Triangle calculator SAS

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


Obtuse isosceles triangle.

Sides: a = 60   b = 60   c = 103.9233048454

Area: T = 1558.846572681
Perimeter: p = 223.9233048454
Semiperimeter: s = 111.9621524227

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 51.96215242271
Height: hb = 51.96215242271
Height: hc = 30

Median: ma = 79.37325393319
Median: mb = 79.37325393319
Median: mc = 30

Inradius: r = 13.92330484541
Circumradius: R = 60

Vertex coordinates: A[103.9233048454; 0] B[0; 0] C[51.96215242271; 30]
Centroid: CG[51.96215242271; 10]
Coordinates of the circumscribed circle: U[51.96215242271; -30]
Coordinates of the inscribed circle: I[51.96215242271; 13.92330484541]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 60° = 2.09443951024 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 = 60 ; ; b = 60 ; ; gamma = 120° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 60**2+60**2 - 2 * 60 * 60 * cos(120° ) } ; ; c = 103.92 ; ;


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 = 60 ; ; b = 60 ; ; c = 103.92 ; ;

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

p = a+b+c = 60+60+103.92 = 223.92 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 223.92 }{ 2 } = 111.96 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 111.96 * (111.96-60)(111.96-60)(111.96-103.92) } ; ; T = sqrt{ 2430000 } = 1558.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1558.85 }{ 60 } = 51.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1558.85 }{ 60 } = 51.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1558.85 }{ 103.92 } = 30 ; ;

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{ 60**2-60**2-103.92**2 }{ 2 * 60 * 103.92 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-60**2-103.92**2 }{ 2 * 60 * 103.92 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 103.92**2-60**2-60**2 }{ 2 * 60 * 60 } ) = 120° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1558.85 }{ 111.96 } = 13.92 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 30° } = 60 ; ;




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