How do you solve a triangle given SSS (three sides)?

Use the Law of Cosines to find each angle from the three known sides. Formula: A = arccos((b² + c² - a²) / (2bc)) Example: a=3, b=4, c=5 - A = arccos((16+25-9)/40) = arccos(0.8) = 36.87 degrees - B = arccos((9+25-16)/30) = arccos(0.6) = 53.13 degrees - C = 180 - 36.87 - 53.13 = 90 degrees This is the 3-4-5 right triangle. The angle opposite the largest side (c=5) is always the largest angle. Check: always verify the triangle inequality first. The sum of any two sides must exceed the third: a+b > c, a+c > b, b+c > a. If any fails, no triangle exists.

How do you solve a triangle given SAS (two sides and included angle)?

Use the Law of Cosines to find the third side, then the Law of Sines for the remaining angles. Formula: c² = a² + b² - 2ab·cos(C) Example: a=5, b=7, C=60 degrees - c² = 25 + 49 - 2(5)(7)cos(60) = 74 - 35 = 39 - c = sqrt(39) = 6.2450 - A = arcsin(5 x sin(60) / 6.2450) = arcsin(0.6934) = 43.79 degrees - B = 180 - 60 - 43.79 = 76.21 degrees - Area = (1/2)(5)(7)sin(60) = 15.1554 Important: C is the angle between sides a and b (the included angle). If you have two sides and a non-included angle, that is the SSA case.

What is the ambiguous case (SSA) in triangle solving?

The SSA case (two sides and a non-included angle) can produce 0, 1, or 2 valid triangles. This is called the ambiguous case. Given sides a and b and angle A (opposite to side a): To find B: sin(B) = b * sin(A) / a Possible outcomes: - If sin(B) > 1: no solution (the triangle is impossible) - If sin(B) = 1: exactly one solution (B = 90 degrees) - If sin(B) = 180, only B1 is valid Example: a=7, b=10, A=30 degrees - sin(B) = 10 x sin(30) / 7 = 0.7143 - B1 = arcsin(0.7143) = 45.58 degrees - B2 = 180 - 45.58 = 134.42 degrees - A + B2 = 30 + 134.42 = 164.42 < 180 so two valid triangles exist This calculator shows both triangles when the ambiguous case applies.

What is the Law of Sines and when is it used?

The Law of Sines states that in any triangle, each side divided by the sine of its opposite angle gives the same value. a/sin(A) = b/sin(B) = c/sin(C) = 2R where R is the circumradius of the triangle. Used for: ASA (find remaining sides), AAS (find remaining sides), SSA (ambiguous case resolution). Not ideal for SSS (use Law of Cosines) because the Law of Sines can give imprecise results when angles are near 0 or 180 degrees. Example (AAS): A=30, B=40, a=8 - C = 180 - 30 - 40 = 110 degrees - b = 8 x sin(40) / sin(30) = 8 x 0.6428 / 0.5 = 10.2846 - c = 8 x sin(110) / sin(30) = 8 x 0.9397 / 0.5 = 15.0351

How do you calculate the area of a triangle from sides and angles?

There are several formulas depending on what is known: From base and height: Area = (1/2) x base x height From two sides and included angle (SAS): Area = (1/2) x a x b x sin(C) Heron's formula (from three sides): Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 Examples: - Triangle with a=3, b=4, C=90 degrees: Area = (1/2)(3)(4)sin(90) = 6 - Triangle 3-4-5: s=6, Area = sqrt(6x3x2x1) = sqrt(36) = 6 (same result) - SAS: a=5, b=7, C=60: Area = (1/2)(5)(7)sin(60) = 15.1554 The circumradius R = abc/(4 x Area) and inradius r = Area/s are additional properties this calculator provides. The inradius is the radius of the largest circle that fits inside the triangle. The circumradius is the radius of the circle that passes through all three vertices.

Triangle Solver — SSS, SAS, ASA, AAS, SSA Calculator

The Triangle Solver calculates all missing sides, angles, area, perimeter, inradius, and circumradius from any three known measurements. Supports all five cases: SSS (3 sides), SAS (2 sides + included angle), ASA (2 angles + included side), AAS (2 angles + non-included side), and SSA (2 sides + non-...

TRIANGLE SOLVER

3 sides known — find all angles

Side a (length)

Side b (length)

Side c (length)

SIDE a

SIDE b

SIDE c

ANGLE A

36.869898°

ANGLE B

53.130102°

ANGLE C

90°

Area

Perimeter

Inradius

Circumradius

2.5

Type

Right Scalene

Law used

Law of Cosines

STEP-BY-STEP

SSS: a=3, b=4, c=5

Law of Cosines: A = arccos((b²+c²−a²)/(2bc)) = arccos(0.8) = 36.869898°

B = arccos((a²+c²−b²)/(2ac)) = 53.130102°

C = 180 − A − B = 90°

TRIANGLE DIAGRAM

Altitude to a

Altitude to b

Altitude to c

2.4

Semi-perimeter

LAWS USED

Law of Cosines

c² = a²+b²−2ab·cos(C)

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

Heron's Area

Area = √(s(s-a)(s-b)(s-c))

Created with❤️byeaglecalculator.com

TRIANGLE FORMULAS

LAW OF SINES, COSINES & AREA FORMULAS

NAME	FORMULA	DESCRIPTION
Law of Cosines	c² = a²+b²−2ab·cos(C)	Find side c when two sides and included angle are known
Law of Cosines (A)	A = arccos((b²+c²−a²) / (2bc))	Find angle A when all three sides are known
Law of Sines	a/sin(A) = b/sin(B) = c/sin(C)	Relates sides and opposite angles
Angle sum	A + B + C = 180°	Sum of interior angles of any triangle
Heron's Area	Area = sqrt(s(s-a)(s-b)(s-c)), s=(a+b+c)/2	Area from three sides using semi-perimeter
SAS Area	Area = (1/2)·a·b·sin(C)	Area when two sides and included angle are known
Inradius	r = Area / s	Radius of inscribed circle
Circumradius	R = (a·b·c) / (4·Area)	Radius of circumscribed circle
Altitude to a	h_a = 2·Area / a	Height of triangle from vertex A to side a
Triangle inequality	a+b > c, a+c > b, b+c > a	Required for a valid triangle to exist

HOW TO USE

1
Select the case that matches what you know: SSS (3 sides), SAS (2 sides + included angle), ASA (2 angles + included side), AAS (2 angles + non-included side), or SSA (2 sides + non-included angle). The input fields update to match your selection.
2
Enter the three known values. Use the preset buttons for quick examples: 3-4-5 right triangle, SAS with 60 degrees, ASA, AAS, the ambiguous SSA case, or an equilateral triangle.
3
All missing sides and angles appear instantly in the result panel, along with area, perimeter, inradius, circumradius, and triangle type (acute/right/obtuse, scalene/isosceles/equilateral).
4
The triangle diagram on the right draws the triangle to scale with all six measurements labelled. For the SSA ambiguous case, both valid triangles are drawn and labelled Triangle 1 and Triangle 2.
5
The step-by-step panel shows which law was applied at each step: Law of Cosines for SSS and SAS, Law of Sines for ASA, AAS, and the SSA resolution, and Heron's formula for area.

WORKED EXAMPLE

SSS 3-4-5: A=36.87 deg, B=53.13 deg, C=90 deg, Area=6, Perimeter=12. SAS a=5,b=7,C=60: c=6.2450, A=43.79 deg, B=76.21 deg, Area=15.1554. ASA A=45,B=60,c=10: C=75, a=7.3205, b=8.9658. AAS A=30,B=40,a=8: C=110, b=10.2846, c=15.0351. SSA a=7,b=10,A=30: two solutions B1=45.58 deg and B2=134.42 deg.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.