Triangle calculator ASA

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

Sides: a = 10.42548001625   b = 9.89327413194   c = 8.8

Area: T = 40.21546853917
Perimeter: p = 29.11875414819
Semiperimeter: s = 14.55987707409

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 61.25° = 61°15' = 1.06990141668 rad
Angle ∠ C = γ = 51.25° = 51°15' = 0.89444812416 rad

Height: ha = 7.71551954502
Height: hb = 8.13301398861
Height: hc = 9.14397012254

Median: ma = 7.7777149272
Median: mb = 8.28220073962
Median: mc = 9.16603162948

Inradius: r = 2.76222308303
Circumradius: R = 5.64218611928

Vertex coordinates: A[8.8; 0] B[0; 0] C[5.01442117964; 9.14397012254]
Centroid: CG[4.60547372655; 3.04765670751]
Coordinates of the circumscribed circle: U[4.4; 3.53113733474]
Coordinates of the inscribed circle: I[4.66660294216; 2.76222308303]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.17880972451 rad
∠ B' = β' = 118.75° = 118°45' = 1.06990141668 rad
∠ C' = γ' = 128.75° = 128°45' = 0.89444812416 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 67° 30' ; ; beta = 61° 15' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 67° 30' - 61° 15' = 51° 15' ; ;

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

c = 8.8 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 8.8 * fraction{ sin 67° 30' }{ sin 51° 15' } = 10.42 ; ;

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 = 8.8 * fraction{ sin 61° 15' }{ sin 51° 15' } = 9.89 ; ;
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 = 10.42 ; ; b = 9.89 ; ; c = 8.8 ; ;

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

p = a+b+c = 10.42+9.89+8.8 = 29.12 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.12 }{ 2 } = 14.56 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.56 * (14.56-10.42)(14.56-9.89)(14.56-8.8) } ; ; T = sqrt{ 1617.22 } = 40.21 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40.21 }{ 10.42 } = 7.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40.21 }{ 9.89 } = 8.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40.21 }{ 8.8 } = 9.14 ; ;

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{ 9.89**2+8.8**2-10.42**2 }{ 2 * 9.89 * 8.8 } ) = 67° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10.42**2+8.8**2-9.89**2 }{ 2 * 10.42 * 8.8 } ) = 61° 15' ; ;
 gamma = 180° - alpha - beta = 180° - 67° 30' - 61° 15' = 51° 15' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40.21 }{ 14.56 } = 2.76 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10.42 }{ 2 * sin 67° 30' } = 5.64 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.89**2+2 * 8.8**2 - 10.42**2 } }{ 2 } = 7.777 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.8**2+2 * 10.42**2 - 9.89**2 } }{ 2 } = 8.282 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.89**2+2 * 10.42**2 - 8.8**2 } }{ 2 } = 9.16 ; ;
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