9 15 20 triangle

Obtuse scalene triangle.

Sides: a = 9   b = 15   c = 20

Area: T = 63.27771680782
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 24.95113010927° = 24°57'5″ = 0.43554823567 rad
Angle ∠ B = β = 44.67546095644° = 44°40'29″ = 0.78797190289 rad
Angle ∠ C = γ = 110.3744089343° = 110°22'27″ = 1.92663912679 rad

Height: ha = 14.06215929063
Height: hb = 8.43769557438
Height: hc = 6.32877168078

Median: ma = 17.09553209973
Median: mb = 13.57438719605
Median: mc = 7.28801098893

Inradius: r = 2.87662349126
Circumradius: R = 10.66773547584

Vertex coordinates: A[20; 0] B[0; 0] C[6.4; 6.32877168078]
Centroid: CG[8.8; 2.10992389359]
Coordinates of the circumscribed circle: U[10; -3.71438198048]
Coordinates of the inscribed circle: I[7; 2.87662349126]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.0498698907° = 155°2'55″ = 0.43554823567 rad
∠ B' = β' = 135.3255390436° = 135°19'31″ = 0.78797190289 rad
∠ C' = γ' = 69.62659106571° = 69°37'33″ = 1.92663912679 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 = 15 ; ; c = 20 ; ;

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

p = a+b+c = 9+15+20 = 44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44 }{ 2 } = 22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22 * (22-9)(22-15)(22-20) } ; ; T = sqrt{ 4004 } = 63.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 63.28 }{ 9 } = 14.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 63.28 }{ 15 } = 8.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 63.28 }{ 20 } = 6.33 ; ;

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**2-15**2-20**2 }{ 2 * 15 * 20 } ) = 24° 57'5" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-9**2-20**2 }{ 2 * 9 * 20 } ) = 44° 40'29" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-9**2-15**2 }{ 2 * 15 * 9 } ) = 110° 22'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 63.28 }{ 22 } = 2.88 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 9 }{ 2 * sin 24° 57'5" } = 10.67 ; ;




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