13 18 21 triangle

Acute scalene triangle.

Sides: a = 13   b = 18   c = 21

Area: T = 116.276553483
Perimeter: p = 52
Semiperimeter: s = 26

Angle ∠ A = α = 37.96875057136° = 37°58'3″ = 0.66326579835 rad
Angle ∠ B = β = 58.41218644948° = 58°24'43″ = 1.01994793577 rad
Angle ∠ C = γ = 83.62106297916° = 83°37'14″ = 1.45994553125 rad

Height: ha = 17.889854382
Height: hb = 12.921950387
Height: hc = 11.074386046

Median: ma = 18.44658667457
Median: mb = 14.96766295471
Median: mc = 11.67326175299

Inradius: r = 4.4722135955
Circumradius: R = 10.56554211937

Vertex coordinates: A[21; 0] B[0; 0] C[6.81095238095; 11.074386046]
Centroid: CG[9.27698412698; 3.691128682]
Coordinates of the circumscribed circle: U[10.5; 1.17439356882]
Coordinates of the inscribed circle: I[8; 4.4722135955]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142.0322494286° = 142°1'57″ = 0.66326579835 rad
∠ B' = β' = 121.5888135505° = 121°35'17″ = 1.01994793577 rad
∠ C' = γ' = 96.37993702084° = 96°22'46″ = 1.45994553125 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 = 18 ; ; c = 21 ; ;

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

p = a+b+c = 13+18+21 = 52 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52 }{ 2 } = 26 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26 * (26-13)(26-18)(26-21) } ; ; T = sqrt{ 13520 } = 116.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 116.28 }{ 13 } = 17.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 116.28 }{ 18 } = 12.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 116.28 }{ 21 } = 11.07 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 18**2+21**2-13**2 }{ 2 * 18 * 21 } ) = 37° 58'3" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 13**2+21**2-18**2 }{ 2 * 13 * 21 } ) = 58° 24'43" ; ; gamma = 180° - alpha - beta = 180° - 37° 58'3" - 58° 24'43" = 83° 37'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 116.28 }{ 26 } = 4.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13 }{ 2 * sin 37° 58'3" } = 10.57 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 21**2 - 13**2 } }{ 2 } = 18.446 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 13**2 - 18**2 } }{ 2 } = 14.967 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 13**2 - 21**2 } }{ 2 } = 11.673 ; ;
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