Triangle calculator - result

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side b, angle α and angle γ.

Acute scalene triangle.

Sides: a = 9.62655856353   b = 22   c = 22.43300561975

Area: T = 104.273286497
Perimeter: p = 54.05656418328
Semiperimeter: s = 27.02878209164

Angle ∠ A = α = 25° = 0.4366332313 rad
Angle ∠ B = β = 75° = 1.3098996939 rad
Angle ∠ C = γ = 80° = 1.39662634016 rad

Height: ha = 21.66657705663
Height: hb = 9.47993513609
Height: hc = 9.29876017583

Median: ma = 21.68884931659
Median: mb = 13.32996112696
Median: mc = 12.74994742698

Inradius: r = 3.85879826799
Circumradius: R = 11.38880379845

Vertex coordinates: A[22.43300561975; 0] B[0; 0] C[2.49112848827; 9.29876017583]
Centroid: CG[8.30771136934; 3.09992005861]
Coordinates of the circumscribed circle: U[11.21550280987; 1.97875120432]
Coordinates of the inscribed circle: I[5.02878209164; 3.85879826799]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155° = 0.4366332313 rad
∠ B' = β' = 105° = 1.3098996939 rad
∠ C' = γ' = 100° = 1.39662634016 rad

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How did we calculate this triangle?

1. Input data entered: side b, angle α and angle γ.

b = 22 ; ; alpha = 25° ; ; gamma = 80° ; ;

2. From angle α and angle γ we calculate angle β:

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 25 ° - 80 ° = 75 ° ; ;

3. From angle α, angle β and side b we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ b } = fraction{ sin alpha }{ sin beta } ; ; ; ; a = b * fraction{ sin alpha }{ sin beta } ; ; ; ; a = 22 * fraction{ sin 25° }{ sin 75° } = 9.63 ; ;

4. Calculation of the third side c of the triangle using a Law of Cosines

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 22**2+9.63**2 - 2 * 22 * 9.63 * cos 80° } ; ; c = 22.43 ; ;
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 = 9.63 ; ; b = 22 ; ; c = 22.43 ; ;

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

p = a+b+c = 9.63+22+22.43 = 54.06 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54.06 }{ 2 } = 27.03 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.03 * (27.03-9.63)(27.03-22)(27.03-22.43) } ; ; T = sqrt{ 10872.83 } = 104.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 104.27 }{ 9.63 } = 21.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 104.27 }{ 22 } = 9.48 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 104.27 }{ 22.43 } = 9.3 ; ;

9. 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{ 22**2+22.43**2-9.63**2 }{ 2 * 22 * 22.43 } ) = 25° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 9.63**2+22.43**2-22**2 }{ 2 * 9.63 * 22.43 } ) = 75° ; ;
 gamma = 180° - alpha - beta = 180° - 25° - 75° = 80° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 104.27 }{ 27.03 } = 3.86 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 9.63 }{ 2 * sin 25° } = 11.39 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 22.43**2 - 9.63**2 } }{ 2 } = 21.688 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22.43**2+2 * 9.63**2 - 22**2 } }{ 2 } = 13.3 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 9.63**2 - 22.43**2 } }{ 2 } = 12.749 ; ;
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