Triangle Calculator

Calculate triangle area, perimeter, angles, and sides. Solve any triangle using SSS, SAS, ASA, AAS, or SSA methods with step-by-step solutions.

Formula:Area = (1/2) × base × height

Triangle Properties

Area

6.0000

square units

Perimeter

12.0000

units

TypeScalene Right
Inradius1.0000

What do you know?

Enter Values

Try These Examples

Common triangle configurations

Triangle Visualization

GIOHA (36.9 deg)B (53.1 deg)C (90.0 deg)a = 3.00b = 4.00c = 5.00
G = CentroidI = IncenterO = CircumcenterH = Orthocenter

Complete Results

Side a

3.000000

Side b

4.000000

Side c

5.000000

Angle A

36.8699 deg

Angle B

53.1301 deg

Angle C

90.0000 deg

Area

6.000000

Perimeter

12.000000

Semi-perimeter

6.000000

Type

Scalene Right

Heights (Altitudes)

h_a (to side a)4.000000
h_b (to side b)3.000000
h_c (to side c)2.400000

Medians

m_a (to vertex A)4.272002
m_b (to vertex B)3.605551
m_c (to vertex C)2.500000

Inradius (inscribed circle)

1.000000

Circumradius (circumscribed circle)

2.500000

Triangle Centers

Coordinates relative to vertex C at origin, with B along positive x-axis.

Centroid (G)

Intersection of medians

(1.0000, 1.3333)

Incenter (I)

Intersection of angle bisectors

(1.0000, 1.0000)

Circumcenter (O)

Intersection of perp. bisectors

(1.5000, 2.0000)

Orthocenter (H)

Intersection of altitudes

(0.0000, 0.0000)

Euler Line: In any non-equilateral triangle, the centroid (G), circumcenter (O), and orthocenter (H) are collinear, with G dividing OH in ratio 1:2.

Step-by-Step Solution

Triangle Formulas

Area Formulas:

Area = (1/2) x base x height

Area = (1/2) x a x b x sin(C)

Area = sqrt(s(s-a)(s-b)(s-c)) (Heron's formula)

Law of Cosines:

c^2 = a^2 + b^2 - 2ab*cos(C)

Used for: SSS (find angles), SAS (find third side)

Law of Sines:

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

Used for: ASA, AAS, SSA (with ambiguous case check)

Radii:

Inradius r = Area / s

Circumradius R = abc / (4 x Area)

Common Special Triangles

3-4-5 Right Triangle

Sides: 3, 4, 5

Angles: 36.87 deg, 53.13 deg, 90 deg

Multiples work too: 6-8-10, 9-12-15

45-45-90 Triangle

Sides ratio: 1 : 1 : sqrt(2)

Isosceles right triangle

Example: 1, 1, 1.414...

30-60-90 Triangle

Sides ratio: 1 : sqrt(3) : 2

Half of equilateral triangle

Example: 1, 1.732, 2

Equilateral Triangle

All sides equal, all angles 60 deg

Area = (sqrt(3)/4) x side^2

Height = (sqrt(3)/2) x side

Triangle Properties

Area

6.0000

square units

Perimeter

12.0000

units

TypeScalene Right
Inradius1.0000

?How to Calculate Triangle Properties

Triangle area = (base x height) / 2, or using Heron's formula: sqrt(s(s-a)(s-b)(s-c)) where s is semi-perimeter. The sum of interior angles always equals 180 degrees. Use Law of Sines (a/sinA = b/sinB = c/sinC) and Law of Cosines (c squared = a squared + b squared - 2ab cosC) to solve triangles.

What is a Triangle?

A triangle is a polygon with three sides and three angles. The sum of interior angles is always 180 degrees. Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). Triangle calculators use trigonometric laws to find unknown measurements from given values.

Key Facts About Triangles

  • Sum of interior angles always equals 180 degrees
  • Area = (base x height) / 2, or use Heron's formula with three sides
  • Perimeter = sum of all three sides (a + b + c)
  • Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
  • Law of Cosines: c squared = a squared + b squared - 2ab*cos(C)
  • Triangle inequality: any side must be less than the sum of the other two
  • Equilateral: all sides and angles equal (60 degrees each)
  • Isosceles: two equal sides and two equal base angles

Practice Triangle Problems

Test your understanding of triangle calculations

Practice with 6 problems to test your understanding.

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Quick Answer

Triangle area = (base x height) / 2, or using Heron's formula: sqrt(s(s-a)(s-b)(s-c)) where s is semi-perimeter. The sum of interior angles always equals 180 degrees. Use Law of Sines (a/sinA = b/sinB = c/sinC) and Law of Cosines (c squared = a squared + b squared - 2ab cosC) to solve triangles.

Frequently Asked Questions

For base and height: Area = (1/2) x base x height. For three sides (Heron's formula): s = (a+b+c)/2, Area = sqrt(s(s-a)(s-b)(s-c)). For two sides and included angle: Area = (1/2) x a x b x sin(C).

When you know two sides and an angle opposite to one of them (SSA), there may be zero, one, or two possible triangles. This happens because the sine function gives the same value for an angle and its supplement. Always check for the second solution!

c^2 = a^2 + b^2 - 2ab*cos(C). This generalizes Pythagorean theorem to any triangle. Use it to find a side when you know two sides and the included angle, or to find angles when you know all three sides.

a/sin(A) = b/sin(B) = c/sin(C). Each side divided by the sine of its opposite angle gives the same ratio. Use it when you know a side-angle pair plus one other measurement.

Centroid: intersection of medians (center of mass). Incenter: intersection of angle bisectors (center of inscribed circle). Circumcenter: intersection of perpendicular bisectors (center of circumscribed circle). Orthocenter: intersection of altitudes.

By sides: Equilateral (all equal), Isosceles (two equal), Scalene (none equal). By angles: Acute (all < 90 deg), Right (one = 90 deg), Obtuse (one > 90 deg). A triangle can be described by both, e.g., "Isosceles Right".

Last updated: 2025-01-15