10 21 27 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 21   c = 27

Area: T = 93.89435567544
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 19.34112748244° = 19°20'29″ = 0.33875689272 rad
Angle ∠ B = β = 44.06876983467° = 44°4'4″ = 0.76991264299 rad
Angle ∠ C = γ = 116.5911026829° = 116°35'28″ = 2.03548972964 rad

Height: ha = 18.77987113509
Height: hb = 8.94222435004
Height: hc = 6.95550782781

Median: ma = 23.66443191324
Median: mb = 17.44327635425
Median: mc = 9.3944147114

Inradius: r = 3.23877088536
Circumradius: R = 15.09768825657

Vertex coordinates: A[27; 0] B[0; 0] C[7.18551851852; 6.95550782781]
Centroid: CG[11.39550617284; 2.3188359426]
Coordinates of the circumscribed circle: U[13.5; -6.75876521961]
Coordinates of the inscribed circle: I[8; 3.23877088536]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.6598725176° = 160°39'31″ = 0.33875689272 rad
∠ B' = β' = 135.9322301653° = 135°55'56″ = 0.76991264299 rad
∠ C' = γ' = 63.40989731711° = 63°24'32″ = 2.03548972964 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 = 10 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 10+21+27 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-10)(29-21)(29-27) } ; ; T = sqrt{ 8816 } = 93.89 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 93.89 }{ 10 } = 18.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 93.89 }{ 21 } = 8.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 93.89 }{ 27 } = 6.96 ; ;

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{ 10**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 19° 20'29" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-10**2-27**2 }{ 2 * 10 * 27 } ) = 44° 4'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-10**2-21**2 }{ 2 * 21 * 10 } ) = 116° 35'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 93.89 }{ 29 } = 3.24 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 19° 20'29" } = 15.1 ; ;




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