Triangle calculator ASA

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

Sides: a = 2408.863323996   b = 9307.133286189   c = 8990

Area: T = 10827840.26436
Perimeter: p = 20705.99661018
Semiperimeter: s = 10352.99880509

Angle ∠ A = α = 15° = 0.26217993878 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 75° = 1.3098996939 rad

Height: ha = 8990
Height: hb = 2326.783321547
Height: hc = 2408.863323996

Median: ma = 9070.323279068
Median: mb = 4653.566643094
Median: mc = 5099.769931918

Inradius: r = 1045.865518903
Circumradius: R = 4653.566643094

Vertex coordinates: A[8990; 0] B[0; 0] C[-0; 2408.863323996]
Centroid: CG[2996.667666667; 802.9544413319]
Coordinates of the circumscribed circle: U[4495; 1204.432161998]
Coordinates of the inscribed circle: I[1045.865518903; 1045.865518903]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165° = 0.26217993878 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 105° = 1.3098996939 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 15° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 15° - 90° = 75° ; ;

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

c = 8990 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 8990 * fraction{ sin(15° ) }{ sin (75° ) } = 2408.86 ; ;

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 = 8990 * fraction{ sin(90° ) }{ sin (75° ) } = 9307.13 ; ;


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 = 2408.86 ; ; b = 9307.13 ; ; c = 8990 ; ;

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

p = a+b+c = 2408.86+9307.13+8990 = 20706 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20706 }{ 2 } = 10353 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10353 * (10353-2408.86)(10353-9307.13)(10353-8990) } ; ; T = sqrt{ 1.172 * 10**{ 14 } } = 10827840.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10827840.26 }{ 2408.86 } = 8990 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10827840.26 }{ 9307.13 } = 2326.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10827840.26 }{ 8990 } = 2408.86 ; ;

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{ 2408.86**2-9307.13**2-8990**2 }{ 2 * 9307.13 * 8990 } ) = 15° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 9307.13**2-2408.86**2-8990**2 }{ 2 * 2408.86 * 8990 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8990**2-2408.86**2-9307.13**2 }{ 2 * 9307.13 * 2408.86 } ) = 75° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10827840.26 }{ 10353 } = 1045.87 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2408.86 }{ 2 * sin 15° } = 4653.57 ; ;




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