10 20 21 triangle

Acute scalene triangle.

Sides: a = 10   b = 20   c = 21

Area: T = 98.90662055687
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 28.09880547134° = 28°5'53″ = 0.49904035682 rad
Angle ∠ B = β = 70.38440208646° = 70°23'2″ = 1.22884329049 rad
Angle ∠ C = γ = 81.5187924422° = 81°31'5″ = 1.42327561806 rad

Height: ha = 19.78112411137
Height: hb = 9.89106205569
Height: hc = 9.42196386256

Median: ma = 19.88771818014
Median: mb = 13.05875648572
Median: mc = 11.82215904175

Inradius: r = 3.87986747282
Circumradius: R = 10.61661185131

Vertex coordinates: A[21; 0] B[0; 0] C[3.35771428571; 9.42196386256]
Centroid: CG[8.1199047619; 3.14398795419]
Coordinates of the circumscribed circle: U[10.5; 1.56658774807]
Coordinates of the inscribed circle: I[5.5; 3.87986747282]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.9021945287° = 151°54'7″ = 0.49904035682 rad
∠ B' = β' = 109.6165979135° = 109°36'58″ = 1.22884329049 rad
∠ C' = γ' = 98.4822075578° = 98°28'55″ = 1.42327561806 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 = 20 ; ; c = 21 ; ;

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

p = a+b+c = 10+20+21 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-10)(25.5-20)(25.5-21) } ; ; T = sqrt{ 9782.44 } = 98.91 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 98.91 }{ 10 } = 19.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 98.91 }{ 20 } = 9.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 98.91 }{ 21 } = 9.42 ; ;

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**2-20**2-21**2 }{ 2 * 20 * 21 } ) = 28° 5'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-10**2-21**2 }{ 2 * 10 * 21 } ) = 70° 23'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-10**2-20**2 }{ 2 * 20 * 10 } ) = 81° 31'5" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 98.91 }{ 25.5 } = 3.88 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 28° 5'53" } = 10.62 ; ;




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