10 13 20 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 13   c = 20

Area: T = 56.14765715783
Perimeter: p = 43
Semiperimeter: s = 21.5

Angle ∠ A = α = 25.58879901207° = 25°35'17″ = 0.44765946766 rad
Angle ∠ B = β = 34.15772224785° = 34°9'26″ = 0.59661559956 rad
Angle ∠ C = γ = 120.2554787401° = 120°15'17″ = 2.09988419814 rad

Height: ha = 11.22993143157
Height: hb = 8.6387934089
Height: hc = 5.61546571578

Median: ma = 16.10990036936
Median: mb = 14.41435353054
Median: mc = 5.87436700622

Inradius: r = 2.61114684455
Circumradius: R = 11.57768422137

Vertex coordinates: A[20; 0] B[0; 0] C[8.275; 5.61546571578]
Centroid: CG[9.425; 1.87215523859]
Coordinates of the circumscribed circle: U[10; -5.8332947423]
Coordinates of the inscribed circle: I[8.5; 2.61114684455]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.4122009879° = 154°24'43″ = 0.44765946766 rad
∠ B' = β' = 145.8432777522° = 145°50'34″ = 0.59661559956 rad
∠ C' = γ' = 59.74552125992° = 59°44'43″ = 2.09988419814 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 = 20 ; ;

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

p = a+b+c = 10+13+20 = 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 * (21.5-10)(21.5-13)(21.5-20) } ; ; T = sqrt{ 3152.44 } = 56.15 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 56.15 }{ 10 } = 11.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 56.15 }{ 13 } = 8.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 56.15 }{ 20 } = 5.61 ; ;

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{ 10**2-13**2-20**2 }{ 2 * 13 * 20 } ) = 25° 35'17" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13**2-10**2-20**2 }{ 2 * 10 * 20 } ) = 34° 9'26" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-10**2-13**2 }{ 2 * 13 * 10 } ) = 120° 15'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 56.15 }{ 21.5 } = 2.61 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 25° 35'17" } = 11.58 ; ;




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