10 21 30 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 21   c = 30

Area: T = 54.49771329521
Perimeter: p = 61
Semiperimeter: s = 30.5

Angle ∠ A = α = 9.96326853346° = 9°57'46″ = 0.17438816614 rad
Angle ∠ B = β = 21.30438736689° = 21°18'14″ = 0.3721822739 rad
Angle ∠ C = γ = 148.7333440996° = 148°44' = 2.59658882532 rad

Height: ha = 10.89994265904
Height: hb = 5.19902031383
Height: hc = 3.63331421968

Median: ma = 25.40766920318
Median: mb = 19.74220870224
Median: mc = 6.74553687816

Inradius: r = 1.78767912443
Circumradius: R = 28.9010602925

Vertex coordinates: A[30; 0] B[0; 0] C[9.31766666667; 3.63331421968]
Centroid: CG[13.10655555556; 1.21110473989]
Coordinates of the circumscribed circle: U[15; -24.70331344049]
Coordinates of the inscribed circle: I[9.5; 1.78767912443]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170.0377314665° = 170°2'14″ = 0.17438816614 rad
∠ B' = β' = 158.6966126331° = 158°41'46″ = 0.3721822739 rad
∠ C' = γ' = 31.26765590035° = 31°16' = 2.59658882532 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 = 30 ; ;

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

p = a+b+c = 10+21+30 = 61 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 61 }{ 2 } = 30.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30.5 * (30.5-10)(30.5-21)(30.5-30) } ; ; T = sqrt{ 2969.94 } = 54.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 54.5 }{ 10 } = 10.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 54.5 }{ 21 } = 5.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 54.5 }{ 30 } = 3.63 ; ;

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**2-30**2 }{ 2 * 21 * 30 } ) = 9° 57'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-10**2-30**2 }{ 2 * 10 * 30 } ) = 21° 18'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-10**2-21**2 }{ 2 * 21 * 10 } ) = 148° 44' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 54.5 }{ 30.5 } = 1.79 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 9° 57'46" } = 28.9 ; ;




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