Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side b, c and angle γ.

Right scalene triangle.

Sides: a = 46.19659348861   b = 80   c = 92.38

Area: T = 1847.837739544
Perimeter: p = 218.5765934886
Semiperimeter: s = 109.2887967443

Angle ∠ A = α = 30.0044250458° = 30°15″ = 0.52436729601 rad
Angle ∠ B = β = 59.9965749542° = 59°59'45″ = 1.04771233667 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 80
Height: hb = 46.19659348861
Height: hc = 40.0055139542

Median: ma = 83.26877374497
Median: mb = 61.10769914167
Median: mc = 46.19

Inradius: r = 16.9087967443
Circumradius: R = 46.19

Vertex coordinates: A[92.38; 0] B[0; 0] C[23.10109352674; 40.0055139542]
Centroid: CG[38.49436450891; 13.3355046514]
Coordinates of the circumscribed circle: U[46.19; 0]
Coordinates of the inscribed circle: I[29.2887967443; 16.9087967443]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.9965749542° = 149°59'45″ = 0.52436729601 rad
∠ B' = β' = 120.0044250458° = 120°15″ = 1.04771233667 rad
∠ C' = γ' = 90° = 1.57107963268 rad

Calculate another triangle


How did we calculate this triangle?

1. Input data entered: side b, c and angle γ.

b = 80 ; ; c = 92.38 ; ; gamma = 90° ; ;

2. From angle γ, side b and side c we calculate side a - by using the law of cosines and quadratic equation:

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 92.38**2 = 80**2 + a**2 - 2 * 80 * a * cos 90° ; ; ; ; ; ; a**2 -2134.064 =0 ; ; p=1; q=-0; r=-2134.064 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-2134.064) = 8536.2576 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 8536.26 } }{ 2 } ; ;
a_{1,2} = ± 46.1959348861 ; ; a_{1} = 46.1959348861 ; ; a_{2} = -46.1959348861 ; ; ; ; text{ Factored form: } ; ; (a -46.1959348861) (a +46.1959348861) = 0 ; ; ; ; a > 0 ; ; ; ; a = 46.196 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 46.2 ; ; b = 80 ; ; c = 92.38 ; ;

3. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 46.2+80+92.38 = 218.58 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 218.58 }{ 2 } = 109.29 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 109.29 * (109.29-46.2)(109.29-80)(109.29-92.38) } ; ; T = sqrt{ 3414503.04 } = 1847.84 ; ;

6. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1847.84 }{ 46.2 } = 80 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1847.84 }{ 80 } = 46.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1847.84 }{ 92.38 } = 40.01 ; ;

7. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 80**2+92.38**2-46.2**2 }{ 2 * 80 * 92.38 } ) = 30° 15" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46.2**2+92.38**2-80**2 }{ 2 * 46.2 * 92.38 } ) = 59° 59'45" ; ;
 gamma = 180° - alpha - beta = 180° - 30° 15" - 59° 59'45" = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1847.84 }{ 109.29 } = 16.91 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 46.2 }{ 2 * sin 30° 15" } = 46.19 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 92.38**2 - 46.2**2 } }{ 2 } = 83.268 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 92.38**2+2 * 46.2**2 - 80**2 } }{ 2 } = 61.107 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 46.2**2 - 92.38**2 } }{ 2 } = 46.19 ; ;
Calculate another triangle


Look also our friend's collection of math examples and problems:

See more information about triangles or more details on solving triangles.