3 12 13 triangle

Obtuse scalene triangle.

Sides: a = 3   b = 12   c = 13

Area: T = 17.55499287748
Perimeter: p = 28
Semiperimeter: s = 14

Angle ∠ A = α = 13.00328244668° = 13°10″ = 0.2276942099 rad
Angle ∠ B = β = 64.15875871263° = 64°9'27″ = 1.12197611355 rad
Angle ∠ C = γ = 102.8439588407° = 102°50'23″ = 1.79548894191 rad

Height: ha = 11.76999525165
Height: hb = 2.92549881291
Height: hc = 2.76999890423

Median: ma = 12.42197423484
Median: mb = 7.28801098893
Median: mc = 5.85223499554

Inradius: r = 1.25435663411
Circumradius: R = 6.66766937229

Vertex coordinates: A[13; 0] B[0; 0] C[1.30876923077; 2.76999890423]
Centroid: CG[4.76992307692; 0.98999963474]
Coordinates of the circumscribed circle: U[6.5; -1.4811487494]
Coordinates of the inscribed circle: I[2; 1.25435663411]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 166.9977175533° = 166°59'50″ = 0.2276942099 rad
∠ B' = β' = 115.8422412874° = 115°50'33″ = 1.12197611355 rad
∠ C' = γ' = 77.16604115931° = 77°9'37″ = 1.79548894191 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 = 12 ; ; c = 13 ; ;

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

p = a+b+c = 3+12+13 = 28 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 28 }{ 2 } = 14 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14 * (14-3)(14-12)(14-13) } ; ; T = sqrt{ 308 } = 17.55 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.55 }{ 3 } = 11.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.55 }{ 12 } = 2.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.55 }{ 13 } = 2.7 ; ;

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**2-12**2-13**2 }{ 2 * 12 * 13 } ) = 13° 10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-3**2-13**2 }{ 2 * 3 * 13 } ) = 64° 9'27" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 13**2-3**2-12**2 }{ 2 * 12 * 3 } ) = 102° 50'23" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.55 }{ 14 } = 1.25 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 13° 10" } = 6.67 ; ;




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