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 β.

Right scalene triangle.

Sides: a = 25.07884284637   b = 10   c = 22.99884254724

Area: T = 114.9922127362
Perimeter: p = 58.0776853936
Semiperimeter: s = 29.0388426968

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 23.5° = 23°30' = 0.41101523742 rad
Angle ∠ C = γ = 66.5° = 66°30' = 1.16106439526 rad

Height: ha = 9.17106007439
Height: hb = 22.99884254724
Height: hc = 10

Median: ma = 12.53992142318
Median: mb = 23.53656660031
Median: mc = 15.23991565893

Inradius: r = 3.96599985043
Circumradius: R = 12.53992142318

Vertex coordinates: A[22.99884254724; 0] B[0; 0] C[22.99884254724; 10]
Centroid: CG[15.33222836482; 3.33333333333]
Coordinates of the circumscribed circle: U[11.49992127362; 5]
Coordinates of the inscribed circle: I[19.0388426968; 3.96599985043]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 156.5° = 156°30' = 0.41101523742 rad
∠ C' = γ' = 113.5° = 113°30' = 1.16106439526 rad

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How did we calculate this triangle?

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

b = 10 ; ; alpha = 90° ; ; beta = 23.5° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 23.5 ° = 66.5 ° ; ;

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 = 10 * fraction{ sin 90° }{ sin 23° 30' } = 25.08 ; ;

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{ 10**2+25.08**2 - 2 * 10 * 25.08 * cos 66° 30' } ; ; c = 23 ; ;
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 = 25.08 ; ; b = 10 ; ; c = 23 ; ;

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

p = a+b+c = 25.08+10+23 = 58.08 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58.08 }{ 2 } = 29.04 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.04 * (29.04-25.08)(29.04-10)(29.04-23) } ; ; T = sqrt{ 13223.19 } = 114.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 114.99 }{ 25.08 } = 9.17 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 114.99 }{ 10 } = 23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 114.99 }{ 23 } = 10 ; ;

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{ 10**2+23**2-25.08**2 }{ 2 * 10 * 23 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 25.08**2+23**2-10**2 }{ 2 * 25.08 * 23 } ) = 23° 30' ; ; gamma = 180° - alpha - beta = 180° - 90° - 23° 30' = 66° 30' ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 114.99 }{ 29.04 } = 3.96 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 25.08 }{ 2 * sin 90° } = 12.54 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10**2+2 * 23**2 - 25.08**2 } }{ 2 } = 12.539 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 23**2+2 * 25.08**2 - 10**2 } }{ 2 } = 23.536 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10**2+2 * 25.08**2 - 23**2 } }{ 2 } = 15.239 ; ;
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