Calculate all triangle properties: sides, angles, area, perimeter, type, inradius and circumradius. Supports SSS, SAS, ASA, AAS, right triangles, and area from base & height. Uses Law of Sines, Law of Cosines, and Heron's formula.
A triangle is the simplest polygon and the foundation of geometry, engineering, navigation, and construction. Our free triangle calculator solves every type of triangle instantly: right triangles using the Pythagorean theorem, oblique triangles using the Law of Sines and Law of Cosines, and any triangle where you know the base and height. Whether you are a student working through GCSE maths in the UK or studying geometry in a US high school, this tool handles all the calculations in seconds.
Enter the values you know, choose your triangle type (Right, SSS, SAS, ASA, AAS, or Base & Height), and the calculator returns all three sides, all three angles, area, perimeter, triangle classification, inradius, and circumradius instantly.
Every triangle problem falls into one of six categories depending on which measurements you know. Here is a complete breakdown:
| Method | Known Values | Formula Used |
|---|---|---|
| Right Triangle | Two legs (a, b) | Pythagorean theorem: c² = a² + b² |
| SSS (Side-Side-Side) | All three sides | Law of Cosines → angles; Heron's formula → area |
| SAS (Side-Angle-Side) | Two sides + included angle | Law of Cosines → third side; Law of Sines → angles |
| ASA (Angle-Side-Angle) | Two angles + side between them | Third angle = 180° − A − B; Law of Sines → sides |
| AAS (Angle-Angle-Side) | Two angles + non-included side | Third angle = 180° − A − B; Law of Sines → sides |
| Base & Height | Base length + perpendicular height | Area = ½ × base × height |
The Pythagorean theorem is the most famous formula in geometry. For any right triangle with legs a and b and hypotenuse c:
c² = a² + b²
This means the square of the hypotenuse equals the sum of the squares of the other two sides. A classic example: a triangle with legs of 3 and 4 has a hypotenuse of exactly 5 (since 9 + 16 = 25). This 3-4-5 triple is used daily by builders on both sides of the Atlantic to check that corners are perfectly square.
In the UK, the Pythagorean theorem is a core topic in GCSE Mathematics at both Foundation and Higher tier. In the US, it appears in the 8th grade Common Core curriculum and again in high school geometry courses.
The Law of Sines states that in any triangle, each side divided by the sine of its opposite angle equals the same value (the diameter of the circumscribed circle):
a / sin(A) = b / sin(B) = c / sin(C)
This law is essential for solving ASA and AAS triangles. If you know angle A = 40°, angle B = 60°, and side a = 8 cm, you can find side b: b = a × sin(B) / sin(A) = 8 × sin(60°) / sin(40°) ≈ 10.77 cm.
The Law of Cosines generalises the Pythagorean theorem to any triangle. It has three forms:
When angle C = 90°, cos(C) = 0 and the formula reduces to c² = a² + b², confirming the Pythagorean theorem as a special case. Use the Law of Cosines for SAS triangles (find the missing side) and SSS triangles (find any angle using the rearranged form: cos(C) = (a² + b² − c²) / (2ab)).
When you know all three sides of a triangle, Heron's formula gives the area without needing the height:
s = (a + b + c) / 2 (semi-perimeter)
Area = √(s(s−a)(s−b)(s−c))
Example: For a triangle with sides 5, 6, and 7: s = (5+6+7)/2 = 9. Area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70 square units.
In a 30-60-90 triangle, the sides are always in the ratio 1 : √3 : 2. If the shortest side (opposite 30°) has length x, then the other leg = x√3 and the hypotenuse = 2x. These triangles appear in roof pitch calculations, equilateral triangle bisections, and trigonometry unit circle derivations.
An isosceles right triangle has two 45° angles. The sides are in ratio 1 : 1 : √2. If each leg = x, the hypotenuse = x√2. These are common in square diagonals, tile patterns, and construction layouts.
For any triangle to exist, the sum of any two sides must be greater than the third side:
a + b > c, a + c > b, b + c > a
If this condition is not met, the triangle cannot be constructed. For example, sides of 2, 3, and 10 cannot form a triangle because 2 + 3 = 5 < 10.
Every triangle has an inscribed circle (incircle) tangent to all three sides, and a circumscribed circle (circumcircle) passing through all three vertices.
The 3-4-5 right triangle method (and its multiples: 6-8-10, 9-12-15) is used by builders worldwide to check that walls and foundations are perfectly square. In the UK, building regulations and British Standards require precise angular accuracy in structural work. In the US, the NFPA and IRC building codes both assume correctly squared foundations. A simple tape measure and the Pythagorean theorem can verify a 90° corner without any special equipment.
Calculating rafter length involves a right triangle where the run is the horizontal distance and the rise is the vertical height difference. US roof pitch is expressed as rise-over-12 (e.g., 6:12 means 6 inches rise for every 12 inches run). UK roof pitch uses degrees. Either way, the rafter length = √(run² + rise²).
Triangulation is the foundation of land surveying. The UK's Ordnance Survey used triangulation networks dating back to the 18th century — the Principal Triangulation of Great Britain measured baselines and angles to map the entire island. Modern GPS satellites use the same geometric principles to calculate positions by measuring distances from multiple known points.
Structural engineers decompose forces into components using right triangles. Bridge trusses, crane arms, and roof structures all rely on triangular geometry because a triangle is the only polygon that is inherently rigid — its shape cannot change without changing side lengths. This is why triangular bracing appears in the Eiffel Tower, suspension bridges, and steel-frame buildings.
Angles can be expressed in degrees (a full circle = 360°) or radians (a full circle = 2π ≈ 6.2832 radians). To convert: radians = degrees × π/180; degrees = radians × 180/π. Scientific calculators and programming languages typically work in radians by default. Our calculator accepts both — simply select your preferred unit using the toggle above.
| Property | Formula |
|---|---|
| Area (base & height) | A = ½bh |
| Area (Heron's) | A = √(s(s-a)(s-b)(s-c)) |
| Perimeter | P = a + b + c |
| Pythagorean theorem | c² = a² + b² |
| Law of Sines | a/sinA = b/sinB = c/sinC |
| Law of Cosines | c² = a² + b² − 2ab·cosC |
| Inradius | r = Area / s |
| Circumradius | R = abc / (4·Area) |
The most common method is Area = ½ × base × height, where the height must be perpendicular to the base. If you only know three sides, use Heron's formula: s = (a+b+c)/2, Area = √(s(s-a)(s-b)(s-c)). For a SAS triangle use Area = ½ab·sinC.
The Law of Sines states a/sinA = b/sinB = c/sinC for any triangle. It is used to solve ASA and AAS cases — when you know two angles and one side. Once you know two angles, the third is 180° minus their sum, then apply the sine ratio to find the remaining sides.
The Law of Cosines: c² = a² + b² − 2ab·cosC, relates the three sides to one of the angles. It is used for SAS triangles (find the third side) and SSS triangles (find any angle). When C = 90°, it simplifies to the Pythagorean theorem.
The perimeter of any triangle is simply the sum of all three sides: P = a + b + c. If you only know some sides or angles, use the appropriate formula (Pythagorean theorem for right triangles, Law of Sines or Cosines for oblique triangles) to find all three sides first, then add them.
An equilateral triangle has all three sides equal and all three angles equal to exactly 60°. If the side length is a, the height = (√3/2)a and area = (√3/4)a². Equilateral triangles are the most symmetrical triangle and appear in architecture, logos, and crystallography.
Pythagorean triples are sets of three whole numbers (a, b, c) satisfying a² + b² = c². The smallest is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Any multiple of a triple is also a triple — so (6, 8, 10), (9, 12, 15) etc. all work.
The triangle inequality theorem states that for any triangle to exist, the sum of any two sides must be strictly greater than the third: a + b > c, a + c > b, and b + c > a. If any condition fails, no triangle is possible with those side lengths.
The inradius r is the radius of the largest circle that fits inside the triangle, touching all three sides. r = Area / s where s is the semi-perimeter. The circumradius R is the radius of the circle passing through all three vertices. R = abc / (4·Area). For a right triangle, R = c/2 (half the hypotenuse).
Results are estimates based on the inputs provided and standard mathematical formulas. While every effort has been made to ensure accuracy, always verify critical calculations independently. Not intended as a substitute for professional engineering, surveying, or architectural advice.