Triangle calculator ASA

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

Sides: a = 63.64396103068   b = 45   c = 45

Area: T = 1012.5
Perimeter: p = 153.6439610307
Semiperimeter: s = 76.82198051534

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

Height: ha = 31.82198051534
Height: hb = 45
Height: hc = 45

Median: ma = 31.82198051534
Median: mb = 50.31215294937
Median: mc = 50.31215294937

Inradius: r = 13.18801948466
Circumradius: R = 31.82198051534

Vertex coordinates: A[45; 0] B[0; 0] C[45; 45]
Centroid: CG[30; 15]
Coordinates of the circumscribed circle: U[22.5; 22.5]
Coordinates of the inscribed circle: I[31.82198051534; 13.18801948466]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 135° = 0.78553981634 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 45° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 45° = 45° ; ;

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

c = 45 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 45 * fraction{ sin 90° }{ sin 45° } = 63.64 ; ;

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 = 45 * fraction{ sin 45° }{ sin 45° } = 45 ; ;
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 = 63.64 ; ; b = 45 ; ; c = 45 ; ;

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

p = a+b+c = 63.64+45+45 = 153.64 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 153.64 }{ 2 } = 76.82 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 76.82 * (76.82-63.64)(76.82-45)(76.82-45) } ; ; T = sqrt{ 1025156.25 } = 1012.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1012.5 }{ 63.64 } = 31.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1012.5 }{ 45 } = 45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1012.5 }{ 45 } = 45 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 45**2+45**2-63.64**2 }{ 2 * 45 * 45 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 63.64**2+45**2-45**2 }{ 2 * 63.64 * 45 } ) = 45° ; ; gamma = 180° - alpha - beta = 180° - 90° - 45° = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1012.5 }{ 76.82 } = 13.18 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 63.64 }{ 2 * sin 90° } = 31.82 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 45**2 - 63.64**2 } }{ 2 } = 31.82 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 63.64**2 - 45**2 } }{ 2 } = 50.312 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 45**2+2 * 63.64**2 - 45**2 } }{ 2 } = 50.312 ; ;
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