Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 3185   b = 5516.582182211   c = 6370

Area: T = 8785156.552171
Perimeter: p = 15071.58218221
Semiperimeter: s = 7535.791091105

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 5516.582182211
Height: hb = 3185
Height: hc = 2758.291091105

Median: ma = 5741.844040618
Median: mb = 4213.359896287
Median: mc = 3185

Inradius: r = 1165.791091105
Circumradius: R = 3185

Vertex coordinates: A[6370; 0] B[0; 0] C[1592.5; 2758.291091105]
Centroid: CG[2654.167666667; 919.4330303685]
Coordinates of the circumscribed circle: U[3185; -0]
Coordinates of the inscribed circle: I[2019.209908895; 1165.791091105]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 30° ; ; beta = 60° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 30° - 60° = 90° ; ;

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

c = 6370 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 6370 * fraction{ sin(30° ) }{ sin (90° ) } = 3185 ; ;

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 = 6370 * fraction{ sin(60° ) }{ sin (90° ) } = 5516.58 ; ;


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 = 3185 ; ; b = 5516.58 ; ; c = 6370 ; ;

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

p = a+b+c = 3185+5516.58+6370 = 15071.58 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15071.58 }{ 2 } = 7535.79 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7535.79 * (7535.79-3185)(7535.79-5516.58)(7535.79-6370) } ; ; T = sqrt{ 7.718 * 10**{ 13 } } = 8785156.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8785156.55 }{ 3185 } = 5516.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8785156.55 }{ 5516.58 } = 3185 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8785156.55 }{ 6370 } = 2758.29 ; ;

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{ 3185**2-5516.58**2-6370**2 }{ 2 * 5516.58 * 6370 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5516.58**2-3185**2-6370**2 }{ 2 * 3185 * 6370 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6370**2-3185**2-5516.58**2 }{ 2 * 5516.58 * 3185 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8785156.55 }{ 7535.79 } = 1165.79 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3185 }{ 2 * sin 30° } = 3185 ; ;




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