9 27 28 triangle

Acute scalene triangle.

Sides: a = 9   b = 27   c = 28

Area: T = 121.3266007105
Perimeter: p = 64
Semiperimeter: s = 32

Angle ∠ A = α = 18.721149202° = 18°43'17″ = 0.32767516766 rad
Angle ∠ B = β = 74.34551782964° = 74°20'43″ = 1.29875681443 rad
Angle ∠ C = γ = 86.93333296836° = 86°56' = 1.51772728327 rad

Height: ha = 26.96113349122
Height: hb = 8.98771116374
Height: hc = 8.66661433646

Median: ma = 27.13439271024
Median: mb = 15.81992920196
Median: mc = 14.45768322948

Inradius: r = 3.7911437722
Circumradius: R = 14.02200773156

Vertex coordinates: A[28; 0] B[0; 0] C[2.42985714286; 8.66661433646]
Centroid: CG[10.14328571429; 2.88987144549]
Coordinates of the circumscribed circle: U[14; 0.75500452885]
Coordinates of the inscribed circle: I[5; 3.7911437722]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 161.279850798° = 161°16'43″ = 0.32767516766 rad
∠ B' = β' = 105.6554821704° = 105°39'17″ = 1.29875681443 rad
∠ C' = γ' = 93.06766703164° = 93°4' = 1.51772728327 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 = 9 ; ; b = 27 ; ; c = 28 ; ;

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

p = a+b+c = 9+27+28 = 64 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 64 }{ 2 } = 32 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32 * (32-9)(32-27)(32-28) } ; ; T = sqrt{ 14720 } = 121.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 121.33 }{ 9 } = 26.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 121.33 }{ 27 } = 8.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 121.33 }{ 28 } = 8.67 ; ;

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{ 9**2-27**2-28**2 }{ 2 * 27 * 28 } ) = 18° 43'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-9**2-28**2 }{ 2 * 9 * 28 } ) = 74° 20'43" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-9**2-27**2 }{ 2 * 27 * 9 } ) = 86° 56' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 121.33 }{ 32 } = 3.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 18° 43'17" } = 14.02 ; ;




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