Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 42   b = 42   c = 59.39769696197

Area: T = 882
Perimeter: p = 143.397696962
Semiperimeter: s = 71.69884848098

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 42
Height: hb = 42
Height: hc = 29.69884848098

Median: ma = 46.95774275275
Median: mb = 46.95774275275
Median: mc = 29.69884848098

Inradius: r = 12.30215151902
Circumradius: R = 29.69884848098

Vertex coordinates: A[59.39769696197; 0] B[0; 0] C[29.69884848098; 29.69884848098]
Centroid: CG[29.69884848098; 9.89994949366]
Coordinates of the circumscribed circle: U[29.69884848098; 0]
Coordinates of the inscribed circle: I[29.69884848098; 12.30215151902]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 42 ; ; b = 42 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 42**2+42**2 - 2 * 42 * 42 * cos(90° ) } ; ; c = 59.4 ; ;


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 = 42 ; ; b = 42 ; ; c = 59.4 ; ;

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

p = a+b+c = 42+42+59.4 = 143.4 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 143.4 }{ 2 } = 71.7 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 71.7 * (71.7-42)(71.7-42)(71.7-59.4) } ; ; T = sqrt{ 777924 } = 882 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 882 }{ 42 } = 42 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 882 }{ 42 } = 42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 882 }{ 59.4 } = 29.7 ; ;

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{ 42**2-42**2-59.4**2 }{ 2 * 42 * 59.4 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42**2-42**2-59.4**2 }{ 2 * 42 * 59.4 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 59.4**2-42**2-42**2 }{ 2 * 42 * 42 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 882 }{ 71.7 } = 12.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 42 }{ 2 * sin 45° } = 29.7 ; ;




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