Calculate area, perimeter, angles, and side lengths of a triangle.
All sides (SSS)
Two sides and included angle (SAS)
Two angles and included side (ASA)
Two angles and non-included side (AAS)
Right Triangle (legs a, b)
Results will appear here.
Understanding Triangle Properties and Calculations
Triangles are fundamental geometric shapes with three sides and three angles. Understanding how to calculate their properties is essential in various fields, including geometry, trigonometry, engineering, architecture, and physics. This calculator helps you determine key attributes of a triangle based on the information you provide.
Triangle Types and Input Methods:
This calculator supports several common scenarios for defining a triangle:
Side-Side-Side (SSS): Given the lengths of all three sides (a, b, c).
Side-Angle-Side (SAS): Given the lengths of two sides (e.g., a, b) and the measure of the angle between them (C).
Angle-Side-Angle (ASA): Given the measures of two angles (e.g., A, C) and the length of the side between them (b).
Angle-Angle-Side (AAS): Given the measures of two angles (e.g., A, B) and the length of a non-included side (e.g., c).
Right Triangle (Legs): Given the lengths of the two legs (a, b) of a right-angled triangle. The hypotenuse and angles can then be derived.
Key Triangle Formulas Used:
The calculator employs several fundamental geometric and trigonometric formulas:
1. Area Calculation:
Heron's Formula (for SSS):
First, calculate the semi-perimeter (s): s = (a + b + c) / 2
Then, the area (A) is: A = sqrt(s * (s - a) * (s - b) * (s - c))
SAS Formula:
Area (A) = (1/2) * a * b * sin(C)
ASA/AAS Formula:
First, find the third angle (e.g., for ASA with A, b, C, find B = 180 – A – C). Then use the Law of Sines to find another side if needed, and proceed like SAS or use a direct formula if available, or calculate third angle and use another side with sine.
A simpler approach for ASA (A, b, C) and AAS (A, B, c) is to calculate the third angle first (e.g., B = 180 – A – C for ASA). Then, the area can be calculated using derived sides and angles, or more directly. For instance, with ASA (A, b, C), we can find side 'a' using the Law of Sines: a = b * sin(A) / sin(B). Then use the SAS area formula: Area = 0.5 * a * b * sin(C).
Right Triangle (Legs):
Area (A) = (1/2) * legA * legB
2. Perimeter Calculation:
The perimeter (P) is simply the sum of the lengths of all sides: P = a + b + c.
Can be used to find the unknown side 'c': c = sqrt(a^2 + b^2 - 2ab * cos(C)). Once 'c' is known, SSS methods can find the other angles.
ASA/AAS: Law of Sines
Once all angles are known (third angle = 180 – sum of two known angles), use the Law of Sines to find unknown sides.
Right Triangle (Legs):
tan(A) = legA / legB => A = atan(legA / legB)
tan(B) = legB / legA => B = atan(legB / legA)
Angle C is always 90 degrees.
4. Side Calculation:
SAS: Law of Cosines
c = sqrt(a^2 + b^2 - 2ab * cos(C))
ASA/AAS: Law of Sines
a / sin(A) = b / sin(B) = c / sin(C). If one side and all angles are known, other sides can be found.
Right Triangle (Legs):
Hypotenuse (c): Pythagorean Theorem: c = sqrt(legA^2 + legB^2)
Important Considerations:
Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this condition is not met, a valid triangle cannot be formed.
Angle Sum: The sum of the interior angles of any triangle is always 180 degrees.
Units: Ensure all length inputs are in the same unit (e.g., meters, inches, cm) and angles are in degrees as specified. The output units will correspond to the input units.
Example Usage:
Suppose you have a triangle with sides a = 5, b = 6, and c = 7. Using the SSS method:
function showInputs() {
var selectedType = document.getElementById("inputType").value;
document.getElementById("sss-inputs").style.display = "none";
document.getElementById("sas-inputs").style.display = "none";
document.getElementById("asa-inputs").style.display = "none";
document.getElementById("aas-inputs").style.display = "none";
document.getElementById("right-inputs").style.display = "none";
if (selectedType === "side-side-side") {
document.getElementById("sss-inputs").style.display = "flex";
} else if (selectedType === "side-angle-side") {
document.getElementById("sas-inputs").style.display = "flex";
} else if (selectedType === "angle-side-angle") {
document.getElementById("asa-inputs").style.display = "flex";
} else if (selectedType === "angle-angle-side") {
document.getElementById("aas-inputs").style.display = "flex";
} else if (selectedType === "right-triangle") {
document.getElementById("right-inputs").style.display = "flex";
}
}
function degreesToRadians(degrees) {
return degrees * Math.PI / 180;
}
function radiansToDegrees(radians) {
return radians * 180 / Math.PI;
}
function calculateTriangle() {
var selectedType = document.getElementById("inputType").value;
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = 'Results will appear here.'; // Clear previous results
var a, b, c, angleA, angleB, angleC;
var area, perimeter;
try {
if (selectedType === "side-side-side") {
a = parseFloat(document.getElementById("sideA_sss").value);
b = parseFloat(document.getElementById("sideB_sss").value);
c = parseFloat(document.getElementById("sideC_sss").value);
if (isNaN(a) || isNaN(b) || isNaN(c) || a <= 0 || b <= 0 || c <= 0) {
throw new Error("Please enter valid positive lengths for all sides.");
}
if (a + b <= c || a + c <= b || b + c <= a) {
throw new Error("The triangle inequality theorem is not satisfied. These side lengths cannot form a triangle.");
}
perimeter = a + b + c;
var s = perimeter / 2;
area = Math.sqrt(s * (s – a) * (s – b) * (s – c));
angleA = radiansToDegrees(Math.acos((b*b + c*c – a*a) / (2*b*c)));
angleB = radiansToDegrees(Math.acos((a*a + c*c – b*b) / (2*a*c)));
angleC = 180 – angleA – angleB; // Third angle
} else if (selectedType === "side-angle-side") {
a = parseFloat(document.getElementById("sideA_sas").value);
b = parseFloat(document.getElementById("sideB_sas").value);
angleC = parseFloat(document.getElementById("angleC_sas").value);
if (isNaN(a) || isNaN(b) || isNaN(angleC) || a <= 0 || b <= 0 || angleC = 180) {
throw new Error("Please enter valid positive lengths for sides and an angle between 0 and 180 degrees.");
}
var angleCRad = degreesToRadians(angleC);
c = Math.sqrt(a*a + b*b – 2*a*b*Math.cos(angleCRad));
perimeter = a + b + c;
area = 0.5 * a * b * Math.sin(angleCRad);
angleA = radiansToDegrees(Math.asin((a * Math.sin(angleCRad)) / c));
angleB = 180 – angleA – angleC;
} else if (selectedType === "angle-side-angle") {
angleA = parseFloat(document.getElementById("angleA_asa").value);
b = parseFloat(document.getElementById("sideB_asa").value);
angleC = parseFloat(document.getElementById("angleC_asa").value);
if (isNaN(angleA) || isNaN(b) || isNaN(angleC) || angleA <= 0 || b <= 0 || angleC = 180) {
throw new Error("Please enter valid positive length for the side and angles that sum to less than 180 degrees.");
}
angleB = 180 – angleA – angleC;
var angleARad = degreesToRadians(angleA);
var angleBRad = degreesToRadians(angleB);
var angleCRad = degreesToRadians(angleC);
a = b * Math.sin(angleARad) / Math.sin(angleBRad);
c = b * Math.sin(angleCRad) / Math.sin(angleBRad);
perimeter = a + b + c;
area = 0.5 * a * b * Math.sin(angleCRad); // Using sides a, b and angle C
} else if (selectedType === "angle-angle-side") {
angleA = parseFloat(document.getElementById("angleA_aas").value);
angleB = parseFloat(document.getElementById("angleB_aas").value);
c = parseFloat(document.getElementById("sideC_aas").value);
if (isNaN(angleA) || isNaN(angleB) || isNaN(c) || angleA <= 0 || angleB <= 0 || c = 180) {
throw new Error("Please enter valid positive length for the side and angles that sum to less than 180 degrees.");
}
angleC = 180 – angleA – angleB;
var angleARad = degreesToRadians(angleA);
var angleCRad = degreesToRadians(angleC);
var angleBRad = degreesToRadians(angleB);
a = c * Math.sin(angleARad) / Math.sin(angleBRad);
b = c * Math.sin(angleCRad) / Math.sin(angleBRad);
perimeter = a + b + c;
area = 0.5 * a * b * Math.sin(angleCRad); // Using sides a, b and angle C
} else if (selectedType === "right-triangle") {
var legA = parseFloat(document.getElementById("legA_rt").value);
var legB = parseFloat(document.getElementById("legB_rt").value);
if (isNaN(legA) || isNaN(legB) || legA <= 0 || legB <= 0) {
throw new Error("Please enter valid positive lengths for both legs.");
}
a = legA; // Assign to a and b for consistency in output
b = legB;
c = Math.sqrt(legA*legA + legB*legB); // Hypotenuse
angleC = 90;
perimeter = a + b + c;
area = 0.5 * legA * legB;
angleA = radiansToDegrees(Math.atan(legA / legB));
angleB = radiansToDegrees(Math.atan(legB / legA));
}
var resultHTML = '