Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
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Acute isosceles triangle.

Sides: a = 3591.227714932   b = 3591.227714932   c = 4806

Area: T = 6413130.901114
Perimeter: p = 11988.45442986
Semiperimeter: s = 5994.227714932

Angle ∠ A = α = 48° = 0.8387758041 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 84° = 1.46660765717 rad

Height: ha = 3571.554403124
Height: hb = 3571.554403124
Height: hc = 2668.802187313

Median: ma = 3843.572205078
Median: mb = 3843.572205078
Median: mc = 2668.802187313

Inradius: r = 1069.88545308
Circumradius: R = 2416.236639579

Vertex coordinates: A[4806; 0] B[0; 0] C[2403; 2668.802187313]
Centroid: CG[2403; 889.6010624378]
Coordinates of the circumscribed circle: U[2403; 252.5655477343]
Coordinates of the inscribed circle: I[2403; 1069.88545308]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132° = 0.8387758041 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 96° = 1.46660765717 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 48° ; ; beta = 48° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 48° - 48° = 84° ; ;

2. By using the law of sines, we calculate unknown side a

c = 4806 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 4806 * fraction{ sin(48° ) }{ sin (84° ) } = 3591.23 ; ;

3. By using the law of sines, we calculate last unknown side b

 fraction{ b }{ c } = fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = c * fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = 4806 * fraction{ sin(48° ) }{ sin (84° ) } = 3591.23 ; ;


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 = 3591.23 ; ; b = 3591.23 ; ; c = 4806 ; ;

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

p = a+b+c = 3591.23+3591.23+4806 = 11988.45 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11988.45 }{ 2 } = 5994.23 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5994.23 * (5994.23-3591.23)(5994.23-3591.23)(5994.23-4806) } ; ; T = sqrt{ 4.113 * 10**{ 13 } } = 6413130.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6413130.9 }{ 3591.23 } = 3571.55 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6413130.9 }{ 3591.23 } = 3571.55 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6413130.9 }{ 4806 } = 2668.8 ; ;

8. 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{ 3591.23**2-3591.23**2-4806**2 }{ 2 * 3591.23 * 4806 } ) = 48° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3591.23**2-3591.23**2-4806**2 }{ 2 * 3591.23 * 4806 } ) = 48° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4806**2-3591.23**2-3591.23**2 }{ 2 * 3591.23 * 3591.23 } ) = 84° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6413130.9 }{ 5994.23 } = 1069.88 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3591.23 }{ 2 * sin 48° } = 2416.24 ; ;




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