10 13 22 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 13   c = 22

Area: T = 36.5550478793
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 14.80990077489° = 14°48'32″ = 0.25884659442 rad
Angle ∠ B = β = 19.40770433824° = 19°24'25″ = 0.33987168051 rad
Angle ∠ C = γ = 145.7843948869° = 145°47'2″ = 2.54444099043 rad

Height: ha = 7.31100957586
Height: hb = 5.62331505835
Height: hc = 3.32327707994

Median: ma = 17.36437553542
Median: mb = 15.80334806293
Median: mc = 3.67442346142

Inradius: r = 1.62444657241
Circumradius: R = 19.56219872464

Vertex coordinates: A[22; 0] B[0; 0] C[9.43218181818; 3.32327707994]
Centroid: CG[10.47772727273; 1.10875902665]
Coordinates of the circumscribed circle: U[11; -16.17662586845]
Coordinates of the inscribed circle: I[9.5; 1.62444657241]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.1910992251° = 165°11'28″ = 0.25884659442 rad
∠ B' = β' = 160.5932956618° = 160°35'35″ = 0.33987168051 rad
∠ C' = γ' = 34.21660511313° = 34°12'58″ = 2.54444099043 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 = 13 ; ; c = 22 ; ;

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

p = a+b+c = 10+13+22 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-10)(22.5-13)(22.5-22) } ; ; T = sqrt{ 1335.94 } = 36.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 36.55 }{ 10 } = 7.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 36.55 }{ 13 } = 5.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 36.55 }{ 22 } = 3.32 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 13**2+22**2-10**2 }{ 2 * 13 * 22 } ) = 14° 48'32" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 10**2+22**2-13**2 }{ 2 * 10 * 22 } ) = 19° 24'25" ; ; gamma = 180° - alpha - beta = 180° - 14° 48'32" - 19° 24'25" = 145° 47'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 36.55 }{ 22.5 } = 1.62 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 10 }{ 2 * sin 14° 48'32" } = 19.56 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 22**2 - 10**2 } }{ 2 } = 17.364 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 22**2+2 * 10**2 - 13**2 } }{ 2 } = 15.803 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 13**2+2 * 10**2 - 22**2 } }{ 2 } = 3.674 ; ;
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