3 19 20 triangle

Obtuse scalene triangle.

Sides: a = 3   b = 19   c = 20

Area: T = 27.49554541697
Perimeter: p = 42
Semiperimeter: s = 21

Angle ∠ A = α = 8.32106551878° = 8°19'14″ = 0.14552228289 rad
Angle ∠ B = β = 66.42218215218° = 66°25'19″ = 1.15992794807 rad
Angle ∠ C = γ = 105.258752329° = 105°15'27″ = 1.83770903439 rad

Height: ha = 18.33303027798
Height: hb = 2.89442583337
Height: hc = 2.7549545417

Median: ma = 19.44986503388
Median: mb = 10.68987791632
Median: mc = 9.22195444573

Inradius: r = 1.30993073414
Circumradius: R = 10.36553497862

Vertex coordinates: A[20; 0] B[0; 0] C[1.2; 2.7549545417]
Centroid: CG[7.06766666667; 0.9176515139]
Coordinates of the circumscribed circle: U[10; -2.72877236279]
Coordinates of the inscribed circle: I[2; 1.30993073414]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 171.6799344812° = 171°40'46″ = 0.14552228289 rad
∠ B' = β' = 113.5788178478° = 113°34'41″ = 1.15992794807 rad
∠ C' = γ' = 74.74224767095° = 74°44'33″ = 1.83770903439 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 = 3 ; ; b = 19 ; ; c = 20 ; ;

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

p = a+b+c = 3+19+20 = 42 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 42 }{ 2 } = 21 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21 * (21-3)(21-19)(21-20) } ; ; T = sqrt{ 756 } = 27.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 27.5 }{ 3 } = 18.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 27.5 }{ 19 } = 2.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 27.5 }{ 20 } = 2.75 ; ;

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{ 3**2-19**2-20**2 }{ 2 * 19 * 20 } ) = 8° 19'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-3**2-20**2 }{ 2 * 3 * 20 } ) = 66° 25'19" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-3**2-19**2 }{ 2 * 19 * 3 } ) = 105° 15'27" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 27.5 }{ 21 } = 1.31 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 8° 19'14" } = 10.37 ; ;




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