Aas Triangle Calculator - Law of Sines Solver

An aas triangle calculator solves a triangle when you know two interior angles and one side that is not between them. "AAS" stands for Angle-Angle-Side — the congruence case where two triangles with matching data are always identical, not just similar. The form takes angles A and B plus side a (opposite A) and returns the third angle, the other two sides, the area, and the perimeter. It fits the angle-and-known-leg data that triangulation, navigation, and roof framing work actually produce.

• • Triangulation to an inaccessible point: Stand at two accessible stations, measure the angle at each, and use the non-included leg to recover the distance to a point you cannot reach directly.
• • Navigation and chart legs: Combine a chart-measured leg with bearings from each end of that leg to plot the line of sight to a buoy, light, or landmark.
• • Roof and truss framing: Compute rafter length and sheathing area when the pitch angles and one rafter side are known.
• • Physics force problems: Solve the geometry under a force diagram when two angles and a non-adjacent side are known.

AAS is the data shape produced when you have two angles plus a non-included side — opposite one of the known angles, not the side between them. That is the structural difference from ASA, where the known side sits between the two angles.

Enter angles A and B and side a into the form, and the result panel shows the third angle, the other two sides, the area, and the perimeter. The form rejects combinations that would force a non-positive third angle.

For the general SSS, SAS, and ASA workflows that complement AAS, the triangle calculator handles all three of the other common congruence cases in one tool.

The calculator takes your two angle inputs (A and B) and one non-included reference side a, computes the third angle C as 180 - A - B, and applies the Law of Sines to recover the missing sides. Area and perimeter follow without any further measurement.

• A, B, C: The three interior angles in degrees. C is computed from A and B.
• a: Reference side opposite angle A. Sets the scale.
• b, c: Sides opposite B and C, from the Law of Sines.

The Law of Sines is the only trigonometric identity you need. In any triangle, a/sin(A) = b/sin(B) = c/sin(C), so knowing one full ratio (side a and angle A) is enough to derive the other two sides. The non-included side is what locks down the scale and turns a family of similar triangles into one specific triangle.

According to Wolfram MathWorld, the Law of Sines states a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius, so AAS data alone is enough to lock down the missing sides. When the data is angles-only with no side, the AAA triangle calculator covers that case with the same engine.

These four ideas cover what makes AAS different from ASA, SSS, and AAA, and why the Law of Sines is the right tool for it.

The congruence criterion tells you the inputs uniquely identify a triangle, and the included-vs-non-included distinction is the cue for the right formula path.

If your third angle works out to 90°, the right triangle calculator covers the special Pythagorean relationships that apply to that case.

Follow these steps in order. The result panel updates as you type, so you can treat it as a scratch pad while you try different angles.

1. 1 Enter angle A: Type the first interior angle in degrees. Use 0 < A < 180, otherwise the form rejects the input.
2. 2 Enter angle B: Type the second interior angle. C is computed as 180 - A - B.
3. 3 Enter side a: Type the length of the side opposite angle A. This non-included side fixes the scale.
4. 4 Read the third angle: Confirm C is positive. If it is zero or negative, your A and B do not form a valid triangle.
5. 5 Read sides b and c: The lengths opposite B and C, from the Law of Sines.
6. 6 Use the area and perimeter: Copy the area for material estimates and the perimeter for trim or framing totals.

A navigator needs the distance from position A to a remote, inaccessible point C. They set up at a second accessible station B, record the angle at A (60°) between the line of sight to B and to C, the angle at B (50°) between the line of sight to A and to C, and the length of BC (7 m) — the non-included side opposite angle A. Enter A = 60, B = 50, side a = 7. C comes out to 70°, side b ≈ 6.19 m as the distance from A to C, side c ≈ 7.60 m as the distance from A to B, area ≈ 20.36 sq m, and perimeter ≈ 20.79 m.

When you already have two sides and the angle between them, the area of an oblique triangle calculator computes the same area with the ½ab sin(C) formula used here.

Solving AAS triangles by hand is fast once you know the Law of Sines, but the aas triangle calculator removes a few classes of error that are easy to make on paper, especially when the third angle works out to obtuse or the side measurement is small.

• • Catches impossible angle combinations: If A and B do not leave a positive third angle, the calculator flags the error.
• • Skips the unit conversion step: You enter degrees, the calculator converts to radians, and you read the sides in the same length unit.
• • Works for any AAS triangle: The Law of Sines applies to acute, right, and obtuse triangles without a separate path.
• • Mirrors a typical triangulation workflow: Two angles at accessible stations plus a leg that is not the included baseline are a common field-data shape, and the form accepts that trio directly.
• • Lets you re-run the math quickly: Each new angle or side value updates the result panel in real time, so you can iterate on a measurement.
• • Useful as a congruence check: When two triangles share the same AAS data, the side outputs let you confirm they are identical, not similar.

These benefits matter most when the inputs come from a real measurement. Angle data and one measured leg tend to come first, and the AAS-first workflow keeps the math in step with that.

If your source data is in gradians or radians instead of degrees, the angle converter lets you convert each angle before entering it here.

Five things change the answer the calculator returns, and the limitations below cover the assumptions behind the Law of Sines itself.

AAS information uniquely identifies a triangle only because a non-included side is present. With only two angles and no side, the data describes a family of similar triangles. Floating-point rounding can display C as 60.00 when the true value is 59.9999, and the Law of Sines itself only applies to planar Euclidean triangles (not spherical longitude/latitude on a globe).

A real measurement of a physical triangle almost always gives you one accurate distance plus angles, which is exactly the AAS data the tool needs.

According to Wolfram MathWorld, two triangles are congruent whenever two angles and a non-included side of one are equal to two angles and a non-included side of the other, which is the AAS congruence criterion.

Once you have the three side lengths, the triangle area calculator cross-checks the area with Heron's formula for an independent verification.