10 12 15 triangle

Acute scalene triangle.

Sides: a = 10   b = 12   c = 15

Area: T = 59.81216836412
Perimeter: p = 37
Semiperimeter: s = 18.5

Angle ∠ A = α = 41.65496722739° = 41°38'59″ = 0.72769239136 rad
Angle ∠ B = β = 52.89109950542° = 52°53'28″ = 0.92331220084 rad
Angle ∠ C = γ = 85.45993326719° = 85°27'34″ = 1.49215467317 rad

Height: ha = 11.96223367282
Height: hb = 9.96986139402
Height: hc = 7.97548911522

Median: ma = 12.62993309403
Median: mb = 11.24772218792
Median: mc = 8.10986373701

Inradius: r = 3.23330639806
Circumradius: R = 7.52436136588

Vertex coordinates: A[15; 0] B[0; 0] C[6.03333333333; 7.97548911522]
Centroid: CG[7.01111111111; 2.65882970507]
Coordinates of the circumscribed circle: U[7.5; 0.59656194147]
Coordinates of the inscribed circle: I[6.5; 3.23330639806]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.3550327726° = 138°21'1″ = 0.72769239136 rad
∠ B' = β' = 127.1099004946° = 127°6'32″ = 0.92331220084 rad
∠ C' = γ' = 94.54106673281° = 94°32'26″ = 1.49215467317 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 = 12 ; ; c = 15 ; ;

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

p = a+b+c = 10+12+15 = 37 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37 }{ 2 } = 18.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.5 * (18.5-10)(18.5-12)(18.5-15) } ; ; T = sqrt{ 3577.44 } = 59.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 59.81 }{ 10 } = 11.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 59.81 }{ 12 } = 9.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 59.81 }{ 15 } = 7.97 ; ;

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**2-12**2-15**2 }{ 2 * 12 * 15 } ) = 41° 38'59" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12**2-10**2-15**2 }{ 2 * 10 * 15 } ) = 52° 53'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 15**2-10**2-12**2 }{ 2 * 12 * 10 } ) = 85° 27'34" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 59.81 }{ 18.5 } = 3.23 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 41° 38'59" } = 7.52 ; ;




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