Triangle Calculator: Solve Sides, Angles, Area & More - Free Online Calculator

1. Introduction to the Triangle Calculator

A triangle calculator helps you solve the unknown measurements of a triangle from the sides and angles you already know. Depending on the information provided, you can determine missing side lengths, missing angles, area, perimeter, semiperimeter, altitudes, medians, inradius, and circumradius.

This calculator is designed as a complete triangle solver rather than a single-purpose formula tool. Enter the known measurements, and it determines the appropriate geometric relationships, checks whether the values can form a valid triangle, and presents the calculated measurements with step-by-step mathematical details.

For a general triangle, the three sides are commonly written as a, b, and c, while the opposite angles are A, B, and C. The standard correspondence is important: side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The three interior angles of a Euclidean triangle always add to 180°.

The calculator can also be used when you already know all three sides, when you know two sides and an included angle, or when your problem gives angles together with one or more sides. The underlying methods include the Law of Sines, Law of Cosines, the Pythagorean theorem for right triangles, and Heron's formula for area.

2. How to Use the Triangle Calculator

Start by entering the measurements you know. The calculator works from valid side and angle combinations and determines the remaining quantities when the information is sufficient.

Step-by-Step Procedure:

Enter the known side lengths and/or angle measurements.
Make sure each measurement is associated with the correct side or opposite angle.
Select degrees or radians when entering angles.
Review the solved sides and angles in real time.
Check the calculated area, perimeter, and other geometric measurements.
Use the step-by-step derivation and interactive triangle diagram to verify the result.

For example, if the three sides are 3, 4, and 5, the calculator identifies a right triangle. The resulting angles are approximately 36.8699°, 53.1301°, and 90°, the area is 6, the perimeter is 12, the inradius is 1, and the circumradius is 2.5.

Because rounding is a presentation choice rather than a change to the underlying geometry, increasing the display precision changes the number of digits shown without altering the mathematical calculation.

3. Which Triangle Information Do You Need?

A triangle has six primary dimensions (three sides and three angles). To uniquely define a Euclidean triangle, you generally require at least three independent measurements, with at least one being a side length.

SSSThree Sides Known

When all three sides are known, the triangle is determined if the side lengths satisfy the triangle inequality:

a + b > c, a + c > b, b + c > a

Angles are calculated via the Law of Cosines, and area is found using Heron's formula.

SASTwo Sides and Included Angle

When two sides and the angle between them are known, the Law of Cosines computes the missing third side:

c = √(a² + b² − 2ab·cos C)

The remaining angles are solved using the Law of Sines or Law of Cosines.

ASA / AASTwo Angles and One Side

First find the third angle via the angle sum rule (A + B + C = 180°), then calculate missing sides using the Law of Sines:

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

This yields a unique triangle whenever the angle sum is strictly below 180°.

SSATwo Sides and Non-Included Angle

SSA is known as the ambiguous case. Depending on the geometry, it may yield:

No triangle (side too short to reach base)
Exactly one unique triangle
Two distinct valid triangles

This calculator evaluates the altitude h = b·sin A to identify and present both solutions when two exist.

4. Core Triangle Formulas

Every triangle problem relies on three fundamental relationships connecting angles and boundary dimensions:

Angle Sum

A + B + C = 180°

Interior angles add to 180° (π rad)

Perimeter

P = a + b + c

Total boundary length around triangle

Semiperimeter

s = (a + b + c) / 2

Used in Heron's formula and inradius

5. Calculating Triangle Area

Triangle area (denoted K or Area) can be evaluated using different mathematical formulations depending on the available information:

Base and Height Method

The classical triangle area formula is:

K = ½ · b · h

The height h must be strictly perpendicular to the chosen base b. For general multi-polygon area computations, you can also refer to our Area Calculator.

Heron's Formula (SSS Area)

When all three side lengths are known without an altitude, Heron's formula calculates the area directly using the semiperimeter s = (a + b + c) / 2:

K = √[ s(s − a)(s − b)(s − c) ]

For a 5-6-7 triangle, the semiperimeter is s = (5 + 6 + 7) / 2 = 9, giving:

K = √[ 9 · (9 − 5) · (9 − 6) · (9 − 7) ] = √[ 9 · 4 · 3 · 2 ] = √216 ≈ 14.6969

6. The Law of Sines

The Law of Sines establishes that the ratio of each side length to the sine of its opposite angle is constant and equals twice the circumradius R:

a / sin A = b / sin B = c / sin C = 2R

Matching lowercase sides with uppercase opposite angles is essential: side a faces angle A, b faces B, and c faces C.

Worked Example:

Suppose A = 40°, B = 60°, and a = 10. First, find the third angle:

C = 180° − 40° − 60° = 80°

Then apply the Law of Sines to solve for side b:

10 / sin(40°) = b / sin(60°) &implies; b = 10 · sin(60°) / sin(40°) ≈ 13.4729

7. The Law of Cosines

The Law of Cosines generalizes the Pythagorean theorem to any oblique or obtuse triangle. It relates all three sides to one interior angle:

a² = b² + c² − 2bc·cos A

b² = a² + c² − 2ac·cos B

c² = a² + b² − 2ab·cos C

When solving an SSS triangle for an angle, the formula is rearranged into:

cos C = (a² + b² − c²) / (2ab) &implies; C = arccos(cos C)

When C = 90°, cos 90° = 0, and the expression simplifies directly to c² = a² + b², reproducing the Pythagorean theorem. Explore our Pythagorean Theorem Calculator for dedicated right-triangle hypotenuse modeling.

8. Right Triangles and Trigonometric Ratios

A right triangle has one interior angle equal to exactly 90°. For legs a and b with hypotenuse c:

a² + b² = c² &implies; c = √(a² + b²)

For a triangle with legs 6 and 8:

c = √(6² + 8²) = √(36 + 64) = √100 = 10, Area = ½·6·8 = 24, Perimeter = 6 + 8 + 10 = 24

The fundamental trigonometric ratios for angle A in this 6-8-10 triangle are:

SINEsin A = 6 / 10 = 0.6000

COSINEcos A = 8 / 10 = 0.8000

TANGENTtan A = 6 / 8 = 0.7500

For dedicated right-triangle calculations, you can also use the Right Triangle Calculator available on this site.

9. Triangle Altitudes

An altitude is a perpendicular line segment dropped from a vertex to the opposite side. Because area K = ½·a·h_a, the three altitudes are:

h_a = 2K / a

h_b = 2K / b

h_c = 2K / c

For a 3-4-5 triangle with area K = 6:

h_a = 12 / 3 = 4.0000, h_b = 12 / 4 = 3.0000, h_c = 12 / 5 = 2.4000

Internal consistency check: ½·a·h_a = ½·b·h_b = ½·c·h_c = K.

10. Triangle Medians

A median is a line segment connecting a vertex to the midpoint of the opposite side. By Apollonius's theorem, the lengths of the three medians are:

m_a = ½√(2b² + 2c² − a²)

m_b = ½√(2a² + 2c² − b²)

m_c = ½√(2a² + 2b² − c²)

For a 5-6-7 triangle, the median to side a = 5 is:

m_a = ½√[2(6²) + 2(7²) − 5²] = ½√[72 + 98 − 25] = ½√145 ≈ 6.0208

11. Inradius (r) and Circumradius (R)

Every triangle has two unique concentric circles associated with its boundary geometry:

Inradius (r = Area / s)

The inradius r is the radius of the incircle inscribed inside the triangle, tangent to all three sides:

r = K / s

For a 3-4-5 triangle: K = 6, s = 6 &implies; r = 6 / 6 = 1.0000.

Circumradius (R = abc / 4K)

The circumradius R is the radius of the circumcircle passing through all three vertices:

R = (a · b · c) / (4K)

For a 3-4-5 triangle: (3 · 4 · 5) / (4 · 6) = 60 / 24 &implies; R = 2.5000.

Standard Notation: Always use lowercase r for inradius and uppercase R for circumradius. By Euler's inequality, R ≥ 2r for all planar triangles, with equality holding if and only if the triangle is equilateral.

12. Triangle Inequality & Existence Verification

Not every triplet of positive numbers can form a geometric triangle. For three segments to connect and form a closed polygon, they must strictly satisfy the Triangle Inequality:

a + b > c | a + c > b | b + c > a

Degenerate Case (1, 2, 3): 1 + 2 = 3. Because the sum equals the third side, the vertices collapse into a single straight line with zero area.
Impossible Case (1, 2, 4): 1 + 2 < 4. The two shorter sides cannot span the distance between the endpoints of side 4.

A robust mathematical engine must cleanly reject degenerate and impossible inputs rather than returning distorted zero-area configurations.

13. Degrees vs. Radians & Exact Expressions

Angles may be represented in degrees (°) or radians (rad). A full circle corresponds to 360° = 2π radians:

π / 6 = 30°

π / 4 = 45°

π / 3 = 60°

π / 2 = 90°

2π / 3 = 120°

3π / 4 = 135°

5π / 6 = 150°

2π = 360°

When entering radians, numerator coefficients are crucial: 2*pi/3 evaluates to 120°, whereas pi/3 evaluates to 60°.

14. Complete Triangle Classification Guide

Classification by Side Lengths:

Equilateral: All 3 sides are equal (a = b = c), all angles 60°.
Isosceles: 2 sides are equal, with equal opposite base angles.
Scalene: All 3 sides have distinct lengths (a ≠ b ≠ c).

Classification by Interior Angles:

Acute: All three angles are strictly less than 90°.
Right: Exactly one interior angle equals 90° (π/2 rad).
Obtuse: Exactly one interior angle is greater than 90°.

15. Complete Worked Example: Solving the 3-4-5 Triangle

Consider a triangle with side lengths a = 3, b = 4, and c = 5:

Step 1: Check Triangle Inequality: 3 + 4 > 5 (7 > 5). A valid closed triangle exists.

Step 2: Identify Triangle Type: 3² + 4² = 9 + 16 = 25 = 5². By the converse of Pythagoras, it is a right triangle with right angle at C = 90°.

Step 3: Solve Angles: sin A = 3/5 = 0.6 &implies; A ≈ 36.8699°; sin B = 4/5 = 0.8 &implies; B ≈ 53.1301°.

Step 4: Area & Perimeter: Area K = ½·3·4 = 6; Perimeter P = 3 + 4 + 5 = 12; Semiperimeter s = 12 / 2 = 6.

Step 5: Altitudes: h_a = 12/3 = 4, h_b = 12/4 = 3, h_c = 12/5 = 2.4.

Step 6: Inradius & Circumradius: Inradius r = 6 / 6 = 1.0000; Circumradius R = (3 · 4 · 5) / (4 · 6) = 2.5000.

A ≈ 36.8699°, B ≈ 53.1301°, C = 90.0000° | Area = 6, P = 12, r = 1, R = 2.5

16. How to Verify a Triangle Calculation

Even with an automated solver, cross-checking results ensures geometric consistency:

Angle Sum: Sum computed angles: A + B + C = 180° within numerical rounding.
Side-Angle Ordering: The largest angle must always face the longest side, and the smallest angle faces the shortest side.
Area Consistency: Verify that K = ½·a·h_a = ½·b·h_b = ½·c·h_c matches Heron's formula.
Radius Cross-Check: Confirm that Euler's inequality holds: R ≥ 2r.

17. Geometric Vector Visualization

Numerical outputs alone can mask transposition errors. The interactive SVG visualizer in this calculator uses uniform isometric scaling—meaning angles, side proportions, and circular tangencies are rendered true to scale without anisotropic distortion.

When toggled, the circumcircle accurately intersects all three vertices (A, B, C), and the incircle maintains precise internal tangency to all three sides.

18. Common Triangle-Calculation Mistakes

Transposing Opposite Pairs:

Associating side a with angle B or C invalidates the Law of Sines and Law of Cosines. Side a must always face angle A.

Applying Pythagoras to Non-Right Triangles:

a² + b² = c² holds strictly when C = 90°. For general acute or obtuse triangles, use the Law of Cosines.

Ignoring the SSA Ambiguous Case:

Two sides and a non-included angle can produce two valid geometric configurations. Ensure both solutions are reviewed.

Confusing Inradius (r) with Circumradius (R):

Lowercase r is the internal tangent circle (r = K / s); uppercase R is the external circumcircle passing through vertices (R = abc / 4K).

19. Applications in Surveying, Engineering & Physics

Triangulation forms the foundation of modern geodesy, global navigation (GPS), robotics kinematics, and structural engineering. In static mechanics, truss systems are decomposed into triangular networks because the triangle is the only rigid 2D polygon whose angles cannot deform without changing side lengths. In physics, vector addition and orthogonal force decomposition rely directly on right-triangle and oblique-triangle trigonometry.

20. Precision, Rounding, and Units

Geometrical solutions often involve irrational numbers such as √3 or non-terminating decimals. This calculator maintains full 64-bit IEEE floating-point precision internally and rounds strictly at the display layer according to the user's active precision setting (2, 4, or 6 decimals), preserving exact trailing zeros. Physical units are preserved: if input sides are in meters, perimeter and radii are in meters, and area is in square meters.

21. Systematic Method for Solving Any Triangle

Identify Knowns: List all given sides and angles with their corresponding opposite pairs (a-A, b-B, c-C).
Classify Case: Identify whether the problem is SSS, SAS, ASA, AAS, or SSA.
Select Theorem: Use the Law of Cosines for SSS/SAS, the Law of Sines for ASA/AAS, and the Pythagorean theorem for right triangles.
Verify Bounds: Confirm triangle inequality and angle sums.
Compute Metrics: Calculate secondary metrics: area, perimeter, altitudes, medians, inradius r, and circumradius R.

22. Summary & Key Takeaways

Solving a triangle is an exercise in identifying the geometric dependencies between given parameters. Use the triangle inequality to verify physical possibility, the Law of Cosines for three sides or two sides with an included angle, the Law of Sines when side-angle pairs are available, and Heron's formula for robust area calculations. Once the core boundary is determined, all secondary properties—from altitudes and medians to inradii and circumradii—follow deterministically.

Frequently Asked Questions

A triangle calculator is a tool for finding unknown sides, angles and other geometric measurements from known triangle data. Depending on the available measurements, it can use relationships such as the Law of Sines, Law of Cosines, the Pythagorean theorem and Heron's formula.

Enter all three side lengths. First make sure they satisfy the triangle inequality. When they do, the three angles can be determined, typically using the Law of Cosines, and the area can be calculated using Heron's formula.

The method depends on what is known. With two sides and their included angle, use the Law of Cosines. With a known side-angle pair and another angle or side, the Law of Sines may be appropriate. In a right triangle, the Pythagorean theorem can find an unknown side.

For two known angles, subtract their sum from 180°. For three known sides, use the Law of Cosines to calculate an angle. Other configurations may be solved with the Law of Sines.

The Law of Sines connects sides with their opposite angles: a/sin(A) = b/sin(B) = c/sin(C). The Law of Cosines relates three sides to an included angle: c² = a² + b² - 2ab cos(C). As a practical rule, SSS and SAS problems commonly point toward the Law of Cosines, while ASA/AAS and suitable SSA problems commonly use the Law of Sines.

No. A proper triangle requires the sum of any two sides to be strictly greater than the third side. Here, 1 + 2 = 3, so the configuration is degenerate rather than a proper triangle.

Heron's formula calculates triangle area when all three side lengths are known: K = √[s(s-a)(s-b)(s-c)], where s is the semiperimeter s = (a+b+c)/2.

The inradius r is the radius of the circle inside the triangle that touches its sides: r = K / s. The circumradius R is the radius of the circle passing through the triangle's three vertices: R = abc / (4K).

Yes. A right triangle contains a 90° angle, allowing the Pythagorean theorem and trigonometric relationships to be used. For example, a 6-8-10 triangle has hypotenuse 10, area 24, and perimeter 24.

SSA gives two sides and a non-included angle. Depending on the dimensions, the known side can sometimes swing into two geometrically different positions while satisfying the same measurements. This is the ambiguous case of the Law of Sines, and it can result in zero, one or two valid triangles.