10 15 21 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 15   c = 21

Area: T = 69.16664658632
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 26.05497983877° = 26°2'59″ = 0.45546547513 rad
Angle ∠ B = β = 41.20329536038° = 41°12'11″ = 0.71991272019 rad
Angle ∠ C = γ = 112.7477248008° = 112°44'50″ = 1.96878107003 rad

Height: ha = 13.83332931726
Height: hb = 9.22221954484
Height: hc = 6.58772824632

Median: ma = 17.55499287748
Median: mb = 14.63772811683
Median: mc = 7.22884161474

Inradius: r = 3.00772376462
Circumradius: R = 11.38655752231

Vertex coordinates: A[21; 0] B[0; 0] C[7.52438095238; 6.58772824632]
Centroid: CG[9.50879365079; 2.19657608211]
Coordinates of the circumscribed circle: U[10.5; -4.40224224196]
Coordinates of the inscribed circle: I[8; 3.00772376462]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153.9550201612° = 153°57'1″ = 0.45546547513 rad
∠ B' = β' = 138.7977046396° = 138°47'49″ = 0.71991272019 rad
∠ C' = γ' = 67.25327519916° = 67°15'10″ = 1.96878107003 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 = 15 ; ; c = 21 ; ;

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

p = a+b+c = 10+15+21 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-10)(23-15)(23-21) } ; ; T = sqrt{ 4784 } = 69.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 69.17 }{ 10 } = 13.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 69.17 }{ 15 } = 9.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 69.17 }{ 21 } = 6.59 ; ;

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**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 26° 2'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-10**2-21**2 }{ 2 * 10 * 21 } ) = 41° 12'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-10**2-15**2 }{ 2 * 15 * 10 } ) = 112° 44'50" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 69.17 }{ 23 } = 3.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 26° 2'59" } = 11.39 ; ;




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