3 10 12 triangle

Obtuse scalene triangle.

Sides: a = 3   b = 10   c = 12

Area: T = 12.1833492931
Perimeter: p = 25
Semiperimeter: s = 12.5

Angle ∠ A = α = 11.71658523949° = 11°42'57″ = 0.2044480199 rad
Angle ∠ B = β = 42.59988128925° = 42°35'56″ = 0.74334895424 rad
Angle ∠ C = γ = 125.6855334713° = 125°41'7″ = 2.19436229122 rad

Height: ha = 8.12223286207
Height: hb = 2.43766985862
Height: hc = 2.03105821552

Median: ma = 10.94330343141
Median: mb = 7.17663500472
Median: mc = 4.30111626335

Inradius: r = 0.97546794345
Circumradius: R = 7.3877044135

Vertex coordinates: A[12; 0] B[0; 0] C[2.20883333333; 2.03105821552]
Centroid: CG[4.73661111111; 0.67768607184]
Coordinates of the circumscribed circle: U[6; -4.30991090788]
Coordinates of the inscribed circle: I[2.5; 0.97546794345]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168.2844147605° = 168°17'3″ = 0.2044480199 rad
∠ B' = β' = 137.4011187108° = 137°24'4″ = 0.74334895424 rad
∠ C' = γ' = 54.31546652873° = 54°18'53″ = 2.19436229122 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 = 3 ; ; b = 10 ; ; c = 12 ; ;

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

p = a+b+c = 3+10+12 = 25 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 25 }{ 2 } = 12.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.5 * (12.5-3)(12.5-10)(12.5-12) } ; ; T = sqrt{ 148.44 } = 12.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 12.18 }{ 3 } = 8.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 12.18 }{ 10 } = 2.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 12.18 }{ 12 } = 2.03 ; ;

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{ 3**2-10**2-12**2 }{ 2 * 10 * 12 } ) = 11° 42'57" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10**2-3**2-12**2 }{ 2 * 3 * 12 } ) = 42° 35'56" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12**2-3**2-10**2 }{ 2 * 10 * 3 } ) = 125° 41'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 12.18 }{ 12.5 } = 0.97 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 11° 42'57" } = 7.39 ; ;




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