Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 400   b = 400   c = 565.6855424949

Area: T = 80000
Perimeter: p = 1365.685542495
Semiperimeter: s = 682.8432712475

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

Height: ha = 400
Height: hb = 400
Height: hc = 282.8432712475

Median: ma = 447.21435955
Median: mb = 447.21435955
Median: mc = 282.8432712475

Inradius: r = 117.1577287525
Circumradius: R = 282.8432712475

Vertex coordinates: A[565.6855424949; 0] B[0; 0] C[282.8432712475; 282.8432712475]
Centroid: CG[282.8432712475; 94.28109041582]
Coordinates of the circumscribed circle: U[282.8432712475; 0]
Coordinates of the inscribed circle: I[282.8432712475; 117.1577287525]

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 = 400 ; ; b = 400 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 400**2+400**2 - 2 * 400 * 400 * cos(90° ) } ; ; c = 565.69 ; ;


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 = 400 ; ; b = 400 ; ; c = 565.69 ; ;

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

p = a+b+c = 400+400+565.69 = 1365.69 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1365.69 }{ 2 } = 682.84 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 682.84 * (682.84-400)(682.84-400)(682.84-565.69) } ; ; T = sqrt{ 6400000000 } = 80000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80000 }{ 400 } = 400 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80000 }{ 400 } = 400 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80000 }{ 565.69 } = 282.84 ; ;

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{ 400**2-400**2-565.69**2 }{ 2 * 400 * 565.69 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 400**2-400**2-565.69**2 }{ 2 * 400 * 565.69 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 565.69**2-400**2-400**2 }{ 2 * 400 * 400 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80000 }{ 682.84 } = 117.16 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 400 }{ 2 * sin 45° } = 282.84 ; ;

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