What is the ASA rule in triangle geometry?

ASA stands for Angle-Side-Angle. It is a triangle congruence rule stating that if you know two angles of a triangle and the side between them (the included side), the triangle is fully determined. You can find the third angle by subtracting the two known angles from 180 degrees, then use the Law of Sines to calculate the remaining two sides.

How do you solve an ASA triangle?

To solve an ASA triangle: (1) Find the third angle: B = 180 - A - C, where A and C are the two known angles. (2) Use the Law of Sines to find the unknown sides: a = b x sin(A) / sin(B) and c = b x sin(C) / sin(B), where b is the known included side opposite angle B. (3) Calculate the area using: Area = (b squared x sin(A) x sin(C)) / (2 x sin(B)). (4) Add all three sides for the perimeter.

What is the included side in an ASA triangle?

The included side in an ASA triangle is the side that lies directly between the two known angles. For example, if angle A is at vertex A and angle C is at vertex C, then side b (which connects vertex A to vertex C) is the included side. In standard triangle notation, side b is always opposite angle B, and it connects the vertices where angles A and C are located.

ASA Triangle Calculator

Solve a triangle from two angles and the side between them (Angle-Side-Angle). Enter angle A, the included side b, and angle C to find all sides, the third angle, area, and perimeter using the Law of Sines.

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Standard geometry Law of Sines and Euclidean triangle properties.

1. Known Angles

Angle A (degrees, at vertex A)

Angle C (degrees, at vertex C)

Angles A and C must each be greater than 0° and their sum must be less than 180°.

2. Included Side

Side b (the side between angles A and C)

units

Side b connects vertex A to vertex C. It is opposite angle B (the unknown angle). Enter any consistent unit (cm, m, mm, etc.).

Please check your inputs. Both angles must be positive, and their sum must be less than 180°. Side b must be greater than zero.

Triangle Solution

Angle B (found)

= 180 - A - C

Area
-
square units

Perimeter

a + b + c

Triangle type

by angles

All Sides

Side a (opposite angle A)-

Side b (included, given)-

Side c (opposite angle C)-

Perimeter (a + b + c)-

All Angles

Angle A (given)-

Angle B (calculated)-

Angle C (given)-

Sum of angles180.00°

Worked example (defaults): With A = 45°, side b = 10 units, and C = 45°:
Angle B = 180 - 45 - 45 = 90°
Side a = 10 × sin(45°) / sin(90°) = 10 × 0.7071 / 1.0000 = 7.07 units
Side c = 10 × sin(45°) / sin(90°) = 7.07 units
Area = (10² × sin(45°) × sin(45°)) / (2 × sin(90°)) = (100 × 0.5) / 2 = 25.00 sq units
Perimeter = 7.07 + 10 + 7.07 = 24.14 units

How to Solve an ASA Triangle

The ASA (Angle-Side-Angle) case occurs when you know two angles of a triangle and the side that lies between them. Because the angles of any triangle sum to 180 degrees, knowing two angles immediately tells you the third. Once all three angles are known, the Law of Sines links every angle to its opposite side, so the full triangle can be solved from just one known side.

The Formulas

Label the triangle with angles A, B, C at the respective vertices, and sides a, b, c opposite to those vertices. In the ASA case you know angles A and C and the side b (which lies between them, opposite angle B).

Step	Formula	Notes
Find angle B	B = 180 - A - C	Angles in a triangle sum to 180°
Find side a	a = b × sin(A) / sin(B)	Law of Sines
Find side c	c = b × sin(C) / sin(B)	Law of Sines
Area	Area = (b² × sin(A) × sin(C)) / (2 × sin(B))	Derived from half base times height
Perimeter	P = a + b + c	Sum of all sides

The Law of Sines

The Law of Sines states that in any triangle: a / sin(A) = b / sin(B) = c / sin(C). This ratio is constant for a given triangle and equals twice the circumradius (the radius of the circle that passes through all three vertices). Once you know one side and all three angles, you can rearrange this equality to find any unknown side.

Triangle Types by Angle

Knowing all three angles lets you classify the triangle:

Acute triangle: all three angles are less than 90°.
Right triangle: one angle is exactly 90°.
Obtuse triangle: one angle is greater than 90°.
Equilateral triangle: all three angles equal 60° (and all sides are equal).
Isosceles triangle: two angles are equal (and their opposite sides are equal).

When ASA Cannot Be Solved

The ASA case always produces a valid, unique triangle provided: both given angles are greater than 0°, their sum is strictly less than 180°, and the included side is positive. If the two angles sum to 180° or more, no triangle is possible. Unlike the SSA (side-side-angle) case, ASA never produces an ambiguous solution.

Related Calculators

Maths and Stats Calculators: full hub for maths tools.
Triangle Calculator (general): solve from any combination of sides and angles.
AAS Triangle Calculator: two angles and a non-included side.
AAA Triangle Calculator: three angles known (similar triangles).
SOHCAHTOA Solver: right-triangle trigonometry.

Method: Standard Euclidean geometry. Third angle derived from the angle sum property (A + B + C = 180°). Unknown sides calculated using the Law of Sines (a / sin(A) = b / sin(B)). Area formula: (b² × sin(A) × sin(C)) / (2 × sin(B)). All calculations use double-precision floating-point arithmetic.

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