Triangle calculator ASA

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

Sides: a = 220.6144059481   b = 280.6110005542   c = 230

Area: T = 24720.37695744
Perimeter: p = 731.2244065023
Semiperimeter: s = 365.6122032512

Angle ∠ A = α = 50° = 0.8732664626 rad
Angle ∠ B = β = 77° = 1.3443903524 rad
Angle ∠ C = γ = 53° = 0.92550245036 rad

Height: ha = 224.1055114901
Height: hb = 176.1990221917
Height: hc = 214.9659735429

Median: ma = 231.6321920933
Median: mb = 176.3511319297
Median: mc = 224.6880371251

Inradius: r = 67.61436652411
Circumradius: R = 143.9965600688

Vertex coordinates: A[230; 0] B[0; 0] C[49.62773652834; 214.9659735429]
Centroid: CG[93.20991217611; 71.65332451431]
Coordinates of the circumscribed circle: U[115; 86.65987157618]
Coordinates of the inscribed circle: I[85.00220269695; 67.61436652411]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130° = 0.8732664626 rad
∠ B' = β' = 103° = 1.3443903524 rad
∠ C' = γ' = 127° = 0.92550245036 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 50° ; ; beta = 77° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 50° - 77° = 53° ; ;

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

c = 230 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 230 * fraction{ sin(50° ) }{ sin (53° ) } = 220.61 ; ;

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 = 230 * fraction{ sin(77° ) }{ sin (53° ) } = 280.61 ; ;


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 = 220.61 ; ; b = 280.61 ; ; c = 230 ; ;

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

p = a+b+c = 220.61+280.61+230 = 731.22 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 731.22 }{ 2 } = 365.61 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 365.61 * (365.61-220.61)(365.61-280.61)(365.61-230) } ; ; T = sqrt{ 611096671.89 } = 24720.37 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 24720.37 }{ 220.61 } = 224.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 24720.37 }{ 280.61 } = 176.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 24720.37 }{ 230 } = 214.96 ; ;

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{ 220.61**2-280.61**2-230**2 }{ 2 * 280.61 * 230 } ) = 50° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 280.61**2-220.61**2-230**2 }{ 2 * 220.61 * 230 } ) = 77° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 230**2-220.61**2-280.61**2 }{ 2 * 280.61 * 220.61 } ) = 53° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 24720.37 }{ 365.61 } = 67.61 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 220.61 }{ 2 * sin 50° } = 144 ; ;




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