Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 110.8511251684   b = 55.42656258422   c = 96

Area: T = 2660.433004043
Perimeter: p = 262.2776877527
Semiperimeter: s = 131.1388438763

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 48
Height: hb = 96
Height: hc = 55.42656258422

Median: ma = 55.42656258422
Median: mb = 99.92199679744
Median: mc = 73.32112111193

Inradius: r = 20.28771870789
Circumradius: R = 55.42656258422

Vertex coordinates: A[96; 0] B[0; 0] C[96; 55.42656258422]
Centroid: CG[64; 18.47552086141]
Coordinates of the circumscribed circle: U[48; 27.71328129211]
Coordinates of the inscribed circle: I[75.71328129211; 20.28771870789]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 120° = 1.04771975512 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 30° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 30° = 60° ; ;

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

c = 96 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 96 * fraction{ sin 90° }{ sin 60° } = 110.85 ; ;

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 = 96 * fraction{ sin 30° }{ sin 60° } = 55.43 ; ;
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 = 110.85 ; ; b = 55.43 ; ; c = 96 ; ;

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

p = a+b+c = 110.85+55.43+96 = 262.28 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 262.28 }{ 2 } = 131.14 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 131.14 * (131.14-110.85)(131.14-55.43)(131.14-96) } ; ; T = sqrt{ 7077888 } = 2660.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2660.43 }{ 110.85 } = 48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2660.43 }{ 55.43 } = 96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2660.43 }{ 96 } = 55.43 ; ;

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{ 55.43**2+96**2-110.85**2 }{ 2 * 55.43 * 96 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 110.85**2+96**2-55.43**2 }{ 2 * 110.85 * 96 } ) = 30° ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 30° = 60° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2660.43 }{ 131.14 } = 20.29 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 110.85 }{ 2 * sin 90° } = 55.43 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 55.43**2+2 * 96**2 - 110.85**2 } }{ 2 } = 55.426 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 96**2+2 * 110.85**2 - 55.43**2 } }{ 2 } = 99.92 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 55.43**2+2 * 110.85**2 - 96**2 } }{ 2 } = 73.321 ; ;
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