Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Acute scalene triangle.

Sides: a = 165.4   b = 175.7   c = 166.3440489043

Area: T = 12359.95112579
Perimeter: p = 507.4440489043
Semiperimeter: s = 253.7220244521

Angle ∠ A = α = 57.76597038965° = 57°45'35″ = 1.0088097008 rad
Angle ∠ B = β = 63.96602961035° = 63°57'37″ = 1.11663177576 rad
Angle ∠ C = γ = 58.28° = 58°16'48″ = 1.01771778881 rad

Height: ha = 149.4555275186
Height: hb = 140.6943810562
Height: hc = 148.6110255134

Median: ma = 149.7688268159
Median: mb = 140.6976612068
Median: mc = 148.9855017456

Inradius: r = 48.71548799702
Circumradius: R = 97.77551500858

Vertex coordinates: A[166.3440489043; 0] B[0; 0] C[72.61095866195; 148.6110255134]
Centroid: CG[79.65500252207; 49.53767517113]
Coordinates of the circumscribed circle: U[83.17702445213; 51.40771045729]
Coordinates of the inscribed circle: I[78.02202445213; 48.71548799702]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122.2440296103° = 122°14'25″ = 1.0088097008 rad
∠ B' = β' = 116.0439703897° = 116°2'23″ = 1.11663177576 rad
∠ C' = γ' = 121.72° = 121°43'12″ = 1.01771778881 rad

Calculate another triangle




How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

a = 165.4 ; ; b = 175.7 ; ; gamma = 58° 16'48" ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 165.4**2+175.7**2 - 2 * 165.4 * 175.7 * cos(58° 16'48") } ; ; c = 166.34 ; ;


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 = 165.4 ; ; b = 175.7 ; ; c = 166.34 ; ;

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

p = a+b+c = 165.4+175.7+166.34 = 507.44 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 507.44 }{ 2 } = 253.72 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 253.72 * (253.72-165.4)(253.72-175.7)(253.72-166.34) } ; ; T = sqrt{ 152768395.1 } = 12359.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12359.95 }{ 165.4 } = 149.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12359.95 }{ 175.7 } = 140.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12359.95 }{ 166.34 } = 148.61 ; ;

6. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 165.4**2-175.7**2-166.34**2 }{ 2 * 175.7 * 166.34 } ) = 57° 45'35" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 175.7**2-165.4**2-166.34**2 }{ 2 * 165.4 * 166.34 } ) = 63° 57'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 166.34**2-165.4**2-175.7**2 }{ 2 * 175.7 * 165.4 } ) = 58° 16'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12359.95 }{ 253.72 } = 48.71 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 165.4 }{ 2 * sin 57° 45'35" } = 97.78 ; ;




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

See more informations about triangles or more information about solving triangles.