4 18 21 triangle

Obtuse scalene triangle.

Sides: a = 4   b = 18   c = 21

Area: T = 25.66600370226
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 7.80329967597° = 7°48'11″ = 0.1366187985 rad
Angle ∠ B = β = 37.65884620062° = 37°39'30″ = 0.65772641532 rad
Angle ∠ C = γ = 134.5398541234° = 134°32'19″ = 2.34881405154 rad

Height: ha = 12.83300185113
Height: hb = 2.85111152247
Height: hc = 2.44438130498

Median: ma = 19.45550764583
Median: mb = 12.14549578015
Median: mc = 7.73298124169

Inradius: r = 1.19334900941
Circumradius: R = 14.73110777326

Vertex coordinates: A[21; 0] B[0; 0] C[3.16766666667; 2.44438130498]
Centroid: CG[8.05655555556; 0.81546043499]
Coordinates of the circumscribed circle: U[10.5; -10.3322214243]
Coordinates of the inscribed circle: I[3.5; 1.19334900941]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 172.197700324° = 172°11'49″ = 0.1366187985 rad
∠ B' = β' = 142.3421537994° = 142°20'30″ = 0.65772641532 rad
∠ C' = γ' = 45.46114587659° = 45°27'41″ = 2.34881405154 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 = 4 ; ; b = 18 ; ; c = 21 ; ;

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

p = a+b+c = 4+18+21 = 43 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 43 }{ 2 } = 21.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 21.5 * (21.5-4)(21.5-18)(21.5-21) } ; ; T = sqrt{ 658.44 } = 25.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 25.66 }{ 4 } = 12.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 25.66 }{ 18 } = 2.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 25.66 }{ 21 } = 2.44 ; ;

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{ 4**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 7° 48'11" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-4**2-21**2 }{ 2 * 4 * 21 } ) = 37° 39'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-4**2-18**2 }{ 2 * 18 * 4 } ) = 134° 32'19" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 25.66 }{ 21.5 } = 1.19 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4 }{ 2 * sin 7° 48'11" } = 14.73 ; ;




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