2026-08-07 11 min

Triangle Calculator

Solve any triangle from 3 values including a side. Computes sides, angles, area, perimeter, heights, medians, inradius, circumradius, and classification with a labeled diagram.

Calculator

Sides

Side a

Side b

Side c

Angles

Angle A °

Angle B °

Angle C °

Angle Unit

Provide 3 values including at least one side. Radians accept pi/2, pi/4, etc.

Results update automatically

Results update live as you type

Status:Sides 3.00 · 4.00 · 5.00Scalene · RightArea 6.0000

Triangle Summary

Sides a · b · c3.0000 · 4.0000 · 5.0000

Angles A · B · C36.87° · 53.13° · 90.00°

Area6.0000

Perimeter12.0000

Inradius / Circumradius1.0000 / 2.5000

3.0000² + 4.0000² = 5.0000² => 9.0000 + 16.0000 = 25.0000 => 25.0000 = 25.0000 ✓ Verified

Heights & Medians

From side	Altitude	Median
a	4.0000	4.2720
b	3.0000	3.6056
c	2.4000	2.5000

Results

Optimal

Area

6.0000

Heron’s formula from a, b, c

Perimeter

12.0000

a + b + c

Angles A · B · C

36.87° · 53.13° · 90.00°

Sum exactly 180°

Classification

Scalene · Right

By sides and by angles

Altitudes ha · hb · hc

4.000 · 3.000 · 2.400

2·Area divided by each side

Inradius

1.0000

Area / semi-perimeter

Circumradius

2.5000

a / (2·sin A)

Medians ma · mb · mc

4.272 · 3.606 · 2.500

Vertex-to-midpoint lengths

Law-of-sines/cosines solution closed by A+B+C=180° with Heron area. SSS and SAS are exact up to rounding; SSA returns the principal acute branch — verify supplements on site.

Visualizations

Side Lengths

Angle Split (degrees)

A 36.9°20%

B 53.1°30%

C 90.0°50%

Largest Angle vs 180°

Max angle90°

Inradius vs Circumradius

Inradius1

Formula

Unknown parts come from the cosine law (SSS, SAS) and the sine law (ASA, AAS, SSA) closed by A+B+C=180°. Area then follows from Heron’s formula, altitudes from 2·Area/side, medians from Apollonius, and radii from Area/s and a/(2·sin A).

c² = a² + b² − 2ab·cos(C); a/sin(A) = b/sin(B) = c/sin(C)

Alternative Form

Area = √(s(s−a)(s−b)(s−c)), s = (a+b+c)/2 (Heron)

Variables

Symbol	Name	Unit	Description
a, b, c	Side lengths	length	a opposite A, b opposite B, c opposite C
A, B, C	Interior angles	° or rad	Always sum to 180°
s	Semi-perimeter	length	(a+b+c)/2
hₐ	Altitude to a	length	2·Area/a
mₐ	Median to a	length	½√(2b²+2c²−a²)
r, R	Inradius, circumradius	length	r=Area/s; R=a/(2·sin A)

Explanation

Sines scale sides against opposite angles; cosines correct Pythagoras for non-right included angles. Once all six parts are known, area and its derivatives are pure algebra.

Derivation

SSS: cos A=(b²+c²−a²)/(2bc), arccos each angle. SAS: c=√(a²+b²−2ab·cos C), then sine law. ASA/AAS: third angle by subtraction, sides by sine law. SSA: sin B=b·sin A/a with supplement check. Then s, Heron area, h=2A/side, r=A/s, R=a/(2 sin A).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Start from the entered values and complete the six parts.

Law of sines / cosines + A+B+C=180°

a:3.0000

b:4.0000

c:5.0000

A:36.87°

B:53.13°

C:90.00°

Intermediate: sides 3.0000, 4.0000, 5.0000

Result: angles 36.87°, 53.13°, 90.00°

Step 2

Check closure: sides satisfy the inequality and angles sum to 180°.

a+b>c; A+B+C=180°

a + b:7.0000

c:5.0000

A+B+C:180.0000°

Intermediate: 7.0000 > 5.0000

Result: angle sum 180.00° ✓

Step 3

Compute the semi-perimeter and apply Heron’s formula.

s=(a+b+c)/2; Area=√(s(s−a)(s−b)(s−c))

s:6.0000

s−a:3.0000

s−b:2.0000

s−c:1.0000

Intermediate: √(6.0000·3.0000·2.0000·1.0000)

Result: area = 6.0000

Step 4

Drop the three altitudes from area.

hₐ=2·Area/a; h_b=2·Area/b; h_c=2·Area/c

Area:6.0000

a:3.0000

b:4.0000

c:5.0000

Intermediate: 2·6.0000 ÷ sides

Result: h = 4.0000 · 3.0000 · 2.4000

Step 5

Compute medians and both radii.

mₐ=½√(2b²+2c²−a²); r=Area/s; R=a/(2·sin A)

Inradius r:1.0000

Circumradius R:2.5000

Intermediate: medians 4.2720 · 3.6056 · 2.5000

Result: r=1.0000, R=2.5000

Step 6

Classify the triangle and verify Pythagoras when right.

Sides: equilateral/isosceles/scalene; Angles: acute/right/obtuse

By sides:Scalene

By angles:Right

Intermediate: 3.0000² + 4.0000² = 5.0000² => 9.0000 + 16.0000 = 25.0000 => 25.0000 = 25.0000 ✓ Verified

Result: Scalene · Right triangle solved

Final Answer

6.0000

Worked Examples

See how the calculator works with different scenarios.

Scenario

Sides a=3, b=4, c=5 with no angles entered.

Inputs

a:3

b:4

c:5

Calculation

cos A=(16+25−9)/(2·4·5)=0.8 → A≈36.87° cos B=(9+25−16)/(2·3·5)=0.6 → B≈53.13° C=180−A−B=90° s=6, area=√(6·3·2·1)=6

Result

Angles 36.87° · 53.13° · 90°; area 6; scalene right; r=1, R=2.5

Interpretation

The solver recognizes 3²+4²=5² and prints the verified identity. Altitudes are 4, 3, and 2.4; medians about 4.27, 3.61, 2.5.

Understanding Your Result

Your Result: Area: 6.0000

Typical range: Valid triangles satisfy a+b>c in all permutations and A+B+C=180°. Altitudes never exceed their adjacent sides; circumradius is at least half the longest side; inradius is well under the shortest altitude.

What This Means

The dashboard reports the full solved triangle: three sides, three angles summing to 180°, Heron area, perimeter, the three altitudes and medians, both circle radii, and the side/angle classification.

Why It Matters

One entry of SSS, SAS, ASA, AAS, or SSA determines everything else. Reading all derivatives together catches mislabeled sides and impossible sets before they reach construction or grading.

Professional Interpretation

Use SSS for fabricated parts, SAS for two-station surveys, ASA/AAS for angle-forward traversing, and treat SSA as provisional until the supplement is ruled out in the field.

Recommendations

•Sketch which side opposes which angle before typing
•Prefer SSS or SAS when distances are more reliable than angles
•Confirm SSA supplements against the site sketch
•Keep degrees selected unless the source is genuinely in radians
•Re-run after rounding field notes to check sensitivity
•Export the diagram with the computation for records

Educational Content

Introduction

A triangle is fixed by three data including a side. This solver walks the standard cases — SSS by cosines, SAS by cosines then sines, ASA/AAS by subtraction then sines, SSA with an ambiguity check — and derives area, heights, medians, and radii.

Concept Explanation

Sine law equates side-to-sine ratios; cosine law adjusts Pythagoras by −2ab·cos(C). Heron’s formula converts the three sides into area, and area converts into altitudes and the incircle while a/(2·sin A) gives the circumcircle.

Applications

▸Setting out and checking property boundaries

▸Sizing roof trusses and bridge bracing

▸Triangulating positions in GPS and surveying

▸Meshing surfaces in 3D graphics and FEM

▸Resolving force vectors on inclined planes

▸Measuring stellar parallax distances

Benefits

•Accepts all five solvable configurations
•Parses degrees, radians, and pi expressions
•Reports heights, medians, and both radii together
•Verifies Pythagoras numerically on right triangles
•Draws a true-shape labeled diagram

Limitations

•AAA alone cannot fix scale
•SSA may hide a second valid triangle supplement
•Flat-plane geometry only — no spherical excess
•Display rounds to 4/2 decimals; keep raw values for machining

Best Practices

•Enter exactly the measured values; leave the rest blank
•Match the angle-unit switch to the instrument output
•Plot the diagram against the site sketch as a sanity check
•Carry extra digits through chained calculations
•Photograph the setup when SSA yields two candidates

Common Misconceptions

Myth: Any three lengths make a triangle.

Truth: They must pass a+b>c in every permutation. Lengths 2, 3, 10 leave a gap that cannot close.

Myth: SSA always gives one answer.

Truth: Two sides plus a non-included angle can yield zero, one, or two triangles — the sine supplement must be tested.

Myth: Pythagoras works on every triangle.

Truth: a²+b²=c² holds only at 90°. The cosine law is the general form for all other angles.

Myth: Three angles determine size.

Truth: AAA fixes shape only. A 30-60-90 triangle can be 1:√3:2 at any scale until a side is given.

Practical Tips

•Give exactly 3 values with at least one side
•Leave unknown fields blank rather than zeroing them
•Use SSS when all distances are taped accurately
•Type pi/2 style angles in radian mode
•Suspect SSA whenever two sides meet one lonely angle
•Compare the diagram to your sketch before building
•Re-run with rounded inputs to test sensitivity

Common Mistakes

Entering AAA with no side

Add any side length. Angles alone describe infinite similar triangles at every scale.

Mislabeling opposite pairs

Side a must oppose angle A. Relabel the sketch so each side faces its angle before solving.

Ignoring the SSA supplement

When SSA returns an acute angle, check 180° minus that angle against the remaining data and the site.

Mixing degrees and radians

Set the unit switch first. A 1.57 entry means 1.57° in degree mode but π/2 in radian mode.

Forcing impossible sides through

Respect inequality errors. Remeasure the longest side — 2,3,10 can never close.

Assumptions

This calculator makes the following assumptions to provide estimates:

Euclidean plane: Angles sum to exactly 180°; no spherical or hyperbolic excess.

Positive lengths: Zero, negative, or non-numeric side entries are treated as unknown.

Positive angles: Zero or negative angle entries are treated as unknown.

Single principal solution: SSA returns the acute configuration; the supplement needs manual review.

Display rounding: Internals stay full-precision while sides show 4 and angles 2 decimals.

Limitations

Be aware of these limitations when using this calculator:

No AAA scaling: Three angles without a side leave size undetermined by similarity.

SSA ambiguity: The second supplement is flagged by reasoning, not enumerated as a twin solution.

Planar only: Geodetic triangles need spherical excess corrections this solver omits.

No construction output: Results are numeric plus diagram; no compass-and-straightedge steps.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a triangle?

A triangle is a three-sided polygon with three vertices and three interior angles that always sum to 180 degrees (π radians) in Euclidean geometry.

What are the types of triangles by sides?

Equilateral has three equal sides, isosceles has two equal sides, and scalene has three different side lengths.

What are the types of triangles by angles?

Acute has all angles under 90°, right has one 90° angle, and obtuse has one angle over 90°. A triangle can never have two obtuse or two right angles.

What do SSS, SAS, ASA, AAS, and SSA mean?

They name the known data: Side-Side-Side, Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Side-Side-Angle. SSS, SAS, ASA, and AAS determine a unique triangle; SSA can give 0, 1, or 2 solutions.

How do I solve a triangle?

Provide any 3 values including at least one side. The solver applies the law of cosines for SSS and SAS, the law of sines for ASA/AAS/SSA, and A+B+C=180° to finish the angles.

How do I find angles from three sides (SSS)?

Use the law of cosines: cos(A)=(b²+c²−a²)/(2bc), then A=arccos of that value. Repeat for B and C, or use 180°−A−B for the last angle.

What is the law of sines?

a/sin(A)=b/sin(B)=c/sin(C)=2R, where R is the circumradius. It links each side to the sine of its opposite angle.

What is the law of cosines?

c²=a²+b²−2ab·cos(C). It generalizes Pythagoras to any triangle and finds a side from two sides plus the included angle.

What is the Pythagorean theorem?

In a right triangle with legs a,b and hypotenuse c: a²+b²=c². The solver verifies it numerically whenever it detects a right triangle.

What is the triangle inequality?

Any two sides must sum to more than the third side (a+b>c and permutations). Violations such as 2,3,10 cannot form a triangle.

How do I find the area with base and height?

Area = ½·base·height. Any side can serve as the base with its matching altitude: A=½·a·hₐ=½·b·h_b=½·c·h_c.

What is Heron’s formula?

With semi-perimeter s=(a+b+c)/2, area=√(s(s−a)(s−b)(s−c)). It needs only the three sides, which is why this solver uses it.

How do I find area from two sides and the included angle?

Use A=½·ab·sin(C). For sides 4 and 5 with included angle 60°, area=½·4·5·sin60°≈8.66.

What is the ambiguous case (SSA)?

Given two sides and a non-included angle, the sine equation can yield an acute and an obtuse supplement. Depending on lengths there are zero, one, or two valid triangles.

Do angles always sum to 180°?

Yes on a flat plane. The solver rejects inputs whose angles sum to 180° or more for the two known angles, and checks the final sum within 0.001 rad.

How do I find the hypotenuse?

For legs a,b the hypotenuse is c=√(a²+b²). A 3-4 right triangle has hypotenuse √(9+16)=5.

What is the median of a triangle?

A segment from a vertex to the midpoint of the opposite side. Length mₐ=½√(2b²+2c²−a²). The three medians meet at the centroid.

What is the altitude (height)?

The perpendicular distance from a vertex to its opposite side: hₐ=2·Area/a. This solver reports all three altitudes.

What is the inradius?

Radius of the incircle touching all three sides: r=Area/s. For a 3-4-5 triangle, area 6 and s 6 give r=1.

What is the circumradius?

Radius of the circle through all vertices: R=a/(2·sin A). For 3-4-5, R=2.5, half the hypotenuse.

What are centroid, orthocenter, circumcenter, incenter?

Centroid is the medians’ meeting point (center of mass); orthocenter the altitudes’ meeting point; circumcenter the perpendicular bisectors’ meeting point; incenter the angle bisectors’ meeting point.

Can I enter radians or pi expressions?

Yes. Switch Angle Unit to radian and type decimals or expressions like pi/2 and pi/4, which convert to 90° and 45°.

Why does the solver say "not enough information"?

Three angles alone (AAA) fix shape but not scale — infinitely many similar triangles share them. Add at least one side.

What is a 30-60-90 triangle?

A right triangle with angles 30°, 60°, 90° and side ratio 1:√3:2. The hypotenuse is twice the shorter leg.

What is a 45-45-90 triangle?

An isosceles right triangle with legs equal and hypotenuse leg·√2. Angles are 45°, 45°, 90°.

How are triangles used in real life?

Roof trusses, bridge bracing, GPS triangulation, 3D-mesh modeling, parallax astronomy, and force-vector physics all rest on triangle geometry.

How do I classify a 3-4-5 triangle?

Sides 3,4,5 are all different so it is scalene; 3²+4²=5² so it is right. The solver reports "Scalene · Right" plus the verified identity.

What precision does the solver use?

Sides, area, heights, medians, and radii display to 4 decimals; angles to 2 decimals. Right-angle detection tolerates 0.01°.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Choosing SSS, SAS, ASA, or SSA

Match your measurements to the right law.

Heron’s Formula Without Height

Area from three taped sides.

Surviving the Ambiguous Case

Testing SSA supplements in the field.

Right-Triangle Shortcuts

30-60-90 and 45-45-90 ratios.

Glossary

Technical terms used in mortgage calculations explained in plain language.

SSS / SAS / ASA / AAS / SSA: Shorthand for which sides and angles are known going in.
Law of sines: a/sin A=b/sin B=c/sin C=2R.
Law of cosines: c²=a²+b²−2ab·cos C.
Heron’s formula: Area from sides: √(s(s−a)(s−b)(s−c)).
Semi-perimeter: s=(a+b+c)/2, the Heron input.
Altitude: Perpendicular height hₐ=2·Area/a.
Median: Vertex-to-midpoint segment meeting at the centroid.
Inradius: Incircle radius r=Area/s.
Circumradius: Circumcircle radius R=a/(2·sin A).
Triangle inequality: a+b>c in every permutation, else no triangle.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Triangle properties Wolfram MathWorld — Triangles Wikipedia — Triangle Wikipedia — Law of sines Wikipedia — Law of cosines Wikipedia — Heron’s formula Brilliant — Triangles GeoGebra — Triangle interactive

Applicable Standards

Euclid’s Elements triangle postulates (planarity, angle sum)NIST SI length and plane-angle guidance

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Page Information

Last Updated:

2026-08-07

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.