10 21 28 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 21   c = 28

Area: T = 85.64113305595
Perimeter: p = 59
Semiperimeter: s = 29.5

Angle ∠ A = α = 16.93656240404° = 16°56'8″ = 0.29655824004 rad
Angle ∠ B = β = 37.71442488191° = 37°42'51″ = 0.65882378168 rad
Angle ∠ C = γ = 125.3550127141° = 125°21' = 2.18877724364 rad

Height: ha = 17.12882661119
Height: hb = 8.15663171961
Height: hc = 6.11772378971

Median: ma = 24.23883992871
Median: mb = 18.21440056001
Median: mc = 8.63113382508

Inradius: r = 2.90330959512
Circumradius: R = 17.16546095454

Vertex coordinates: A[28; 0] B[0; 0] C[7.91107142857; 6.11772378971]
Centroid: CG[11.97702380952; 2.0399079299]
Coordinates of the circumscribed circle: U[14; -9.93109526655]
Coordinates of the inscribed circle: I[8.5; 2.90330959512]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.064437596° = 163°3'52″ = 0.29655824004 rad
∠ B' = β' = 142.2865751181° = 142°17'9″ = 0.65882378168 rad
∠ C' = γ' = 54.65498728595° = 54°39' = 2.18877724364 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 = 21 ; ; c = 28 ; ;

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

p = a+b+c = 10+21+28 = 59 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 59 }{ 2 } = 29.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29.5 * (29.5-10)(29.5-21)(29.5-28) } ; ; T = sqrt{ 7334.44 } = 85.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 85.64 }{ 10 } = 17.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 85.64 }{ 21 } = 8.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 85.64 }{ 28 } = 6.12 ; ;

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**2-21**2-28**2 }{ 2 * 21 * 28 } ) = 16° 56'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-10**2-28**2 }{ 2 * 10 * 28 } ) = 37° 42'51" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 28**2-10**2-21**2 }{ 2 * 21 * 10 } ) = 125° 21' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 85.64 }{ 29.5 } = 2.9 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 16° 56'8" } = 17.16 ; ;




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