10 12 21 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 12   c = 21

Area: T = 34.2770067114
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 15.78223998562° = 15°46'57″ = 0.27554548414 rad
Angle ∠ B = β = 19.04992993211° = 19°2'57″ = 0.33224729934 rad
Angle ∠ C = γ = 145.1688300823° = 145°10'6″ = 2.53436648189 rad

Height: ha = 6.85440134228
Height: hb = 5.71216778523
Height: hc = 3.26438159156

Median: ma = 16.35554272338
Median: mb = 15.31333928311
Median: mc = 3.42878273002

Inradius: r = 1.594395661
Circumradius: R = 18.38333897349

Vertex coordinates: A[21; 0] B[0; 0] C[9.45223809524; 3.26438159156]
Centroid: CG[10.15107936508; 1.08879386385]
Coordinates of the circumscribed circle: U[10.5; -15.09896990741]
Coordinates of the inscribed circle: I[9.5; 1.594395661]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.2187600144° = 164°13'3″ = 0.27554548414 rad
∠ B' = β' = 160.9510700679° = 160°57'3″ = 0.33224729934 rad
∠ C' = γ' = 34.83216991773° = 34°49'54″ = 2.53436648189 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 = 12 ; ; c = 21 ; ;

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

p = a+b+c = 10+12+21 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-10)(21.5-12)(21.5-21) } ; ; T = sqrt{ 1174.44 } = 34.27 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34.27 }{ 10 } = 6.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34.27 }{ 12 } = 5.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34.27 }{ 21 } = 3.26 ; ;

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**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 15° 46'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-10**2-21**2 }{ 2 * 10 * 21 } ) = 19° 2'57" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-10**2-12**2 }{ 2 * 12 * 10 } ) = 145° 10'6" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34.27 }{ 21.5 } = 1.59 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 15° 46'57" } = 18.38 ; ;




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