21 23 30 triangle

Acute scalene triangle.

Sides: a = 21   b = 23   c = 30

Area: T = 240.8655107477
Perimeter: p = 74
Semiperimeter: s = 37

Angle ∠ A = α = 44.2879544862° = 44°16'46″ = 0.77328238491 rad
Angle ∠ B = β = 49.87659654065° = 49°52'33″ = 0.8770499814 rad
Angle ∠ C = γ = 85.84444897315° = 85°50'40″ = 1.49882689905 rad

Height: ha = 22.94395340454
Height: hb = 20.94547919545
Height: hc = 16.05876738318

Median: ma = 24.58114971066
Median: mb = 23.22002155162
Median: mc = 16.12545154966

Inradius: r = 6.51098677697
Circumradius: R = 15.04395382625

Vertex coordinates: A[30; 0] B[0; 0] C[13.53333333333; 16.05876738318]
Centroid: CG[14.51111111111; 5.35325579439]
Coordinates of the circumscribed circle: U[15; 1.09898216132]
Coordinates of the inscribed circle: I[14; 6.51098677697]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.7220455138° = 135°43'14″ = 0.77328238491 rad
∠ B' = β' = 130.1244034594° = 130°7'27″ = 0.8770499814 rad
∠ C' = γ' = 94.15655102685° = 94°9'20″ = 1.49882689905 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 = 21 ; ; b = 23 ; ; c = 30 ; ;

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

p = a+b+c = 21+23+30 = 74 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74 }{ 2 } = 37 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37 * (37-21)(37-23)(37-30) } ; ; T = sqrt{ 58016 } = 240.87 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 240.87 }{ 21 } = 22.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 240.87 }{ 23 } = 20.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 240.87 }{ 30 } = 16.06 ; ;

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{ 21**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 44° 16'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-21**2-30**2 }{ 2 * 21 * 30 } ) = 49° 52'33" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-21**2-23**2 }{ 2 * 23 * 21 } ) = 85° 50'40" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 240.87 }{ 37 } = 6.51 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 21 }{ 2 * sin 44° 16'46" } = 15.04 ; ;




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