12 15 21 triangle

Obtuse scalene triangle.

Sides: a = 12   b = 15   c = 21

Area: T = 88.18216307402
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 34.048773237° = 34°2'52″ = 0.59442450327 rad
Angle ∠ B = β = 44.41553085972° = 44°24'55″ = 0.77551933733 rad
Angle ∠ C = γ = 101.5376959033° = 101°32'13″ = 1.77221542476 rad

Height: ha = 14.69769384567
Height: hb = 11.75875507654
Height: hc = 8.39882505467

Median: ma = 17.23436879396
Median: mb = 15.37704261489
Median: mc = 8.61768439698

Inradius: r = 3.67442346142
Circumradius: R = 10.71765176247

Vertex coordinates: A[21; 0] B[0; 0] C[8.57114285714; 8.39882505467]
Centroid: CG[9.85771428571; 2.79994168489]
Coordinates of the circumscribed circle: U[10.5; -2.14333035249]
Coordinates of the inscribed circle: I[9; 3.67442346142]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.952226763° = 145°57'8″ = 0.59442450327 rad
∠ B' = β' = 135.5854691403° = 135°35'5″ = 0.77551933733 rad
∠ C' = γ' = 78.46330409672° = 78°27'47″ = 1.77221542476 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 = 12 ; ; b = 15 ; ; c = 21 ; ;

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

p = a+b+c = 12+15+21 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-12)(24-15)(24-21) } ; ; T = sqrt{ 7776 } = 88.18 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 88.18 }{ 12 } = 14.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 88.18 }{ 15 } = 11.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 88.18 }{ 21 } = 8.4 ; ;

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{ 12**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 34° 2'52" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-12**2-21**2 }{ 2 * 12 * 21 } ) = 44° 24'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-12**2-15**2 }{ 2 * 15 * 12 } ) = 101° 32'13" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 88.18 }{ 24 } = 3.67 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 12 }{ 2 * sin 34° 2'52" } = 10.72 ; ;




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