Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 59.50109088688   b = 42.14768641326   c = 42

Area: T = 885.0844146784
Perimeter: p = 143.6487773001
Semiperimeter: s = 71.82438865007

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 45.1° = 45°6' = 0.78771434926 rad
Angle ∠ C = γ = 44.9° = 44°54' = 0.78436528341 rad

Height: ha = 29.75502731844
Height: hb = 42
Height: hc = 42.14768641326

Median: ma = 29.75504544344
Median: mb = 46.9990313247
Median: mc = 47.08988326061

Inradius: r = 12.32329776319
Circumradius: R = 29.75504544344

Vertex coordinates: A[42; 0] B[0; 0] C[42; 42.14768641326]
Centroid: CG[28; 14.04989547109]
Coordinates of the circumscribed circle: U[21; 21.07334320663]
Coordinates of the inscribed circle: I[29.67770223681; 12.32329776319]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 134.9° = 134°54' = 0.78771434926 rad
∠ C' = γ' = 135.1° = 135°6' = 0.78436528341 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 45° 6' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 45° 6' = 44° 54' ; ;

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

c = 42 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 42 * fraction{ sin 90° }{ sin 44° 54' } = 59.5 ; ;

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 = 42 * fraction{ sin 45° 6' }{ sin 44° 54' } = 42.15 ; ;
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 = 59.5 ; ; b = 42.15 ; ; c = 42 ; ;

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

p = a+b+c = 59.5+42.15+42 = 143.65 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 143.65 }{ 2 } = 71.82 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 71.82 * (71.82-59.5)(71.82-42.15)(71.82-42) } ; ; T = sqrt{ 783373.95 } = 885.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 885.08 }{ 59.5 } = 29.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 885.08 }{ 42.15 } = 42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 885.08 }{ 42 } = 42.15 ; ;

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{ 42.15**2+42**2-59.5**2 }{ 2 * 42.15 * 42 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 59.5**2+42**2-42.15**2 }{ 2 * 59.5 * 42 } ) = 45° 6' ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 45° 6' = 44° 54' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 885.08 }{ 71.82 } = 12.32 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 59.5 }{ 2 * sin 90° } = 29.75 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.15**2+2 * 42**2 - 59.5**2 } }{ 2 } = 29.75 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 42**2+2 * 59.5**2 - 42.15**2 } }{ 2 } = 46.99 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 42.15**2+2 * 59.5**2 - 42**2 } }{ 2 } = 47.089 ; ;
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