13 17 21 triangle

Acute scalene triangle.

Sides: a = 13   b = 17   c = 21

Area: T = 110.4188238983
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 38.21332107017° = 38°12'48″ = 0.66769463445 rad
Angle ∠ B = β = 53.99110168022° = 53°59'28″ = 0.94223210097 rad
Angle ∠ C = γ = 87.7965772496° = 87°47'45″ = 1.53223252994 rad

Height: ha = 16.98774213819
Height: hb = 12.99903810568
Height: hc = 10.51660227602

Median: ma = 17.96552442232
Median: mb = 15.25661463024
Median: mc = 10.89772473589

Inradius: r = 4.33301270189
Circumradius: R = 10.50877748993

Vertex coordinates: A[21; 0] B[0; 0] C[7.64328571429; 10.51660227602]
Centroid: CG[9.54876190476; 3.50553409201]
Coordinates of the circumscribed circle: U[10.5; 0.40441451884]
Coordinates of the inscribed circle: I[8.5; 4.33301270189]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 141.7876789298° = 141°47'12″ = 0.66769463445 rad
∠ B' = β' = 126.0098983198° = 126°32″ = 0.94223210097 rad
∠ C' = γ' = 92.2044227504° = 92°12'15″ = 1.53223252994 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 = 13 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 13+17+21 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-13)(25.5-17)(25.5-21) } ; ; T = sqrt{ 12192.19 } = 110.42 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 110.42 }{ 13 } = 16.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 110.42 }{ 17 } = 12.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 110.42 }{ 21 } = 10.52 ; ;

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{ 13**2-17**2-21**2 }{ 2 * 17 * 21 } ) = 38° 12'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-13**2-21**2 }{ 2 * 13 * 21 } ) = 53° 59'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-13**2-17**2 }{ 2 * 17 * 13 } ) = 87° 47'45" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 110.42 }{ 25.5 } = 4.33 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 38° 12'48" } = 10.51 ; ;




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