10 18 21 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 18   c = 21

Area: T = 89.98995967733
Perimeter: p = 49
Semiperimeter: s = 24.5

Angle ∠ A = α = 28.40222819014° = 28°24'8″ = 0.49657133343 rad
Angle ∠ B = β = 58.89110774895° = 58°53'28″ = 1.02878432022 rad
Angle ∠ C = γ = 92.70766406091° = 92°42'24″ = 1.61880361171 rad

Height: ha = 17.98799193547
Height: hb = 9.98988440859
Height: hc = 8.56218663594

Median: ma = 18.90876704012
Median: mb = 13.76658998979
Median: mc = 10.08771205009

Inradius: r = 3.66993712969
Circumradius: R = 10.51217267921

Vertex coordinates: A[21; 0] B[0; 0] C[5.16766666667; 8.56218663594]
Centroid: CG[8.72222222222; 2.85439554531]
Coordinates of the circumscribed circle: U[10.5; -0.49663870985]
Coordinates of the inscribed circle: I[6.5; 3.66993712969]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 151.5987718099° = 151°35'52″ = 0.49657133343 rad
∠ B' = β' = 121.1098922511° = 121°6'32″ = 1.02878432022 rad
∠ C' = γ' = 87.29333593909° = 87°17'36″ = 1.61880361171 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 = 18 ; ; c = 21 ; ;

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

p = a+b+c = 10+18+21 = 49 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 49 }{ 2 } = 24.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24.5 * (24.5-10)(24.5-18)(24.5-21) } ; ; T = sqrt{ 8081.94 } = 89.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 89.9 }{ 10 } = 17.98 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 89.9 }{ 18 } = 9.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 89.9 }{ 21 } = 8.56 ; ;

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**2-18**2-21**2 }{ 2 * 18 * 21 } ) = 28° 24'8" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 18**2-10**2-21**2 }{ 2 * 10 * 21 } ) = 58° 53'28" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-10**2-18**2 }{ 2 * 18 * 10 } ) = 92° 42'24" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 89.9 }{ 24.5 } = 3.67 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 28° 24'8" } = 10.51 ; ;




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