GEOMETRY • TRIANGLES

Triangle calculator

Solve triangle sides and angles from compatible measurements, then explore area, perimeter, radii, heights and the geometry behind the result.

⚑ Report error
Area
—
square units
Perimeter
—
a + b + c
Triangle type
—
by sides + angles
LIVE GEOMETRY

Scaled triangle sketch

LIVE
Angle A—
Angle B—
Angle C—
Semiperimeter—

Side-length profile

Relative size of a, b and c
Live
Three sidesUpdates with every input

Angle distribution

A + B + C always equals 180°
Live
Angle share180° total
LIVE GEOMETRY INTELLIGENCE

What the current triangle is telling you

Updates live
Inradius—distance from incenter to each side
Circumradius—radius through all three vertices
Largest angle—opposite the longest side
Altitudes—hₐ, hᵦ, h𝒸
Enter compatible triangle measurements to see the geometry update live.

Triangle geometry table

Live
QuantityValueMeaning

Solve for

Enter a compatible set of measurements. Use at least one side; the calculator detects SSS, SAS, ASA/AAS and the SSA case automatically.
∠ Angles
↔ Sides
Tip: the lettered side and angle pairs are opposite each other: a ↔ A, b ↔ B, c ↔ C.
THE CONCEPT

What is a triangle?

A triangle is a three-sided polygon whose three interior angles always add to 180°. The key to solving triangle problems is the pairing between each side and its opposite angle: side a belongs with angle A, side b with B, and side c with C.

Once that pairing is understood, triangle geometry becomes a connected system rather than a collection of unrelated formulas. Side lengths determine angles, angles determine proportions, and area links the shape to heights, radii and practical measurements.

C A B b a c height h
The conventional notation pairs each side with the angle opposite it. The altitude is perpendicular to the chosen base.
Σ

Angle sum

A + B + C = 180°. This is the fastest check for any solved triangle.

△

Three sides

The triangle inequality requires every two sides to add to more than the third.

↔

Opposite pairs

The longest side faces the largest angle; the shortest side faces the smallest.

CLASSIFICATION

Triangle types: sides and angles

Triangles can be classified in two independent ways. By side lengths they are equilateral, isosceles or scalene. By angles they are acute, right or obtuse.

EquilateralIsoscelesScalene three equal sides two equal sides all different

By angle

01 · ACUTE

All angles < 90°

No right angle; the triangle is “pointed” at all three vertices.

02 · RIGHT

One angle = 90°

The opposite side is the hypotenuse and is always the longest side.

03 · OBTUSE

One angle > 90°

Only one angle can be obtuse because the total is exactly 180°.

04 · CHECK

Largest side ↔ largest angle

This relationship is a fast sanity check for a numerical result.

FORMULAS

Triangle formulas: the core toolkit

Different sets of known measurements call for different formulas. The formulas below are connected: the Law of Cosines generalizes Pythagoras, while the Law of Sines connects proportional sides and opposite angles.

Angle sumA + B + C = 180°
Area from two sides and included angleK = ½ab sin(C)
Heron's formulas = (a+b+c)/2   ·   K = √[s(s−a)(s−b)(s−c)]
Law of Cosinesc² = a² + b² − 2ab cos(C)
Law of Sinesa/sin(A) = b/sin(B) = c/sin(C)
Radiir = K/s   ·   R = abc/(4K)
Altitude to side ahₐ = 2K/a

How the formulas connect

If all three sides are known, Heron's formula gives the area and the Law of Cosines gives the angles. If two sides and their included angle are known, the Law of Cosines gives the missing side and the sine-area formula gives the area directly. If two angles and one side are known, the angle sum gives the third angle and the Law of Sines gives the remaining sides.

Important: in the formula ½ab sin(C), angle C must be the angle between sides a and b. The same rule applies cyclically: ½bc sin(A) and ½ac sin(B).
SPECIAL CASE

Right triangles and the Pythagorean theorem

When one angle is exactly 90°, the Law of Cosines becomes the Pythagorean theorem because cos(90°) = 0.

Pythagorean relationship

a² + b² = c²

If c is the hypotenuse, it is opposite the right angle and must be the longest side.

Classic 3–4–5 triangle

3² + 4² = 5²

The same relationship scales: 6–8–10, 9–12–15, and so on.

AREA

Three useful ways to find triangle area

Area is not tied to one formula. The best method depends on what you know.

½

Base × height

K = ½bh. The height must meet the selected base at 90°.

sin

Two sides + angle

K = ½ab sin(C). The angle must be included between the two sides.

√

Three sides

Heron's formula uses only a, b and c, so no height or angle is required.

base b height h 90°
The altitude is the perpendicular distance from a vertex to the line containing the chosen base.
WORKED EXAMPLES

Worked triangle examples

SSS · sides 5, 6, 7

First find the semiperimeter:

s = (5 + 6 + 7) / 2 = 9

Then Heron's formula gives

K = √(9·4·3·2) = √216 ≈ 14.70
Area ≈ 14.70

SAS · a = 7, b = 9, C = 60°

Use the Law of Cosines:

c² = 7² + 9² − 2·7·9·cos60° = 67
c ≈ 8.19

Right triangle · 3 and 4

With the two legs known:

c = √(3² + 4²) = 5
Area = 6

Two angles + one side

Suppose A = 40°, B = 60°, and a = 10. Then C = 80°. The Law of Sines scales the other sides from a/sin(A).

c ≈ 15.32
REAL-WORLD GEOMETRY

Where triangular geometry is used

Triangles appear in structures because a triangular frame resists shape change: once the three side lengths are fixed, the geometry is rigid. That property makes triangular arrangements useful in trusses, roof frames, towers, surveying networks and temporary structures.

Historical Warren triangular truss diagram showing repeated triangular members
A historical Warren truss illustration: repeated triangles turn straight members into a stable structural system. The image is in the public domain.

In a roof truss, triangle geometry helps distribute loads along connected members. In bridge trusses, repeated triangular panels can span distance while keeping the structure comparatively light. In surveying, triangles help establish positions from measured angles and a known baseline. In navigation and engineering, the same side-angle relationships become a practical language for distance and direction.

GEOMETRY CHECKS

How to catch a wrong triangle answer

  • Angle sum: A + B + C must equal 180°.
  • Triangle inequality: every pair of sides must add to more than the third side.
  • Opposite pairing: a is opposite A, b is opposite B, c is opposite C.
  • Largest ↔ largest: the longest side must face the largest angle.
  • Area units: if lengths are in centimeters, area is in square centimeters.
  • Included angle: in ½ab sin(C), C must lie between a and b.
FAQ

Triangle calculator FAQ

How many values do I need to solve a triangle?
In general, three compatible measurements are needed, and at least one must be a side length. Three sides, two sides plus their included angle, or two angles plus one side are common solvable combinations.
Can I solve a triangle from three sides?
Yes. This is the SSS case. The side lengths are first checked with the triangle inequality, then the angles can be found with the Law of Cosines and the area with Heron's formula.
What is the difference between the inradius and circumradius?
The inradius r is the radius of the circle tangent to all three sides from inside the triangle. The circumradius R is the radius of the circle passing through all three vertices.
What happens in the SSA case?
Two sides and a non-included angle can sometimes produce two different triangles. This is the ambiguous case. The calculator identifies when the supplied measurements admit an alternate solution.
What is the fastest way to recognize a right triangle?
Let c be the longest side. If c² = a² + b², the triangle is right-angled. If c² is smaller, it is acute; if larger, it is obtuse.