12 18 23 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 18   c = 23

Area: T = 106.9187900746
Perimeter: p = 53
Semiperimeter: s = 26.5

Angle ∠ A = α = 31.09985459112° = 31°5'55″ = 0.54327720187 rad
Angle ∠ B = β = 50.78439487166° = 50°47'2″ = 0.88663471123 rad
Angle ∠ C = γ = 98.11875053723° = 98°7'3″ = 1.71224735226 rad

Height: ha = 17.82196501244
Height: hb = 11.88797667496
Height: hc = 9.29772087605

Median: ma = 19.76110728454
Median: mb = 15.98443673631
Median: mc = 10.08771205009

Inradius: r = 4.0354637764
Circumradius: R = 11.61663896909

Vertex coordinates: A[23; 0] B[0; 0] C[7.58769565217; 9.29772087605]
Centroid: CG[10.19656521739; 3.09990695868]
Coordinates of the circumscribed circle: U[11.5; -1.6440277248]
Coordinates of the inscribed circle: I[8.5; 4.0354637764]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.9011454089° = 148°54'5″ = 0.54327720187 rad
∠ B' = β' = 129.2166051283° = 129°12'58″ = 0.88663471123 rad
∠ C' = γ' = 81.88224946277° = 81°52'57″ = 1.71224735226 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 = 12 ; ; b = 18 ; ; c = 23 ; ;

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

p = a+b+c = 12+18+23 = 53 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 53 }{ 2 } = 26.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.5 * (26.5-12)(26.5-18)(26.5-23) } ; ; T = sqrt{ 11431.44 } = 106.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.92 }{ 12 } = 17.82 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.92 }{ 18 } = 11.88 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.92 }{ 23 } = 9.3 ; ;

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{ 12**2-18**2-23**2 }{ 2 * 18 * 23 } ) = 31° 5'55" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-12**2-23**2 }{ 2 * 12 * 23 } ) = 50° 47'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-12**2-18**2 }{ 2 * 18 * 12 } ) = 98° 7'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.92 }{ 26.5 } = 4.03 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 31° 5'55" } = 11.62 ; ;




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