12 27 28 triangle

Acute scalene triangle.

Sides: a = 12   b = 27   c = 28

Area: T = 160.4654754697
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 25.12196477175° = 25°7'11″ = 0.43884205596 rad
Angle ∠ B = β = 72.7754633383° = 72°46'29″ = 1.27701569645 rad
Angle ∠ C = γ = 82.10657188995° = 82°6'21″ = 1.43330151295 rad

Height: ha = 26.74441257828
Height: hb = 11.88662781257
Height: hc = 11.46217681926

Median: ma = 26.84221310629
Median: mb = 16.78554103316
Median: mc = 15.50880624193

Inradius: r = 4.79899926775
Circumradius: R = 14.1343944892

Vertex coordinates: A[28; 0] B[0; 0] C[3.55435714286; 11.46217681926]
Centroid: CG[10.51878571429; 3.82105893975]
Coordinates of the circumscribed circle: U[14; 1.94112362583]
Coordinates of the inscribed circle: I[6.5; 4.79899926775]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.8880352282° = 154°52'49″ = 0.43884205596 rad
∠ B' = β' = 107.2255366617° = 107°13'31″ = 1.27701569645 rad
∠ C' = γ' = 97.89442811005° = 97°53'39″ = 1.43330151295 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 = 12 ; ; b = 27 ; ; c = 28 ; ;

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

p = a+b+c = 12+27+28 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-12)(33.5-27)(33.5-28) } ; ; T = sqrt{ 25748.94 } = 160.46 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 160.46 }{ 12 } = 26.74 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 160.46 }{ 27 } = 11.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 160.46 }{ 28 } = 11.46 ; ;

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{ 12**2-27**2-28**2 }{ 2 * 27 * 28 } ) = 25° 7'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-12**2-28**2 }{ 2 * 12 * 28 } ) = 72° 46'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-12**2-27**2 }{ 2 * 27 * 12 } ) = 82° 6'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 160.46 }{ 33.5 } = 4.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 25° 7'11" } = 14.13 ; ;




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