Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 19.62766877351   b = 62.17988297691   c = 59

Area: T = 578.9877288186
Perimeter: p = 140.8065517504
Semiperimeter: s = 70.40327587521

Angle ∠ A = α = 18.4° = 18°24' = 0.32111405824 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 71.6° = 71°36' = 1.25496557444 rad

Height: ha = 59
Height: hb = 18.62332931799
Height: hc = 19.62766877351

Median: ma = 59.81105485501
Median: mb = 31.08994148845
Median: mc = 35.43224268355

Inradius: r = 8.2243928983
Circumradius: R = 31.08994148845

Vertex coordinates: A[59; 0] B[0; 0] C[-0; 19.62766877351]
Centroid: CG[19.66766666667; 6.5422229245]
Coordinates of the circumscribed circle: U[29.5; 9.81333438676]
Coordinates of the inscribed circle: I[8.2243928983; 8.2243928983]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.6° = 161°36' = 0.32111405824 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 108.4° = 108°24' = 1.25496557444 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 18° 24' ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 18° 24' - 90° = 71° 36' ; ;

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

c = 59 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 59 * fraction{ sin 18° 24' }{ sin 71° 36' } = 19.63 ; ;

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 = 59 * fraction{ sin 90° }{ sin 71° 36' } = 62.18 ; ;
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 = 19.63 ; ; b = 62.18 ; ; c = 59 ; ;

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

p = a+b+c = 19.63+62.18+59 = 140.81 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 140.81 }{ 2 } = 70.4 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 70.4 * (70.4-19.63)(70.4-62.18)(70.4-59) } ; ; T = sqrt{ 335226.28 } = 578.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 578.99 }{ 19.63 } = 59 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 578.99 }{ 62.18 } = 18.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 578.99 }{ 59 } = 19.63 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 62.18**2+59**2-19.63**2 }{ 2 * 62.18 * 59 } ) = 18° 24' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 19.63**2+59**2-62.18**2 }{ 2 * 19.63 * 59 } ) = 90° ; ;
 gamma = 180° - alpha - beta = 180° - 18° 24' - 90° = 71° 36' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 578.99 }{ 70.4 } = 8.22 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 19.63 }{ 2 * sin 18° 24' } = 31.09 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62.18**2+2 * 59**2 - 19.63**2 } }{ 2 } = 59.811 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 59**2+2 * 19.63**2 - 62.18**2 } }{ 2 } = 31.089 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62.18**2+2 * 19.63**2 - 59**2 } }{ 2 } = 35.432 ; ;
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