10 27 28 triangle

Acute scalene triangle.

Sides: a = 10   b = 27   c = 28

Area: T = 134.533043336
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 20.84986512302° = 20°50'55″ = 0.36438776086 rad
Angle ∠ B = β = 73.9321540617° = 73°55'54″ = 1.29903488048 rad
Angle ∠ C = γ = 85.22198081528° = 85°13'11″ = 1.48773662402 rad

Height: ha = 26.9066086672
Height: hb = 9.96552172859
Height: hc = 9.60993166686

Median: ma = 27.04662566726
Median: mb = 16.11767614613
Median: mc = 14.78217454991

Inradius: r = 4.13993979495
Circumradius: R = 14.04988657681

Vertex coordinates: A[28; 0] B[0; 0] C[2.76878571429; 9.60993166686]
Centroid: CG[10.2565952381; 3.20331055562]
Coordinates of the circumscribed circle: U[14; 1.1710738814]
Coordinates of the inscribed circle: I[5.5; 4.13993979495]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.151134877° = 159°9'5″ = 0.36438776086 rad
∠ B' = β' = 106.0688459383° = 106°4'6″ = 1.29903488048 rad
∠ C' = γ' = 94.78801918472° = 94°46'49″ = 1.48773662402 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 = 27 ; ; c = 28 ; ;

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

p = a+b+c = 10+27+28 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-10)(32.5-27)(32.5-28) } ; ; T = sqrt{ 18098.44 } = 134.53 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 134.53 }{ 10 } = 26.91 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 134.53 }{ 27 } = 9.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 134.53 }{ 28 } = 9.61 ; ;

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**2-27**2-28**2 }{ 2 * 27 * 28 } ) = 20° 50'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-10**2-28**2 }{ 2 * 10 * 28 } ) = 73° 55'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-10**2-27**2 }{ 2 * 27 * 10 } ) = 85° 13'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 134.53 }{ 32.5 } = 4.14 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 20° 50'55" } = 14.05 ; ;




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