Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 31   b = 31   c = 28.14774109839

Area: T = 388.7332665797
Perimeter: p = 90.14774109839
Semiperimeter: s = 45.07437054919

Angle ∠ A = α = 63° = 1.10995574288 rad
Angle ∠ B = β = 63° = 1.10995574288 rad
Angle ∠ C = γ = 54° = 0.94224777961 rad

Height: ha = 25.08795268256
Height: hb = 25.08795268256
Height: hc = 27.62112022498

Median: ma = 25.22767392373
Median: mb = 25.22767392373
Median: mc = 27.62112022498

Inradius: r = 8.62443778175
Circumradius: R = 17.39660566833

Vertex coordinates: A[28.14774109839; 0] B[0; 0] C[14.07437054919; 27.62112022498]
Centroid: CG[14.07437054919; 9.20770674166]
Coordinates of the circumscribed circle: U[14.07437054919; 10.22551455665]
Coordinates of the inscribed circle: I[14.07437054919; 8.62443778175]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117° = 1.10995574288 rad
∠ B' = β' = 117° = 1.10995574288 rad
∠ C' = γ' = 126° = 0.94224777961 rad

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How did we calculate this triangle?

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

a = 31 ; ; b = 31 ; ; gamma = 54° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 31**2+31**2 - 2 * 31 * 31 * cos(54° ) } ; ; c = 28.15 ; ;


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 = 31 ; ; b = 31 ; ; c = 28.15 ; ;

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

p = a+b+c = 31+31+28.15 = 90.15 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 90.15 }{ 2 } = 45.07 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.07 * (45.07-31)(45.07-31)(45.07-28.15) } ; ; T = sqrt{ 151113.09 } = 388.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 388.73 }{ 31 } = 25.08 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 388.73 }{ 31 } = 25.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 388.73 }{ 28.15 } = 27.62 ; ;

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{ 31**2-31**2-28.15**2 }{ 2 * 31 * 28.15 } ) = 63° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 31**2-31**2-28.15**2 }{ 2 * 31 * 28.15 } ) = 63° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28.15**2-31**2-31**2 }{ 2 * 31 * 31 } ) = 54° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 388.73 }{ 45.07 } = 8.62 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 31 }{ 2 * sin 63° } = 17.4 ; ;

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