20 23 28 triangle

Acute scalene triangle.

Sides: a = 20   b = 23   c = 28

Area: T = 227.1255378371
Perimeter: p = 71
Semiperimeter: s = 35.5

Angle ∠ A = α = 44.85884992844° = 44°51'31″ = 0.783292851 rad
Angle ∠ B = β = 54.21096242387° = 54°12'35″ = 0.94661364292 rad
Angle ∠ C = γ = 80.93218764769° = 80°55'55″ = 1.41325277143 rad

Height: ha = 22.71325378371
Height: hb = 19.75500329018
Height: hc = 16.22332413122

Median: ma = 23.59902522242
Median: mb = 21.44217816424
Median: mc = 16.38659696082

Inradius: r = 6.39878979823
Circumradius: R = 14.17771915719

Vertex coordinates: A[28; 0] B[0; 0] C[11.69664285714; 16.22332413122]
Centroid: CG[13.23221428571; 5.40877471041]
Coordinates of the circumscribed circle: U[14; 2.23444486717]
Coordinates of the inscribed circle: I[12.5; 6.39878979823]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.1421500716° = 135°8'29″ = 0.783292851 rad
∠ B' = β' = 125.7990375761° = 125°47'25″ = 0.94661364292 rad
∠ C' = γ' = 99.06881235231° = 99°4'5″ = 1.41325277143 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 = 28 ; ;

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

p = a+b+c = 20+23+28 = 71 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 71 }{ 2 } = 35.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35.5 * (35.5-20)(35.5-23)(35.5-28) } ; ; T = sqrt{ 51585.94 } = 227.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 227.13 }{ 20 } = 22.71 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 227.13 }{ 23 } = 19.75 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 227.13 }{ 28 } = 16.22 ; ;

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**2-28**2 }{ 2 * 23 * 28 } ) = 44° 51'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-20**2-28**2 }{ 2 * 20 * 28 } ) = 54° 12'35" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-20**2-23**2 }{ 2 * 23 * 20 } ) = 80° 55'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 227.13 }{ 35.5 } = 6.4 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 20 }{ 2 * sin 44° 51'31" } = 14.18 ; ;




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