6 20 21 triangle

Obtuse scalene triangle.

Sides: a = 6   b = 20   c = 21

Area: T = 59.98769777535
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 16.59878421359° = 16°35'52″ = 0.2989686994 rad
Angle ∠ B = β = 72.2088409427° = 72°12'30″ = 1.26602744921 rad
Angle ∠ C = γ = 91.19437484371° = 91°11'37″ = 1.59216311675 rad

Height: ha = 19.99656592512
Height: hb = 5.99986977754
Height: hc = 5.71330455003

Median: ma = 20.28554627751
Median: mb = 11.76986022959
Median: mc = 10.3880269746

Inradius: r = 2.55326373512
Circumradius: R = 10.50222793878

Vertex coordinates: A[21; 0] B[0; 0] C[1.83333333333; 5.71330455003]
Centroid: CG[7.61111111111; 1.90443485001]
Coordinates of the circumscribed circle: U[10.5; -0.21987974872]
Coordinates of the inscribed circle: I[3.5; 2.55326373512]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.4022157864° = 163°24'8″ = 0.2989686994 rad
∠ B' = β' = 107.7921590573° = 107°47'30″ = 1.26602744921 rad
∠ C' = γ' = 88.80662515629° = 88°48'23″ = 1.59216311675 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 = 6 ; ; b = 20 ; ; c = 21 ; ;

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

p = a+b+c = 6+20+21 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-6)(23.5-20)(23.5-21) } ; ; T = sqrt{ 3598.44 } = 59.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 59.99 }{ 6 } = 20 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 59.99 }{ 20 } = 6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 59.99 }{ 21 } = 5.71 ; ;

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{ 6**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 16° 35'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-6**2-21**2 }{ 2 * 6 * 21 } ) = 72° 12'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-6**2-20**2 }{ 2 * 20 * 6 } ) = 91° 11'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 59.99 }{ 23.5 } = 2.55 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 16° 35'52" } = 10.5 ; ;




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