4 20 21 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 20   c = 21

Area: T = 39.50987015732
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 10.84440625637° = 10°50'39″ = 0.1899264596 rad
Angle ∠ B = β = 70.16766380911° = 70°10' = 1.22546388597 rad
Angle ∠ C = γ = 98.98992993452° = 98°59'21″ = 1.72876891978 rad

Height: ha = 19.75443507866
Height: hb = 3.95108701573
Height: hc = 3.76327334832

Median: ma = 20.4088331632
Median: mb = 11.33657840488
Median: mc = 9.88768599666

Inradius: r = 1.75659422921
Circumradius: R = 10.63105695524

Vertex coordinates: A[21; 0] B[0; 0] C[1.35771428571; 3.76327334832]
Centroid: CG[7.45223809524; 1.25442444944]
Coordinates of the circumscribed circle: U[10.5; -1.66110264926]
Coordinates of the inscribed circle: I[2.5; 1.75659422921]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 169.1565937436° = 169°9'21″ = 0.1899264596 rad
∠ B' = β' = 109.8333361909° = 109°50' = 1.22546388597 rad
∠ C' = γ' = 81.01107006548° = 81°39″ = 1.72876891978 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 = 4 ; ; b = 20 ; ; c = 21 ; ;

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

p = a+b+c = 4+20+21 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-4)(22.5-20)(22.5-21) } ; ; T = sqrt{ 1560.94 } = 39.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 39.51 }{ 4 } = 19.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 39.51 }{ 20 } = 3.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 39.51 }{ 21 } = 3.76 ; ;

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{ 4**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 10° 50'39" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-4**2-21**2 }{ 2 * 4 * 21 } ) = 70° 10' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-4**2-20**2 }{ 2 * 20 * 4 } ) = 98° 59'21" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 39.51 }{ 22.5 } = 1.76 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 10° 50'39" } = 10.63 ; ;




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