Triangle calculator ASA

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

Sides: a = 1523.143991783   b = 264.4990471063   c = 1500

Area: T = 198367.8533297
Perimeter: p = 3287.633038889
Semiperimeter: s = 1643.815519445

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 10° = 0.17545329252 rad
Angle ∠ C = γ = 80° = 1.39662634016 rad

Height: ha = 260.47222665
Height: hb = 1500
Height: hc = 264.4990471063

Median: ma = 761.5769958914
Median: mb = 1505.81883165
Median: mc = 795.2710525848

Inradius: r = 120.6755276617
Circumradius: R = 761.5769958914

Vertex coordinates: A[1500; 0] B[0; 0] C[1500; 264.4990471063]
Centroid: CG[1000; 88.16334903542]
Coordinates of the circumscribed circle: U[750; 132.2455235531]
Coordinates of the inscribed circle: I[1379.325472338; 120.6755276617]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 170° = 0.17545329252 rad
∠ C' = γ' = 100° = 1.39662634016 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 10° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 10° = 80° ; ;

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

c = 1500 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 1500 * fraction{ sin(90° ) }{ sin (80° ) } = 1523.14 ; ;

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 = 1500 * fraction{ sin(10° ) }{ sin (80° ) } = 264.49 ; ;


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 = 1523.14 ; ; b = 264.49 ; ; c = 1500 ; ;

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

p = a+b+c = 1523.14+264.49+1500 = 3287.63 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 3287.63 }{ 2 } = 1643.82 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1643.82 * (1643.82-1523.14)(1643.82-264.49)(1643.82-1500) } ; ; T = sqrt{ 39349805221.7 } = 198367.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 198367.85 }{ 1523.14 } = 260.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 198367.85 }{ 264.49 } = 1500 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 198367.85 }{ 1500 } = 264.49 ; ;

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{ 1523.14**2-264.49**2-1500**2 }{ 2 * 264.49 * 1500 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 264.49**2-1523.14**2-1500**2 }{ 2 * 1523.14 * 1500 } ) = 10° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1500**2-1523.14**2-264.49**2 }{ 2 * 264.49 * 1523.14 } ) = 80° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 198367.85 }{ 1643.82 } = 120.68 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1523.14 }{ 2 * sin 90° } = 761.57 ; ;




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