8 21 27 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 21   c = 27

Area: T = 62.610990337
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 12.75987404169° = 12°45'31″ = 0.22326820287 rad
Angle ∠ B = β = 35.43109446873° = 35°25'51″ = 0.61883866419 rad
Angle ∠ C = γ = 131.8110314896° = 131°48'37″ = 2.3010523983 rad

Height: ha = 15.65224758425
Height: hb = 5.963284794
Height: hc = 4.638777062

Median: ma = 23.85437208838
Median: mb = 16.91989243157
Median: mc = 8.38215273071

Inradius: r = 2.23660679775
Circumradius: R = 18.11221506177

Vertex coordinates: A[27; 0] B[0; 0] C[6.51985185185; 4.638777062]
Centroid: CG[11.17328395062; 1.546592354]
Coordinates of the circumscribed circle: U[13.5; -12.07547670785]
Coordinates of the inscribed circle: I[7; 2.23660679775]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 167.2411259583° = 167°14'29″ = 0.22326820287 rad
∠ B' = β' = 144.5699055313° = 144°34'9″ = 0.61883866419 rad
∠ C' = γ' = 48.19896851042° = 48°11'23″ = 2.3010523983 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 = 8 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 8+21+27 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-8)(28-21)(28-27) } ; ; T = sqrt{ 3920 } = 62.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 62.61 }{ 8 } = 15.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 62.61 }{ 21 } = 5.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 62.61 }{ 27 } = 4.64 ; ;

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{ 8**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 12° 45'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-8**2-27**2 }{ 2 * 8 * 27 } ) = 35° 25'51" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-8**2-21**2 }{ 2 * 21 * 8 } ) = 131° 48'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 62.61 }{ 28 } = 2.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 12° 45'31" } = 18.11 ; ;




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