🔺 Triangle Calculator
Calculate area, perimeter, angles, and all triangle properties
Triangle Calculator
3 Sides (SSS)
2 Sides + Angle (SAS)
1 Side + 2 Angles (ASA)
Base + Height
Calculation Results
Understanding Triangle Calculators
A triangle calculator is a powerful mathematical tool that helps you determine various properties of a triangle including its area, perimeter, angles, and side lengths. Whether you're a student learning geometry, an architect designing structures, or an engineer working on projects, understanding how to calculate triangle properties is essential.
What is a Triangle?
A triangle is a polygon with three sides, three vertices, and three angles. The sum of all interior angles in any triangle always equals 180 degrees. Triangles are the simplest polygon and form the basis of many geometric and structural calculations in mathematics, engineering, and architecture.
Types of Triangles
Triangles can be classified based on their sides or angles:
Classification by Sides:
- Equilateral Triangle: All three sides are equal in length, and all three angles are 60 degrees
- Isosceles Triangle: Two sides are equal in length, and two angles are equal
- Scalene Triangle: All three sides have different lengths, and all three angles are different
Classification by Angles:
- Acute Triangle: All three angles are less than 90 degrees
- Right Triangle: One angle is exactly 90 degrees
- Obtuse Triangle: One angle is greater than 90 degrees
Key Triangle Formulas
Area Calculations:
Base-Height Formula:
Area = (1/2) × base × height
Area = (1/2) × base × height
Heron's Formula (using three sides):
s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]
where s is the semi-perimeter
s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]
where s is the semi-perimeter
SAS Formula (two sides and included angle):
Area = (1/2) × a × b × sin(C)
Area = (1/2) × a × b × sin(C)
Perimeter:
Perimeter = a + b + c
(sum of all three sides)
(sum of all three sides)
Law of Cosines:
c² = a² + b² – 2ab × cos(C)
Used to find the third side when two sides and the included angle are known
Used to find the third side when two sides and the included angle are known
Law of Sines:
a/sin(A) = b/sin(B) = c/sin(C)
Used to find unknown sides or angles
Used to find unknown sides or angles
How to Use the Triangle Calculator
This calculator offers multiple input methods to accommodate different scenarios:
- 3 Sides (SSS): Enter the lengths of all three sides. The calculator will verify if a valid triangle can be formed and compute the area, perimeter, and all three angles.
- 2 Sides + Angle (SAS): Enter two sides and the angle between them. The calculator will find the third side, area, perimeter, and remaining angles.
- 1 Side + 2 Angles (ASA): Enter one side and two adjacent angles. The calculator will determine the other two sides, area, and perimeter.
- Base + Height: For quick area calculations, simply enter the base and height of the triangle.
Triangle Inequality Theorem
For any triangle to exist, the sum of any two sides must be greater than the third side. This is known as the Triangle Inequality Theorem:
a + b > c
b + c > a
a + c > b
b + c > a
a + c > b
Our calculator automatically checks this condition to ensure you're working with valid triangle dimensions.
Practical Examples
Example 1: Calculating Area with Base and Height
Problem: A triangular garden has a base of 12 meters and a height of 8 meters. What is its area?
Solution:
Area = (1/2) × base × height
Area = (1/2) × 12 × 8 = 48 square meters
Example 2: Using Three Sides (Heron's Formula)
Problem: A triangle has sides of 5 cm, 6 cm, and 7 cm. Find its area and perimeter.
Solution:
Perimeter = 5 + 6 + 7 = 18 cm
Semi-perimeter s = 18/2 = 9 cm
Area = √[9(9-5)(9-6)(9-7)] = √[9×4×3×2] = √216 ≈ 14.7 square cm
Example 3: Two Sides and Included Angle
Problem: Two sides of a triangle measure 10 m and 15 m, with an angle of 60° between them. Find the area.
Solution:
Area = (1/2) × 10 × 15 × sin(60°)
Area = (1/2) × 10 × 15 × 0.866 = 64.95 square meters
Real-World Applications
Triangle calculations are used extensively in various fields:
- Architecture and Construction: Calculating roof areas, designing triangular structures, and determining load distributions
- Engineering: Structural analysis, bridge design, and truss calculations
- Surveying: Measuring land areas and distances using triangulation
- Navigation: Determining positions and distances using triangulation methods
- Computer Graphics: Rendering 3D objects using triangle meshes
- Physics: Analyzing forces and vectors in mechanical systems
- Astronomy: Calculating distances to celestial objects using parallax
Special Right Triangles
Certain right triangles have special properties that make calculations easier:
45-45-90 Triangle:
In this isosceles right triangle, if the legs have length x, the hypotenuse has length x√2.
30-60-90 Triangle:
If the shortest side (opposite to 30°) has length x, the other leg (opposite to 60°) has length x√3, and the hypotenuse has length 2x.
Tips for Accurate Calculations
- Always ensure your angle measurements are in the correct unit (degrees or radians)
- Double-check that your triangle dimensions satisfy the triangle inequality theorem
- When working with angles, remember that the sum must equal 180 degrees
- Use consistent units for all measurements (don't mix centimeters with meters)
- For greater precision, use more decimal places in intermediate calculations
- Verify your results by using alternative calculation methods when possible
Common Mistakes to Avoid
- Confusing the base-height formula with Heron's formula
- Using the wrong angle in the Law of Cosines
- Forgetting to check the triangle inequality before calculations
- Mixing up degrees and radians in trigonometric functions
- Assuming all triangles are right triangles when they're not
- Incorrectly identifying which angle is opposite which side
Pythagorean Theorem for Right Triangles
For right triangles specifically, the Pythagorean theorem provides a direct relationship between the sides:
a² + b² = c²
where c is the hypotenuse (longest side)
where c is the hypotenuse (longest side)
This theorem only applies to right triangles and is one of the most famous mathematical relationships in geometry.
Altitude and Median
Other important triangle measurements include:
- Altitude: The perpendicular distance from a vertex to the opposite side (or its extension)
- Median: A line segment connecting a vertex to the midpoint of the opposite side
- Angle Bisector: A line that divides an angle into two equal parts
- Circumradius: The radius of the circle that passes through all three vertices
- Inradius: The radius of the circle inscribed within the triangle
Conclusion
Understanding triangle calculations is fundamental to many areas of mathematics, science, and engineering. This triangle calculator simplifies complex calculations and helps you quickly determine all properties of any triangle. Whether you're solving homework problems, planning construction projects, or working on technical designs, mastering triangle geometry will prove invaluable in your academic and professional endeavors.
Triangle Type: ' + triangleType + angleType + '
';
resultHTML += 'Area: ' + area.toFixed(2) + ' square units
';
resultHTML += 'Perimeter: ' + perimeter.toFixed(2) + ' units
';
resultHTML += 'Angle A: ' + angleA.toFixed(2) + '°
';
resultHTML += 'Angle B: ' + angleB.toFixed(2) + '°
';
resultHTML += 'Angle C: ' + angleC.toFixed(2) + '°
';
document.getElementById('resultContent').innerHTML = resultHTML;
document.getElementById('result').classList.add('show');
}
function calculate2Sides1Angle() {
var a = parseFloat(document.getElementById('sasA').value);
var b = parseFloat(document.getElementById('sasB').value);
var angleC = parseFloat(document.getElementById('angleC').value);
if (isNaN(a) || isNaN(b) || isNaN(angleC) || a <= 0 || b <= 0 || angleC = 180) {
alert('Please enter valid positive numbers. Angle must be between 0 and 180 degrees.');
return;
}
var angleCRad = toRadians(angleC);
var c = Math.sqrt(a * a + b * b – 2 * a * b * Math.cos(angleCRad));
var area = 0.5 * a * b * Math.sin(angleCRad);
var perimeter = a + b + c;
var angleA = toDegrees(Math.asin(a * Math.sin(angleCRad) / c));
var angleB = 180 – angleA – angleC;
var resultHTML = 'Side C: ' + c.toFixed(2) + ' units
';
resultHTML += 'Area: ' + area.toFixed(2) + ' square units
';
resultHTML += 'Perimeter: ' + perimeter.toFixed(2) + ' units
';
resultHTML += 'Angle A: ' + angleA.toFixed(2) + '°
';
resultHTML += 'Angle B: ' + angleB.toFixed(2) + '°
';
document.getElementById('resultContent').innerHTML = resultHTML;
document.getElementById('result').classList.add('show');
}
function calculate1Side2Angles() {
var side = parseFloat(document.getElementById('asaSide').value);
var angle1 = parseFloat(document.getElementById('asaAngle1').value);
var angle2 = parseFloat(document.getElementById('asaAngle2').value);
if (isNaN(side) || isNaN(angle1) || isNaN(angle2) || side <= 0 || angle1 <= 0 || angle2 = 180 || angle2 >= 180) {
alert('Please enter valid positive numbers. Angles must be between 0 and 180 degrees.');
return;
}
var angle3 = 180 – angle1 – angle2;
if (angle3 <= 0) {
alert('The sum of the two angles must be less than 180 degrees.');
return;
}
var angle1Rad = toRadians(angle1);
var angle2Rad = toRadians(angle