12 22 25 triangle

Acute scalene triangle.

Sides: a = 12   b = 22   c = 25

Area: T = 131.9987869301
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 28.68548959816° = 28°41'6″ = 0.50106458805 rad
Angle ∠ B = β = 61.64106499718° = 61°38'26″ = 1.07658322951 rad
Angle ∠ C = γ = 89.67444540466° = 89°40'28″ = 1.5655114478 rad

Height: ha = 221.9996448835
Height: hb = 121.9998063001
Height: hc = 10.56598295441

Median: ma = 22.77105950735
Median: mb = 16.23326830807
Median: mc = 12.56598566871

Inradius: r = 4.47545040441
Circumradius: R = 12.5500201774

Vertex coordinates: A[25; 0] B[0; 0] C[5.7; 10.56598295441]
Centroid: CG[10.23333333333; 3.52199431814]
Coordinates of the circumscribed circle: U[12.5; 0.07110238737]
Coordinates of the inscribed circle: I[7.5; 4.47545040441]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.3155104018° = 151°18'54″ = 0.50106458805 rad
∠ B' = β' = 118.3599350028° = 118°21'34″ = 1.07658322951 rad
∠ C' = γ' = 90.32655459534° = 90°19'32″ = 1.5655114478 rad

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How did we calculate this triangle?

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 = 12 ; ; b = 22 ; ; c = 25 ; ;

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

p = a+b+c = 12+22+25 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-12)(29.5-22)(29.5-25) } ; ; T = sqrt{ 17423.44 } = 132 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 132 }{ 12 } = 22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 132 }{ 22 } = 12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 132 }{ 25 } = 10.56 ; ;

5. 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+25**2-12**2 }{ 2 * 22 * 25 } ) = 28° 41'6" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+25**2-22**2 }{ 2 * 12 * 25 } ) = 61° 38'26" ; ; gamma = 180° - alpha - beta = 180° - 28° 41'6" - 61° 38'26" = 89° 40'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 132 }{ 29.5 } = 4.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 28° 41'6" } = 12.5 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 25**2 - 12**2 } }{ 2 } = 22.771 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 12**2 - 22**2 } }{ 2 } = 16.233 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 12**2 - 25**2 } }{ 2 } = 12.56 ; ;
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