16 21 30 triangle

Obtuse scalene triangle.

Sides: a = 16   b = 21   c = 30

Area: T = 160.1511295655
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 30.55884317859° = 30°33'30″ = 0.53333452489 rad
Angle ∠ B = β = 41.85987922056° = 41°51'32″ = 0.73105737449 rad
Angle ∠ C = γ = 107.5832776008° = 107°34'58″ = 1.87876736598 rad

Height: ha = 20.01989119569
Height: hb = 15.25325043481
Height: hc = 10.67767530437

Median: ma = 24.6277220712
Median: mb = 21.62875287539
Median: mc = 11.11330553854

Inradius: r = 4.78106356912
Circumradius: R = 15.73551209036

Vertex coordinates: A[30; 0] B[0; 0] C[11.91766666667; 10.67767530437]
Centroid: CG[13.97222222222; 3.55989176812]
Coordinates of the circumscribed circle: U[15; -4.7533317773]
Coordinates of the inscribed circle: I[12.5; 4.78106356912]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 149.4421568214° = 149°26'30″ = 0.53333452489 rad
∠ B' = β' = 138.1411207794° = 138°8'28″ = 0.73105737449 rad
∠ C' = γ' = 72.41772239916° = 72°25'2″ = 1.87876736598 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 = 16 ; ; b = 21 ; ; c = 30 ; ;

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

p = a+b+c = 16+21+30 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-16)(33.5-21)(33.5-30) } ; ; T = sqrt{ 25648.44 } = 160.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 160.15 }{ 16 } = 20.02 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 160.15 }{ 21 } = 15.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 160.15 }{ 30 } = 10.68 ; ;

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{ 16**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 30° 33'30" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-16**2-30**2 }{ 2 * 16 * 30 } ) = 41° 51'32" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-16**2-21**2 }{ 2 * 21 * 16 } ) = 107° 34'58" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 160.15 }{ 33.5 } = 4.78 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16 }{ 2 * sin 30° 33'30" } = 15.74 ; ;




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