5 21 23 triangle

Obtuse scalene triangle.

Sides: a = 5   b = 21   c = 23

Area: T = 50.08218080744
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 11.96987455547° = 11°58'8″ = 0.20988940173 rad
Angle ∠ B = β = 60.57436513216° = 60°34'25″ = 1.05772096555 rad
Angle ∠ C = γ = 107.4587603124° = 107°27'27″ = 1.87554889808 rad

Height: ha = 20.03327232298
Height: hb = 4.77696960071
Height: hc = 4.35549398326

Median: ma = 21.8880356487
Median: mb = 12.91331715701
Median: mc = 10.03774299499

Inradius: r = 2.04441554316
Circumradius: R = 12.05552756223

Vertex coordinates: A[23; 0] B[0; 0] C[2.45765217391; 4.35549398326]
Centroid: CG[8.48655072464; 1.45216466109]
Coordinates of the circumscribed circle: U[11.5; -3.61765826867]
Coordinates of the inscribed circle: I[3.5; 2.04441554316]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168.0311254445° = 168°1'53″ = 0.20988940173 rad
∠ B' = β' = 119.4266348678° = 119°25'35″ = 1.05772096555 rad
∠ C' = γ' = 72.54223968763° = 72°32'33″ = 1.87554889808 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 = 5 ; ; b = 21 ; ; c = 23 ; ;

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

p = a+b+c = 5+21+23 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-5)(24.5-21)(24.5-23) } ; ; T = sqrt{ 2508.19 } = 50.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 50.08 }{ 5 } = 20.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 50.08 }{ 21 } = 4.77 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 50.08 }{ 23 } = 4.35 ; ;

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{ 5**2-21**2-23**2 }{ 2 * 21 * 23 } ) = 11° 58'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-5**2-23**2 }{ 2 * 5 * 23 } ) = 60° 34'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-5**2-21**2 }{ 2 * 21 * 5 } ) = 107° 27'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 50.08 }{ 24.5 } = 2.04 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5 }{ 2 * sin 11° 58'8" } = 12.06 ; ;




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