10 13 21 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 13   c = 21

Area: T = 48.74442304278
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 20.92222396826° = 20°55'20″ = 0.36551619694 rad
Angle ∠ B = β = 27.66604498993° = 27°39'38″ = 0.48327659233 rad
Angle ∠ C = γ = 131.4177310418° = 131°25'2″ = 2.29436647609 rad

Height: ha = 9.74988460856
Height: hb = 7.49991123735
Height: hc = 4.64223076598

Median: ma = 16.73332005307
Median: mb = 15.10879449297
Median: mc = 4.92444289009

Inradius: r = 2.21656468376
Circumradius: R = 14.00216570989

Vertex coordinates: A[21; 0] B[0; 0] C[8.85771428571; 4.64223076598]
Centroid: CG[9.95223809524; 1.54774358866]
Coordinates of the circumscribed circle: U[10.5; -9.26326346962]
Coordinates of the inscribed circle: I[9; 2.21656468376]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.0787760317° = 159°4'40″ = 0.36551619694 rad
∠ B' = β' = 152.3439550101° = 152°20'22″ = 0.48327659233 rad
∠ C' = γ' = 48.58326895819° = 48°34'58″ = 2.29436647609 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 = 10 ; ; b = 13 ; ; c = 21 ; ;

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

p = a+b+c = 10+13+21 = 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 * (22-10)(22-13)(22-21) } ; ; T = sqrt{ 2376 } = 48.74 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 48.74 }{ 10 } = 9.75 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 48.74 }{ 13 } = 7.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 48.74 }{ 21 } = 4.64 ; ;

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{ 10**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 20° 55'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-10**2-21**2 }{ 2 * 10 * 21 } ) = 27° 39'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-10**2-13**2 }{ 2 * 13 * 10 } ) = 131° 25'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 48.74 }{ 22 } = 2.22 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 20° 55'20" } = 14 ; ;




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