12 23 30 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 23   c = 30

Area: T = 125.7911245721
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 21.3843725332° = 21°23'1″ = 0.37332164134 rad
Angle ∠ B = β = 44.33440313162° = 44°20'3″ = 0.77437748172 rad
Angle ∠ C = γ = 114.2822243352° = 114°16'56″ = 1.99546014231 rad

Height: ha = 20.96552076201
Height: hb = 10.93883691931
Height: hc = 8.38660830481

Median: ma = 26.04880325553
Median: mb = 19.74220870224
Median: mc = 10.5599356041

Inradius: r = 3.87704998683
Circumradius: R = 16.4565835127

Vertex coordinates: A[30; 0] B[0; 0] C[8.58333333333; 8.38660830481]
Centroid: CG[12.86111111111; 2.7955361016]
Coordinates of the circumscribed circle: U[15; -6.7677164083]
Coordinates of the inscribed circle: I[9.5; 3.87704998683]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.6166274668° = 158°36'59″ = 0.37332164134 rad
∠ B' = β' = 135.6665968684° = 135°39'57″ = 0.77437748172 rad
∠ C' = γ' = 65.71877566482° = 65°43'4″ = 1.99546014231 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 = 23 ; ; c = 30 ; ;

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

p = a+b+c = 12+23+30 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-12)(32.5-23)(32.5-30) } ; ; T = sqrt{ 15823.44 } = 125.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 125.79 }{ 12 } = 20.97 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 125.79 }{ 23 } = 10.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 125.79 }{ 30 } = 8.39 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 12**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 21° 23'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-12**2-30**2 }{ 2 * 12 * 30 } ) = 44° 20'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-12**2-23**2 }{ 2 * 23 * 12 } ) = 114° 16'56" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 125.79 }{ 32.5 } = 3.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 21° 23'1" } = 16.46 ; ;




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