Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 500   b = 500   c = 707.1076781187

Area: T = 125000
Perimeter: p = 1707.107678119
Semiperimeter: s = 853.5533390593

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

Height: ha = 500
Height: hb = 500
Height: hc = 353.5533390593

Median: ma = 559.0176994375
Median: mb = 559.0176994375
Median: mc = 353.5533390593

Inradius: r = 146.4476609407
Circumradius: R = 353.5533390593

Vertex coordinates: A[707.1076781187; 0] B[0; 0] C[353.5533390593; 353.5533390593]
Centroid: CG[353.5533390593; 117.8511130198]
Coordinates of the circumscribed circle: U[353.5533390593; 0]
Coordinates of the inscribed circle: I[353.5533390593; 146.4476609407]

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 = 500 ; ; b = 500 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 500**2+500**2 - 2 * 500 * 500 * cos(90° ) } ; ; c = 707.11 ; ;


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 = 500 ; ; b = 500 ; ; c = 707.11 ; ;

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

p = a+b+c = 500+500+707.11 = 1707.11 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1707.11 }{ 2 } = 853.55 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 853.55 * (853.55-500)(853.55-500)(853.55-707.11) } ; ; T = sqrt{ 15625000000 } = 125000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 125000 }{ 500 } = 500 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 125000 }{ 500 } = 500 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 125000 }{ 707.11 } = 353.55 ; ;

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{ 500**2-500**2-707.11**2 }{ 2 * 500 * 707.11 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 500**2-500**2-707.11**2 }{ 2 * 500 * 707.11 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 707.11**2-500**2-500**2 }{ 2 * 500 * 500 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 125000 }{ 853.55 } = 146.45 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 500 }{ 2 * sin 45° } = 353.55 ; ;




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