Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Right scalene triangle.

Sides: a = 70.71106781187   b = 50   c = 50

Area: T = 1250
Perimeter: p = 170.7110678119
Semiperimeter: s = 85.35553390593

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

Height: ha = 35.35553390593
Height: hb = 50
Height: hc = 50

Median: ma = 35.35553390593
Median: mb = 55.90216994375
Median: mc = 55.90216994375

Inradius: r = 14.64546609407
Circumradius: R = 35.35553390593

Vertex coordinates: A[50; 0] B[0; 0] C[50; 50]
Centroid: CG[33.33333333333; 16.66766666667]
Coordinates of the circumscribed circle: U[25; 25]
Coordinates of the inscribed circle: I[35.35553390593; 14.64546609407]

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 = 50 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 50 * fraction{ sin(90° ) }{ sin (45° ) } = 70.71 ; ;

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 = 50 * fraction{ sin(45° ) }{ sin (45° ) } = 50 ; ;


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 = 70.71 ; ; b = 50 ; ; c = 50 ; ;

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

p = a+b+c = 70.71+50+50 = 170.71 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 170.71 }{ 2 } = 85.36 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 85.36 * (85.36-70.71)(85.36-50)(85.36-50) } ; ; T = sqrt{ 1562500 } = 1250 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1250 }{ 70.71 } = 35.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1250 }{ 50 } = 50 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1250 }{ 50 } = 50 ; ;

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{ 70.71**2-50**2-50**2 }{ 2 * 50 * 50 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 50**2-70.71**2-50**2 }{ 2 * 70.71 * 50 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 50**2-70.71**2-50**2 }{ 2 * 50 * 70.71 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1250 }{ 85.36 } = 14.64 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 70.71 }{ 2 * sin 90° } = 35.36 ; ;




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