Asa Triangle Calculator - Solve ASA From Two Angles

The asa triangle calculator solves a triangle when you know two of its interior angles and the side that sits between them. Type the included side a and the two adjacent angles beta and gamma, and the calculator returns the third angle alpha, the two missing sides b and c, the area, and the perimeter using the law of sines. It is built for the case when an angle-side-angle pair is the cleanest description of the triangle, such as a roof rafter or a surveying baseline.

• • Roof rafter checks: Confirm a roof rafter length when the rise angle and the seat cut angle are known and the rafter run is measured.
• • Surveying baseline: Compute the third side of a survey triangle when the included side and the two base angles from a theodolite are logged in the field book.
• • Trigonometry homework: Verify textbook exercises in which the triangle is given by an angle-side-angle pair and the answer must be checked against the law of sines.

An ASA triangle is one of the four standard cases for solving an oblique triangle by hand, alongside SSS, SAS, and AAS. The included side touches both known angles, which is what makes the law of sines solvable in a single pass: the third angle comes from the angle sum, and the two missing sides follow from a / sin(alpha) = b / sin(beta) = c / sin(gamma).

For the general case where the inputs are three sides, two sides and an included angle, or two sides and a non-included angle, the Triangle Calculator covers the matching SSS, SAS, and AAS variants in the same Math & Conversion cluster.

The calculator uses the angle sum of a triangle to recover alpha, then plugs the three angles into the law of sines to recover b and c, and folds both into the standard triangle area formula to recover the area. Every step is a single closed-form expression.

• a: Included side, sitting between angles beta and gamma. Enter it in the chosen length unit; the calculator reports b, c, and the perimeter in the same unit.
• beta, gamma: Two known interior angles in degrees, both adjacent to side a. The third angle is whatever is left of 180.
• alpha: Recovered third angle, equal to 180 - beta - gamma. This angle is opposite the included side a, which is why the law of sines uses it as the constant denominator.
• b, c, Area: Recovered sides and area use a times the sine of each angle divided by sin(beta + gamma), collapsing the area to a squared over two times sin(beta) sin(gamma) over sin(beta + gamma).

The same arithmetic works no matter which two angles are known, because alpha is always whichever angle is missing. The only structural requirement is that beta + gamma stays below 180 deg, so that sin(beta + gamma) never collapses to zero.

According to Wolfram MathWorld, the law of sines relates each side of a triangle to the sine of its opposite angle, so an ASA triangle with included side a and third angle alpha has missing sides b = a * sin(beta)/sin(alpha) and c = a * sin(gamma)/sin(alpha).

When beta + gamma is 90 deg the ASA case collapses to a right triangle, so the result should match the Pythagorean answer from the Right Triangle Calculator for the same two legs and hypotenuse.

These four ideas decide whether the law-of-sines result is the right number for the triangle on your desk.

The law of sines works for any triangle, and the ASA pattern is the cleanest way to feed it the inputs it needs in one pass.

When the same triangle is easier to describe with two sides and the included angle instead of two angles and the included side, the Area Oblique Triangle Calculator applies the law of sines and Heron's formula in the same family of oblique-triangle tools.

Use the asa triangle calculator with a, beta, and gamma in the same units you measured, then read alpha, b, c, area, and perimeter in one pass.

1. 1 Enter the included side a: Type the length of the side that sits between the two known angles. Pick a length unit from the dropdown so the computed b, c, perimeter, and area are labelled in the same unit.
2. 2 Enter angle beta and angle gamma: Type the two known interior angles in degrees. Each must be greater than 0 and less than 180, and beta + gamma must stay strictly less than 180 to leave a positive third angle.
3. 3 Read the third angle alpha: The black results panel shows alpha in degrees as 180 - beta - gamma. If alpha is 90 deg the triangle is a right triangle; if alpha is above 90 deg it is obtuse.
4. 4 Read the two missing sides and the area and perimeter: Side b sits opposite beta, side c sits opposite gamma, area is in square units, and perimeter is a + b + c, all in the same length unit as a.

A roof rafter with a 50 deg rise angle, a 32 deg seat-cut angle, and a 5 cm run gives alpha = 98 deg, b = 3.87 cm, c = 2.68 cm, area = 5.12 cm squared, and perimeter = 11.55 cm.

Once the area is in hand from the ASA inputs, the same triangle can be re-described by its three Cartesian vertices and the area re-checked from the shoelace formula in the Area Triangle Coordinates Calculator in the same Math & Conversion cluster.

The ASA pattern is the shortest path from two measured angles and one side to a fully solved triangle.

• • One pass from inputs to a solved triangle: Side a, beta, and gamma are enough to recover alpha, b, c, area, and perimeter without any iterative search or extra diagram step.
• • Law of sines done correctly: The page applies a / sin(alpha) = b / sin(beta) = c / sin(gamma) directly, so the recovered sides are consistent with the measured angles to double precision.
• • Degenerate input is caught early: When beta + gamma is 180 deg or above, the calculator shows an error explaining that the two angles cannot sum to 180 in a valid triangle.
• • Right, acute, and obtuse in one tool: The ASA formulas cover alpha = 90 deg (right), alpha < 90 deg (acute), and alpha > 90 deg (obtuse) without a separate mode.

The same page is useful for homework, a field-book baseline, and a quick rafter or brace check on a jobsite.

When the same triangle happens to have all three sides of different length, the Scalene Triangle Area Calculator adds the scalene-area workflow in the same category, including Heron's formula and base-height area cross-checks.

The ASA formulas are compact, but a few input choices decide whether the calculator returns the right value.

• • The calculator does not solve for a missing angle when the inputs are two sides and a non-included angle. That is the AAS case, which still uses the law of sines but in a different order.
• • Hand calculation that rounds beta and gamma partway through can differ by a few hundredths of a unit from the calculator, which keeps full double precision until the display step.

When the same triangle is easier to measure in coordinates, the shoelace formula gives the same area from three vertices, and the law of sines can be cross-checked against the side lengths from those coordinates.

According to Wikipedia, the law of sines gives the same constant a/sin(alpha) = b/sin(beta) = c/sin(gamma) for every triangle, and the relationship holds whether the triangle is acute, right, or obtuse.

According to Wikipedia, the interior angles of any triangle always sum to 180 degrees, so given two angles the third is fixed, which is what makes the ASA input pattern a single-step solve.

When the recovered alpha is above 90 deg the triangle is obtuse, and the Area Obtuse Triangle Calculator applies Heron's formula and the SAS area to the same obtuse shape in the same category.