7 21 23 triangle

Obtuse scalene triangle.

Sides: a = 7   b = 21   c = 23

Area: T = 72.85504461208
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 17.55772234277° = 17°33'26″ = 0.30664313563 rad
Angle ∠ B = β = 64.82198469684° = 64°49'11″ = 1.13113197502 rad
Angle ∠ C = γ = 97.62329296039° = 97°37'23″ = 1.7043841547 rad

Height: ha = 20.81444131774
Height: hb = 6.93881377258
Height: hc = 6.33548214018

Median: ma = 21.74328149052
Median: mb = 13.37697419571
Median: mc = 10.61883802908

Inradius: r = 2.857688024
Circumradius: R = 11.60325370469

Vertex coordinates: A[23; 0] B[0; 0] C[2.97882608696; 6.33548214018]
Centroid: CG[8.65994202899; 2.11216071339]
Coordinates of the circumscribed circle: U[11.5; -1.53991120572]
Coordinates of the inscribed circle: I[4.5; 2.857688024]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.4432776572° = 162°26'34″ = 0.30664313563 rad
∠ B' = β' = 115.1880153032° = 115°10'49″ = 1.13113197502 rad
∠ C' = γ' = 82.37770703961° = 82°22'37″ = 1.7043841547 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 = 23 ; ;

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

p = a+b+c = 7+21+23 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-7)(25.5-21)(25.5-23) } ; ; T = sqrt{ 5307.19 } = 72.85 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 72.85 }{ 7 } = 20.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 72.85 }{ 21 } = 6.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 72.85 }{ 23 } = 6.33 ; ;

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**2-23**2 }{ 2 * 21 * 23 } ) = 17° 33'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-7**2-23**2 }{ 2 * 7 * 23 } ) = 64° 49'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-7**2-21**2 }{ 2 * 21 * 7 } ) = 97° 37'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 72.85 }{ 25.5 } = 2.86 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 17° 33'26" } = 11.6 ; ;




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