20 25 27 triangle

Acute scalene triangle.

Sides: a = 20   b = 25   c = 27

Area: T = 238.7976984906
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 45.03656507165° = 45°2'8″ = 0.78660203858 rad
Angle ∠ B = β = 62.18218607153° = 62°10'55″ = 1.08552782045 rad
Angle ∠ C = γ = 72.78224885682° = 72°46'57″ = 1.27702940633 rad

Height: ha = 23.88796984906
Height: hb = 19.10437587924
Height: hc = 17.68986655486

Median: ma = 24.02108242989
Median: mb = 20.20551973512
Median: mc = 18.17327818454

Inradius: r = 6.63332495807
Circumradius: R = 14.13333442771

Vertex coordinates: A[27; 0] B[0; 0] C[9.33333333333; 17.68986655486]
Centroid: CG[12.11111111111; 5.89662218495]
Coordinates of the circumscribed circle: U[13.5; 4.1833469906]
Coordinates of the inscribed circle: I[11; 6.63332495807]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134.9644349284° = 134°57'52″ = 0.78660203858 rad
∠ B' = β' = 117.8188139285° = 117°49'5″ = 1.08552782045 rad
∠ C' = γ' = 107.2187511432° = 107°13'3″ = 1.27702940633 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 = 25 ; ; c = 27 ; ;

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

p = a+b+c = 20+25+27 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-20)(36-25)(36-27) } ; ; T = sqrt{ 57024 } = 238.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 238.8 }{ 20 } = 23.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 238.8 }{ 25 } = 19.1 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 238.8 }{ 27 } = 17.69 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 25**2+27**2-20**2 }{ 2 * 25 * 27 } ) = 45° 2'8" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 20**2+27**2-25**2 }{ 2 * 20 * 27 } ) = 62° 10'55" ; ; gamma = 180° - alpha - beta = 180° - 45° 2'8" - 62° 10'55" = 72° 46'57" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 238.8 }{ 36 } = 6.63 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 20 }{ 2 * sin 45° 2'8" } = 14.13 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 27**2 - 20**2 } }{ 2 } = 24.021 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 27**2+2 * 20**2 - 25**2 } }{ 2 } = 20.205 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 25**2+2 * 20**2 - 27**2 } }{ 2 } = 18.173 ; ;
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