(res_A_deg);
res_B_deg = 90 – res_A_deg;
res_c = res_a / Math.sin(A_rad);
res_b = res_a / Math.tan(A_rad);
}
// Case 5: Side a and Angle B
else if ((val1.type === 'a' && val2.type === 'B') || (val1.type === 'B' && val2.type === 'a')) {
res_a = getVal('a');
res_B_deg = getVal('B');
var B_rad = toRadians(res_B_deg);
res_A_deg = 90 – res_B_deg;
res_c = res_a / Math.cos(B_rad);
res_b = res_a * Math.tan(B_rad);
}
// Case 6: Side b and Angle A
else if ((val1.type === 'b' && val2.type === 'A') || (val1.type === 'A' && val2.type === 'b')) {
res_b = getVal('b');
res_A_deg = getVal('A');
var A_rad = toRadians(res_A_deg);
res_B_deg = 90 – res_A_deg;
res_c = res_b / Math.cos(A_rad);
res_a = res_b * Math.tan(A_rad);
}
// Case 7: Side b and Angle B
else if ((val1.type === 'b' && val2.type === 'B') || (val1.type === 'B' && val2.type === 'b')) {
res_b = getVal('b');
res_B_deg = getVal('B');
var B_rad = toRadians(res_B_deg);
res_A_deg = 90 – res_B_deg;
res_c = res_b / Math.sin(B_rad);
res_a = res_b / Math.tan(B_rad);
}
// Case 8: Side c and Angle A
else if ((val1.type === 'c' && val2.type === 'A') || (val1.type === 'A' && val2.type === 'c')) {
res_c = getVal('c');
res_A_deg = getVal('A');
var A_rad = toRadians(res_A_deg);
res_B_deg = 90 – res_A_deg;
res_a = res_c * Math.sin(A_rad);
res_b = res_c * Math.cos(A_rad);
}
// Case 9: Side c and Angle B
else if ((val1.type === 'c' && val2.type === 'B') || (val1.type === 'B' && val2.type === 'c')) {
res_c = getVal('c');
res_B_deg = getVal('B');
var B_rad = toRadians(res_B_deg);
res_A_deg = 90 – res_B_deg;
res_b = res_c * Math.sin(B_rad);
res_a = res_c * Math.cos(B_rad);
}
// Case 10: Two Angles (A, B)
else if ((val1.type === 'A' && val2.type === 'B') || (val1.type === 'B' && val2.type === 'A')) {
res_A_deg = getVal('A');
res_B_deg = getVal('B');
if (Math.abs(res_A_deg + res_B_deg – 90) < 0.001) { // Allow for floating point inaccuracies
document.getElementById('triangleResult').innerHTML = 'Angles A and B are consistent (sum to 90°), but at least one side length is required to determine the specific dimensions of the triangle.';
} else {
document.getElementById('triangleResult').innerHTML = 'Angles A and B must sum to 90° for a right triangle.';
}
return;
}
else {
document.getElementById('triangleResult').innerHTML = 'Invalid combination of inputs. Please ensure you provide two distinct values that allow for solving a right triangle.';
return;
}
var resultHTML = '
Calculated Triangle Properties:
';
resultHTML += '
Side A (Opposite Angle A): ' + res_a.toFixed(4) + ' units';
resultHTML += '
Side B (Opposite Angle B): ' + res_b.toFixed(4) + ' units';
resultHTML += '
Side C (Hypotenuse): ' + res_c.toFixed(4) + ' units';
resultHTML += '
Angle A: ' + res_A_deg.toFixed(4) + '°';
resultHTML += '
Angle B: ' + res_B_deg.toFixed(4) + '°';
resultHTML += '
Angle C: 90°';
document.getElementById('triangleResult').innerHTML = resultHTML;
}
function clearForm() {
document.getElementById('sideA').value = ";
document.getElementById('sideB').value = ";
document.getElementById('sideC').value = ";
document.getElementById('angleA').value = ";
document.getElementById('angleB').value = ";
document.getElementById('triangleResult').innerHTML = ";
}
Understanding the Right Triangle
A right triangle is a special type of triangle that has one angle measuring exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and it is always the longest side. The other two sides are called legs.
Solving a right triangle means finding the lengths of all its sides and the measures of all its angles when some information is already known. This calculator uses fundamental geometric and trigonometric principles to achieve this.
Key Principles Used:
- Pythagorean Theorem: For any right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b). Mathematically, this is expressed as
a² + b² = c².
- Angle Sum Property: The sum of all angles in any triangle is 180 degrees. Since a right triangle has one 90-degree angle, the sum of the other two acute angles (Angle A and Angle B) must be 90 degrees (
A + B = 90°).
- Trigonometric Ratios (SOH CAH TOA): These ratios relate the angles of a right triangle to the ratios of its side lengths:
- Sine (SOH):
sin(Angle) = Opposite / Hypotenuse
- Cosine (CAH):
cos(Angle) = Adjacent / Hypotenuse
- Tangent (TOA):
tan(Angle) = Opposite / Adjacent
These ratios, along with their inverse functions (arcsin, arccos, arctan), allow us to find unknown angles from side ratios or unknown sides from an angle and a known side.
How to Use the Calculator:
- Identify Known Values: Look at your right triangle and determine which two values (sides or acute angles) you already know.
- Enter Values: Input these two values into the corresponding fields in the calculator. Ensure that angles are entered in degrees.
- Click "Calculate": The calculator will then apply the appropriate formulas to determine the remaining sides and angles.
- Review Results: The calculated values for all sides and angles will be displayed.
- Clear and Repeat: Use the "Clear" button to reset the form for a new calculation.
Examples:
Example 1: Given Two Legs
Suppose you have a right triangle where Side A = 3 units and Side B = 4 units.
- Input: Side A = 3, Side B = 4
- Calculation:
- Using Pythagorean Theorem:
c = √(3² + 4²) = √(9 + 16) = √25 = 5 units
- Using Tangent for Angle A:
A = arctan(3/4) ≈ 36.87°
- Using Angle Sum:
B = 90° - 36.87° = 53.13°
- Result: Side C = 5 units, Angle A ≈ 36.87°, Angle B ≈ 53.13°
Example 2: Given One Leg and the Hypotenuse
Consider a right triangle where Side A = 5 units and the Hypotenuse C = 13 units.
- Input: Side A = 5, Side C = 13
- Calculation:
- Using Pythagorean Theorem:
b = √(13² - 5²) = √(169 - 25) = √144 = 12 units
- Using Sine for Angle A:
A = arcsin(5/13) ≈ 22.62°
- Using Angle Sum:
B = 90° - 22.62° = 67.38°
- Result: Side B = 12 units, Angle A ≈ 22.62°, Angle B ≈ 67.38°
Example 3: Given One Leg and One Acute Angle
Imagine a right triangle with Side A = 10 units and Angle A = 30 degrees.
- Input: Side A = 10, Angle A = 30
- Calculation:
- Using Angle Sum:
B = 90° - 30° = 60°
- Using Sine for Hypotenuse C:
C = A / sin(A) = 10 / sin(30°) = 10 / 0.5 = 20 units
- Using Tangent for Side B:
B = A / tan(A) = 10 / tan(30°) ≈ 10 / 0.5774 ≈ 17.32 units
- Result: Side B ≈ 17.32 units, Side C = 20 units, Angle B = 60°
Applications of Right Triangles:
Right triangles are fundamental in many fields:
- Construction and Architecture: Used for calculating roof pitches, ramp slopes, and ensuring structural stability.
- Navigation: Essential for determining distances, bearings, and positions in air and sea travel.
- Engineering: Applied in mechanical, civil, and electrical engineering for design, stress analysis, and circuit calculations.
- Physics: Used to resolve forces into components, analyze projectile motion, and understand wave phenomena.
- Surveying: Crucial for measuring distances and elevations across land.
- Computer Graphics: Forms the basis for rendering 3D objects and calculating perspectives.
This calculator simplifies these complex calculations, making it a valuable tool for students, educators, and professionals alike.