21 23 28 triangle

Acute scalene triangle.

Sides: a = 21   b = 23   c = 28

Area: T = 236.9811011898
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 47.38988944466° = 47°23'20″ = 0.8277092237 rad
Angle ∠ B = β = 53.71325429149° = 53°42'45″ = 0.93774607235 rad
Angle ∠ C = γ = 78.89985626385° = 78°53'55″ = 1.37770396931 rad

Height: ha = 22.57696201807
Height: hb = 20.60770445128
Height: hc = 16.92772151355

Median: ma = 23.3721991785
Median: mb = 21.91546070008
Median: mc = 17

Inradius: r = 6.5832805886
Circumradius: R = 14.26769658338

Vertex coordinates: A[28; 0] B[0; 0] C[12.42985714286; 16.92772151355]
Centroid: CG[13.47661904762; 5.64224050452]
Coordinates of the circumscribed circle: U[14; 2.74770555332]
Coordinates of the inscribed circle: I[13; 6.5832805886]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.6111105553° = 132°36'40″ = 0.8277092237 rad
∠ B' = β' = 126.2877457085° = 126°17'15″ = 0.93774607235 rad
∠ C' = γ' = 101.1011437362° = 101°6'5″ = 1.37770396931 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 = 21 ; ; b = 23 ; ; c = 28 ; ;

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

p = a+b+c = 21+23+28 = 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 * (36-21)(36-23)(36-28) } ; ; T = sqrt{ 56160 } = 236.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 236.98 }{ 21 } = 22.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 236.98 }{ 23 } = 20.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 236.98 }{ 28 } = 16.93 ; ;

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{ 21**2-23**2-28**2 }{ 2 * 23 * 28 } ) = 47° 23'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 53° 42'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-21**2-23**2 }{ 2 * 23 * 21 } ) = 78° 53'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 236.98 }{ 36 } = 6.58 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 47° 23'20" } = 14.27 ; ;




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