Triangle calculator ASA

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

Sides: a = 424.2644068712   b = 300   c = 300

Area: T = 45000
Perimeter: p = 1024.264406871
Semiperimeter: s = 512.1322034356

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

Height: ha = 212.1322034356
Height: hb = 300
Height: hc = 300

Median: ma = 212.1322034356
Median: mb = 335.4110196625
Median: mc = 335.4110196625

Inradius: r = 87.8687965644
Circumradius: R = 212.1322034356

Vertex coordinates: A[300; 0] B[0; 0] C[300; 300]
Centroid: CG[200; 100]
Coordinates of the circumscribed circle: U[150; 150]
Coordinates of the inscribed circle: I[212.1322034356; 87.8687965644]

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 = 300 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 300 * fraction{ sin(90° ) }{ sin (45° ) } = 424.26 ; ;

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 = 300 * fraction{ sin(45° ) }{ sin (45° ) } = 300 ; ;


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 = 424.26 ; ; b = 300 ; ; c = 300 ; ;

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

p = a+b+c = 424.26+300+300 = 1024.26 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1024.26 }{ 2 } = 512.13 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 512.13 * (512.13-424.26)(512.13-300)(512.13-300) } ; ; T = sqrt{ 2025000000 } = 45000 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 45000 }{ 424.26 } = 212.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 45000 }{ 300 } = 300 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 45000 }{ 300 } = 300 ; ;

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{ 424.26**2-300**2-300**2 }{ 2 * 300 * 300 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 300**2-424.26**2-300**2 }{ 2 * 424.26 * 300 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 300**2-424.26**2-300**2 }{ 2 * 300 * 424.26 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 45000 }{ 512.13 } = 87.87 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 424.26 }{ 2 * sin 90° } = 212.13 ; ;




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