20 23 30 triangle

Acute scalene triangle.

Sides: a = 20   b = 23   c = 30

Area: T = 229.8865705297
Perimeter: p = 73
Semiperimeter: s = 36.5

Angle ∠ A = α = 41.78548536525° = 41°47'5″ = 0.72992832737 rad
Angle ∠ B = β = 50.02215081864° = 50°1'17″ = 0.87330400147 rad
Angle ∠ C = γ = 88.19436381611° = 88°11'37″ = 1.53992693652 rad

Height: ha = 22.98985705297
Height: hb = 19.99900613302
Height: hc = 15.32657136865

Median: ma = 24.78991105125
Median: mb = 22.7544120506
Median: mc = 15.47657875405

Inradius: r = 6.29882385013
Circumradius: R = 15.00774577083

Vertex coordinates: A[30; 0] B[0; 0] C[12.85; 15.32657136865]
Centroid: CG[14.28333333333; 5.10985712288]
Coordinates of the circumscribed circle: U[15; 0.47330611669]
Coordinates of the inscribed circle: I[13.5; 6.29882385013]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.2155146347° = 138°12'55″ = 0.72992832737 rad
∠ B' = β' = 129.9788491814° = 129°58'43″ = 0.87330400147 rad
∠ C' = γ' = 91.80663618389° = 91°48'23″ = 1.53992693652 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 = 20 ; ; b = 23 ; ; c = 30 ; ;

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

p = a+b+c = 20+23+30 = 73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73 }{ 2 } = 36.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.5 * (36.5-20)(36.5-23)(36.5-30) } ; ; T = sqrt{ 52847.44 } = 229.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 229.89 }{ 20 } = 22.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 229.89 }{ 23 } = 19.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 229.89 }{ 30 } = 15.33 ; ;

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{ 20**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 41° 47'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-20**2-30**2 }{ 2 * 20 * 30 } ) = 50° 1'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-20**2-23**2 }{ 2 * 23 * 20 } ) = 88° 11'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 229.89 }{ 36.5 } = 6.3 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 41° 47'5" } = 15.01 ; ;




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