9 20 28 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 20   c = 28

Area: T = 48.6599768518
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 9.99554866967° = 9°59'44″ = 0.17444541532 rad
Angle ∠ B = β = 22.6887977061° = 22°41'17″ = 0.39659799003 rad
Angle ∠ C = γ = 147.3176536242° = 147°19' = 2.57111586001 rad

Height: ha = 10.87999485595
Height: hb = 4.86599768518
Height: hc = 3.4711412037

Median: ma = 23.91112944024
Median: mb = 18.23545825288
Median: mc = 6.67108320321

Inradius: r = 1.70552550357
Circumradius: R = 25.92660494118

Vertex coordinates: A[28; 0] B[0; 0] C[8.30435714286; 3.4711412037]
Centroid: CG[12.10111904762; 1.15771373457]
Coordinates of the circumscribed circle: U[14; -21.82110915883]
Coordinates of the inscribed circle: I[8.5; 1.70552550357]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.0054513303° = 170°16″ = 0.17444541532 rad
∠ B' = β' = 157.3122022939° = 157°18'43″ = 0.39659799003 rad
∠ C' = γ' = 32.68334637577° = 32°41' = 2.57111586001 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 = 20 ; ; c = 28 ; ;

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

p = a+b+c = 9+20+28 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-9)(28.5-20)(28.5-28) } ; ; T = sqrt{ 2361.94 } = 48.6 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 48.6 }{ 9 } = 10.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 48.6 }{ 20 } = 4.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 48.6 }{ 28 } = 3.47 ; ;

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**2-20**2-28**2 }{ 2 * 20 * 28 } ) = 9° 59'44" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-9**2-28**2 }{ 2 * 9 * 28 } ) = 22° 41'17" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-9**2-20**2 }{ 2 * 20 * 9 } ) = 147° 19' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 48.6 }{ 28.5 } = 1.71 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 9° 59'44" } = 25.93 ; ;




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