7 21 27 triangle

Obtuse scalene triangle.

Sides: a = 7   b = 21   c = 27

Area: T = 42.80440593869
Perimeter: p = 55
Semiperimeter: s = 27.5

Angle ∠ A = α = 8.68439745947° = 8°41'2″ = 0.15215639488 rad
Angle ∠ B = β = 26.93333044068° = 26°56' = 0.47700748403 rad
Angle ∠ C = γ = 144.3832720999° = 144°22'58″ = 2.52199538644 rad

Height: ha = 12.23297312534
Height: hb = 4.07765770845
Height: hc = 3.17106710657

Median: ma = 23.93221958875
Median: mb = 16.69658078571
Median: mc = 7.92114897589

Inradius: r = 1.55765112504
Circumradius: R = 23.18112125815

Vertex coordinates: A[27; 0] B[0; 0] C[6.24107407407; 3.17106710657]
Centroid: CG[11.08802469136; 1.05768903552]
Coordinates of the circumscribed circle: U[13.5; -18.84545911802]
Coordinates of the inscribed circle: I[6.5; 1.55765112504]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 171.3166025405° = 171°18'58″ = 0.15215639488 rad
∠ B' = β' = 153.0676695593° = 153°4' = 0.47700748403 rad
∠ C' = γ' = 35.61772790015° = 35°37'2″ = 2.52199538644 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 = 7 ; ; b = 21 ; ; c = 27 ; ;

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

p = a+b+c = 7+21+27 = 55 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 55 }{ 2 } = 27.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27.5 * (27.5-7)(27.5-21)(27.5-27) } ; ; T = sqrt{ 1832.19 } = 42.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.8 }{ 7 } = 12.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.8 }{ 21 } = 4.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.8 }{ 27 } = 3.17 ; ;

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{ 7**2-21**2-27**2 }{ 2 * 21 * 27 } ) = 8° 41'2" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-7**2-27**2 }{ 2 * 7 * 27 } ) = 26° 56' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-7**2-21**2 }{ 2 * 21 * 7 } ) = 144° 22'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.8 }{ 27.5 } = 1.56 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 8° 41'2" } = 23.18 ; ;




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