23 29 30 triangle

Acute scalene triangle.

Sides: a = 23   b = 29   c = 30

Area: T = 312.1155363287
Perimeter: p = 82
Semiperimeter: s = 41

Angle ∠ A = α = 45.84990010993° = 45°50'56″ = 0.88002160279 rad
Angle ∠ B = β = 64.78105163713° = 64°46'50″ = 1.13106333018 rad
Angle ∠ C = γ = 69.37704825295° = 69°22'14″ = 1.21107433238 rad

Height: ha = 27.14404663728
Height: hb = 21.52551974681
Height: hc = 20.80876908858

Median: ma = 27.17107563384
Median: mb = 22.45655115729
Median: mc = 21.44876105895

Inradius: r = 7.61325698363
Circumradius: R = 16.02877275278

Vertex coordinates: A[30; 0] B[0; 0] C[9.8; 20.80876908858]
Centroid: CG[13.26766666667; 6.93658969619]
Coordinates of the circumscribed circle: U[15; 5.64769504783]
Coordinates of the inscribed circle: I[12; 7.61325698363]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.1510998901° = 134°9'4″ = 0.88002160279 rad
∠ B' = β' = 115.2199483629° = 115°13'10″ = 1.13106333018 rad
∠ C' = γ' = 110.6329517471° = 110°37'46″ = 1.21107433238 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 = 29 ; ; c = 30 ; ;

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

p = a+b+c = 23+29+30 = 82 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 82 }{ 2 } = 41 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 41 * (41-23)(41-29)(41-30) } ; ; T = sqrt{ 97416 } = 312.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 312.12 }{ 23 } = 27.14 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 312.12 }{ 29 } = 21.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 312.12 }{ 30 } = 20.81 ; ;

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**2-29**2-30**2 }{ 2 * 29 * 30 } ) = 45° 50'56" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 29**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 64° 46'50" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-23**2-29**2 }{ 2 * 29 * 23 } ) = 69° 22'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 312.12 }{ 41 } = 7.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 45° 50'56" } = 16.03 ; ;




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