7 17 21 triangle

Obtuse scalene triangle.

Sides: a = 7   b = 17   c = 21

Area: T = 53.63994211378
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 17.48876943436° = 17°29'16″ = 0.30552178449 rad
Angle ∠ B = β = 46.86986278687° = 46°52'7″ = 0.81880118722 rad
Angle ∠ C = γ = 115.6443677788° = 115°38'37″ = 2.01883629365 rad

Height: ha = 15.32655488965
Height: hb = 6.31105201339
Height: hc = 5.10985162988

Median: ma = 18.7821639971
Median: mb = 13.14334394281
Median: mc = 7.66548548584

Inradius: r = 2.38439742728
Circumradius: R = 11.64772174149

Vertex coordinates: A[21; 0] B[0; 0] C[4.78657142857; 5.10985162988]
Centroid: CG[8.59552380952; 1.70328387663]
Coordinates of the circumscribed circle: U[10.5; -5.04106024947]
Coordinates of the inscribed circle: I[5.5; 2.38439742728]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.5122305656° = 162°30'44″ = 0.30552178449 rad
∠ B' = β' = 133.1311372131° = 133°7'53″ = 0.81880118722 rad
∠ C' = γ' = 64.35663222123° = 64°21'23″ = 2.01883629365 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 = 7 ; ; b = 17 ; ; c = 21 ; ;

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

p = a+b+c = 7+17+21 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-7)(22.5-17)(22.5-21) } ; ; T = sqrt{ 2877.19 } = 53.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 53.64 }{ 7 } = 15.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 53.64 }{ 17 } = 6.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 53.64 }{ 21 } = 5.11 ; ;

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{ 17**2+21**2-7**2 }{ 2 * 17 * 21 } ) = 17° 29'16" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 7**2+21**2-17**2 }{ 2 * 7 * 21 } ) = 46° 52'7" ; ; gamma = 180° - alpha - beta = 180° - 17° 29'16" - 46° 52'7" = 115° 38'37" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 53.64 }{ 22.5 } = 2.38 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 7 }{ 2 * sin 17° 29'16" } = 11.65 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 21**2 - 7**2 } }{ 2 } = 18.782 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 7**2 - 17**2 } }{ 2 } = 13.143 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 17**2+2 * 7**2 - 21**2 } }{ 2 } = 7.665 ; ;
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