Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 4000   b = 4000   c = 5656.854424949

Area: T = 8000000
Perimeter: p = 13656.85442495
Semiperimeter: s = 6828.427712475

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

Height: ha = 4000
Height: hb = 4000
Height: hc = 2828.427712475

Median: ma = 4472.1365955
Median: mb = 4472.1365955
Median: mc = 2828.427712475

Inradius: r = 1171.573287525
Circumradius: R = 2828.427712475

Vertex coordinates: A[5656.854424949; 0] B[0; 0] C[2828.427712475; 2828.427712475]
Centroid: CG[2828.427712475; 942.8099041582]
Coordinates of the circumscribed circle: U[2828.427712475; 0]
Coordinates of the inscribed circle: I[2828.427712475; 1171.573287525]

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 = 4000 ; ; b = 4000 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 4000**2+4000**2 - 2 * 4000 * 4000 * cos(90° ) } ; ; c = 5656.85 ; ;


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 = 4000 ; ; b = 4000 ; ; c = 5656.85 ; ;

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

p = a+b+c = 4000+4000+5656.85 = 13656.85 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13656.85 }{ 2 } = 6828.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6828.43 * (6828.43-4000)(6828.43-4000)(6828.43-5656.85) } ; ; T = sqrt{ 6.4 * 10**{ 13 } } = 8000000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8000000 }{ 4000 } = 4000 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8000000 }{ 4000 } = 4000 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8000000 }{ 5656.85 } = 2828.43 ; ;

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{ 4000**2-4000**2-5656.85**2 }{ 2 * 4000 * 5656.85 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4000**2-4000**2-5656.85**2 }{ 2 * 4000 * 5656.85 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5656.85**2-4000**2-4000**2 }{ 2 * 4000 * 4000 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8000000 }{ 6828.43 } = 1171.57 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4000 }{ 2 * sin 45° } = 2828.43 ; ;




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