13 18 27 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 18   c = 27

Area: T = 101.0354647522
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 24.56884742814° = 24°34'6″ = 0.42988007684 rad
Angle ∠ B = β = 35.14883878921° = 35°8'54″ = 0.61334550955 rad
Angle ∠ C = γ = 120.2833137827° = 120°16'59″ = 2.09993367897 rad

Height: ha = 15.54437919265
Height: hb = 11.22660719469
Height: hc = 7.48440479646

Median: ma = 22.00656810847
Median: mb = 19.18333260933
Median: mc = 8.01656097709

Inradius: r = 3.48439533628
Circumradius: R = 15.63332509563

Vertex coordinates: A[27; 0] B[0; 0] C[10.63296296296; 7.48440479646]
Centroid: CG[12.54332098765; 2.49546826549]
Coordinates of the circumscribed circle: U[13.5; -7.88334342429]
Coordinates of the inscribed circle: I[11; 3.48439533628]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.4321525719° = 155°25'54″ = 0.42988007684 rad
∠ B' = β' = 144.8521612108° = 144°51'6″ = 0.61334550955 rad
∠ C' = γ' = 59.71768621735° = 59°43'1″ = 2.09993367897 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 = 13 ; ; b = 18 ; ; c = 27 ; ;

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

p = a+b+c = 13+18+27 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-13)(29-18)(29-27) } ; ; T = sqrt{ 10208 } = 101.03 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 101.03 }{ 13 } = 15.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 101.03 }{ 18 } = 11.23 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 101.03 }{ 27 } = 7.48 ; ;

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{ 13**2-18**2-27**2 }{ 2 * 18 * 27 } ) = 24° 34'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-13**2-27**2 }{ 2 * 13 * 27 } ) = 35° 8'54" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-13**2-18**2 }{ 2 * 18 * 13 } ) = 120° 16'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 101.03 }{ 29 } = 3.48 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 24° 34'6" } = 15.63 ; ;




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