Triangle calculator ASA

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

Sides: a = 914.7565849435   b = 1127.997724193   c = 520

Area: T = 234223.2549686
Perimeter: p = 2562.753309136
Semiperimeter: s = 1281.377654568

Angle ∠ A = α = 53° = 0.92550245036 rad
Angle ∠ B = β = 100° = 1.7455329252 rad
Angle ∠ C = γ = 27° = 0.4711238898 rad

Height: ha = 512.1100031566
Height: hb = 415.2990465225
Height: hc = 900.8598652637

Median: ma = 749.7966187558
Median: mb = 485.2787948799
Median: mc = 993.468767483

Inradius: r = 182.7990336279
Circumradius: R = 572.6999208792

Vertex coordinates: A[520; 0] B[0; 0] C[-158.8465686264; 900.8598652637]
Centroid: CG[120.3854771245; 300.2866217546]
Coordinates of the circumscribed circle: U[260; 510.2798731431]
Coordinates of the inscribed circle: I[153.3799303755; 182.7990336279]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127° = 0.92550245036 rad
∠ B' = β' = 80° = 1.7455329252 rad
∠ C' = γ' = 153° = 0.4711238898 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 53° ; ; beta = 100° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 53° - 100° = 27° ; ;

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

c = 520 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 520 * fraction{ sin(53° ) }{ sin (27° ) } = 914.76 ; ;

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 = 520 * fraction{ sin(100° ) }{ sin (27° ) } = 1128 ; ;


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 = 914.76 ; ; b = 1128 ; ; c = 520 ; ;

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

p = a+b+c = 914.76+1128+520 = 2562.75 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2562.75 }{ 2 } = 1281.38 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1281.38 * (1281.38-914.76)(1281.38-1128)(1281.38-520) } ; ; T = sqrt{ 54860530693.2 } = 234223.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 234223.25 }{ 914.76 } = 512.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 234223.25 }{ 1128 } = 415.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 234223.25 }{ 520 } = 900.86 ; ;

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{ 914.76**2-1128**2-520**2 }{ 2 * 1128 * 520 } ) = 53° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1128**2-914.76**2-520**2 }{ 2 * 914.76 * 520 } ) = 100° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 520**2-914.76**2-1128**2 }{ 2 * 1128 * 914.76 } ) = 27° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 234223.25 }{ 1281.38 } = 182.79 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 914.76 }{ 2 * sin 53° } = 572.7 ; ;




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