23 27 30 triangle

Acute scalene triangle.

Sides: a = 23   b = 27   c = 30

Area: T = 297.3211374946
Perimeter: p = 80
Semiperimeter: s = 40

Angle ∠ A = α = 47.23334876223° = 47°14'1″ = 0.82443798762 rad
Angle ∠ B = β = 59.51994142744° = 59°31'10″ = 1.03988097479 rad
Angle ∠ C = γ = 73.24770981033° = 73°14'50″ = 1.27884030294 rad

Height: ha = 25.8544032604
Height: hb = 22.02438055516
Height: hc = 19.82114249964

Median: ma = 26.12199157732
Median: mb = 23.07105439901
Median: mc = 20.10997512422

Inradius: r = 7.43330343737
Circumradius: R = 15.66548676902

Vertex coordinates: A[30; 0] B[0; 0] C[11.66766666667; 19.82114249964]
Centroid: CG[13.88988888889; 6.60771416655]
Coordinates of the circumscribed circle: U[15; 4.51553161297]
Coordinates of the inscribed circle: I[13; 7.43330343737]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.7676512378° = 132°45'59″ = 0.82443798762 rad
∠ B' = β' = 120.4810585726° = 120°28'50″ = 1.03988097479 rad
∠ C' = γ' = 106.7532901897° = 106°45'10″ = 1.27884030294 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 = 23 ; ; b = 27 ; ; c = 30 ; ;

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

p = a+b+c = 23+27+30 = 80 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 80 }{ 2 } = 40 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40 * (40-23)(40-27)(40-30) } ; ; T = sqrt{ 88400 } = 297.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 297.32 }{ 23 } = 25.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 297.32 }{ 27 } = 22.02 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 297.32 }{ 30 } = 19.82 ; ;

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{ 23**2-27**2-30**2 }{ 2 * 27 * 30 } ) = 47° 14'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 59° 31'10" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-23**2-27**2 }{ 2 * 27 * 23 } ) = 73° 14'50" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 297.32 }{ 40 } = 7.43 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 47° 14'1" } = 15.66 ; ;




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