Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 50   b = 50   c = 38.26883432365

Area: T = 883.8833476483
Perimeter: p = 138.2688343236
Semiperimeter: s = 69.13441716183

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 35.35553390593
Height: hb = 35.35553390593
Height: hc = 46.19439766256

Median: ma = 36.84106439552
Median: mb = 36.84106439552
Median: mc = 46.19439766256

Inradius: r = 12.78550447296
Circumradius: R = 27.06598050073

Vertex coordinates: A[38.26883432365; 0] B[0; 0] C[19.13441716183; 46.19439766256]
Centroid: CG[19.13441716183; 15.39879922085]
Coordinates of the circumscribed circle: U[19.13441716183; 19.13441716183]
Coordinates of the inscribed circle: I[19.13441716183; 12.78550447296]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.17880972451 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 rad
∠ C' = γ' = 135° = 0.78553981634 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 = 50 ; ; b = 50 ; ; gamma = 45° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 50**2+50**2 - 2 * 50 * 50 * cos(45° ) } ; ; c = 38.27 ; ;


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 = 50 ; ; b = 50 ; ; c = 38.27 ; ;

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

p = a+b+c = 50+50+38.27 = 138.27 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 138.27 }{ 2 } = 69.13 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 69.13 * (69.13-50)(69.13-50)(69.13-38.27) } ; ; T = sqrt{ 781250 } = 883.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 883.88 }{ 50 } = 35.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 883.88 }{ 50 } = 35.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 883.88 }{ 38.27 } = 46.19 ; ;

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{ 50**2-50**2-38.27**2 }{ 2 * 50 * 38.27 } ) = 67° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 50**2-50**2-38.27**2 }{ 2 * 50 * 38.27 } ) = 67° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 38.27**2-50**2-50**2 }{ 2 * 50 * 50 } ) = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 883.88 }{ 69.13 } = 12.79 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 67° 30' } = 27.06 ; ;




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