Triangle Calculator: SSS, SAS, ASA, AAS & SSA Solver

Solving for a triangle’s missing sides, angles, or area is a common challenge in geometry, construction, and even DIY projects. Our comprehensive triangle calculator provides instant and accurate solutions, whether you have two angles and a side, three sides, or another combination.

Provide 3 values, including at least one side, to solve for the remaining properties of a triangle.

Side a Side b Side c

Angle A Angle B Angle C

Angle Unit:

How to Use Our Triangle Calculator

Our tool is designed to be flexible and intuitive. To get started, select the type of information you already know from the options below. Then, enter your known values into the corresponding fields. The calculator will automatically solve for all missing properties.

A standard triangle has three sides (labeled a, b, and c) and three angles (labeled A, B, and C). By convention, angle A is opposite side a, angle B is opposite side b, and angle C is opposite side c.

SSS (Side-Side-Side): Use this option when you know the lengths of all three sides of the triangle.
- Side a: The length of the first side.
- Side b: The length of the second side.
- Side c: The length of the third side.
SAS (Side-Angle-Side): Use this when you know two side lengths and the angle that is between them.
- Side b: The length of the first known side.
- Angle A: The measure of the angle between sides b and c.
- Side c: The length of the second known side.
ASA (Angle-Side-Angle): Use this when you know two angles and the side that is between them.
- Angle B: The measure of the first known angle.
- Side a: The length of the side between angles B and C.
- Angle C: The measure of the second known angle.
AAS (Angle-Angle-Side): Use this when you know two angles and a side that is not between them.
- Angle A: The measure of the first known angle.
- Angle B: The measure of the second known angle.
- Side b: The length of the side that is not between angles A and B.
SSA (Side-Side-Angle): Use this when you know two sides and an angle that is not between them. This is known as the “Ambiguous Case” because it can sometimes result in two possible triangles, one triangle, or no triangle at all. Our calculator will provide all possible solutions.
- Side a: Length of the side opposite Angle A.
- Side b: Length of the side adjacent to Angle A.
- Angle A: Measure of the known angle.

Understanding Your Results

After you input your values, our calculator provides a complete analysis of your triangle’s geometry. Here’s a breakdown of what each result means and the mathematical principles behind them.

Calculated Sides and Angles

The primary results are the missing lengths of the sides (a, b, c) and the measures of the angles (A, B, C) in degrees. The calculator uses two fundamental trigonometric laws for these calculations:

The Law of Cosines: This law is used to find a missing side when you know two sides and the included angle (SAS) or to find the angles when you know all three sides (SSS). The formulas are:
$a^{2} = b^{2} + c^{2} - 2 b c cos (A)$ $b^{2} = a^{2} + c^{2} - 2 a c cos (B)$ $c^{2} = a^{2} + b^{2} - 2 ab cos (C)$
The Law of Sines: This law is used when you know a side and its opposite angle, plus one other piece of information (ASA, AAS). It establishes a ratio between the sides of a triangle and the sines of their opposite angles.
$sin ( A ) a = sin ( B ) b = sin ( C ) c$

Geometric Properties

Beyond the basic sides and angles, the calculator also provides these key properties:

Area: This is the total space enclosed by the triangle. The area can be calculated in several ways depending on the known information. A common method, especially when you know all three sides, is Heron’s Formula.
1. First, calculate the semi-perimeter ( $s$ ), which is half the perimeter: $s = f r a c a + b + c 2$
2. Then, apply Heron’s Formula: Area $= s q r t s (s - a) (s - b) (s - c)$
Perimeter: This is the total length of the boundary of the triangle. It’s the simplest calculation: just the sum of the three side lengths.
$Perimeter = a + b + c$
Semi-perimeter: As mentioned above, this is simply half the perimeter. It’s a crucial intermediate step for calculating the area with Heron’s formula and other properties like the radius of the inscribed circle.
Heights (h_a, h_b, h_c): A triangle has three heights (or altitudes). The height $h_a$ is the perpendicular distance from vertex A to the opposite side a. The heights are calculated using the area:
$h_{a} = a 2 \times Area ∣ h_{b} = b 2 \times Area ∣ h_{c} = c 2 \times Area$

Triangle Classification Chart

Your results will also classify the triangle based on its sides and angles. This helps you visualize and understand its shape and properties.

Classification by Side Lengths	Definition	Classification by Internal Angles	Definition
Equilateral	All three sides are equal in length.	Acute	All three angles are less than $90^\\circ$.
Isosceles	Two sides are equal in length.	Right	One angle is exactly $90^\\circ$.
Scalene	All three sides have different lengths.	Obtuse	One angle is greater than $90^\\circ$.

Frequently Asked Questions

What is the triangle inequality theorem?

The Triangle Inequality Theorem is a fundamental rule of geometry that states the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If you try to input side lengths that violate this rule, a triangle cannot be formed.

$a + b c$
$a + c b$
$b + c a$

For example, you cannot form a triangle with sides of length 3, 4, and 8 because $3 + 4 = 7$ , which is not greater than 8. Our calculator will alert you if your inputs do not form a valid triangle.

Why is SSA called the “ambiguous case”?

SSA (Side-Side-Angle) is called the ambiguous case because the given information—two sides and a non-included angle—can produce zero, one, or even two distinct possible triangles.

Here’s why: Imagine you have side b, side a, and angle A. You swing side a from the end of side b.

No Solution: If side a is too short to reach the third side, no triangle can be formed.
One Solution: If side a just touches the third side (forming a right angle) or if it’s longer than side b, there is only one possible triangle.
Two Solutions: If side a is shorter than side b but long enough to reach the third side, it can intersect it at two different points, creating two valid triangles (one acute and one obtuse).

Our calculator automatically analyzes the SSA case and will show you all valid solutions.

How do I know if my triangle is a right triangle?

A triangle is a right triangle if one of its interior angles is exactly $90^\\circ$. You can also check using the Pythagorean Theorem. If the side lengths a, b, and c (where c is the longest side) satisfy the equation $a^{2} + b^{2} = c^{2}$ , then it is a right triangle. Our calculator will explicitly state “Right” in the angle classification if this is the case.

What is the difference between an equilateral and an isosceles triangle?

The key difference is the number of equal sides and angles.

An Equilateral triangle is the most symmetrical type. All three sides are equal, and all three internal angles are equal, each measuring $60^\\circ$.
An Isosceles triangle has two equal sides. The angles opposite these equal sides are also equal. An equilateral triangle is technically a very specific type of isosceles triangle.

Can a triangle be both right and isosceles?

Yes. A right isosceles triangle has one $90^\\circ$ angle and two sides of equal length (the two legs that form the right angle). This means its other two angles must be equal, and since all angles must sum to $180^\\circ$, they are each $45^\\circ$. The angles in a right isosceles triangle are always 45-45-90.

How are triangles used in the real world?

Triangles are fundamental to engineering, architecture, navigation, and even art due to their inherent strength and rigidity.

Construction & Architecture: Trusses in bridges, roofs, and geodesic domes use interconnected triangles to distribute weight and handle stress efficiently. The Eiffel Tower is a famous example of truss-based construction.
Navigation & GPS: Triangulation is a method used to determine a location by forming a triangle between the unknown point and two known reference points. GPS satellites use a 3D version of this called trilateration.
Computer Graphics: 3D models in video games and movies are built from a mesh of millions of tiny triangles called polygons. The simple, flat surface of a triangle is easy for a computer to render, and combining them creates complex surfaces.

What’s the easiest way to find the area of a triangle?

It depends on what you know.

If you know the base and height, the easiest formula is Area = $f r a c 12 t im es t e x t ba se t im es t e x t h e i g h t$ .
If you know two sides and the included angle (SAS), use the formula Area = $f r a c 12 ab s in (C)$ .
If you know all three sides (SSS), using our calculator to apply Heron’s Formula is the easiest method.

Why do the angles in a triangle always add up to 180 degrees?

This is a core property of Euclidean (flat) geometry. One simple way to visualize it is to draw any triangle on a piece of paper. Tear off the three corners (angles) and line them up next to each other along a straight line. You will see that they perfectly form a straight angle, which is $180^\\circ$.

Can I solve a triangle if I only know the three angles (AAA)?

No, you cannot determine the side lengths of a triangle if you only know the three angles. This is because knowing only the angles defines the triangle’s shape, but not its size. You can have an infinite number of triangles (a tiny one and a massive one) that are similar—meaning they have the exact same angles but different side lengths. You need to know the length of at least one side to determine the scale.

What happens if I enter impossible values?

Our calculator has built-in validation to prevent impossible calculations. For example:

Invalid Sides (SSS): If you enter side lengths that violate the Triangle Inequality Theorem (e.g., sides 2, 3, 6), the calculator will return an error stating that these sides cannot form a triangle.
Invalid Angles (ASA/AAS): If you enter two angles that sum to $180^\\circ$ or more, the calculator will return an error, as there’s no room for a third angle.

Now that you’ve mastered the geometry of your triangle, you might want to explore specific cases. For right triangles, our Pythagorean Theorem Calculator can help you quickly find the hypotenuse or legs. If you’re working with circles related to your triangles, check out our Circle Calculator to find circumference and area.

Creator

Ismael Vargas

An experienced software developer specializing in React, JavaScript, Django and Python, with more than six years’ expertise building full‑stack applications, data visualizations and cloud‑hosted solutions. He has a strong background in API integration, testing, and AWS services, delivering polished web products.

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