10 16 21 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 16   c = 21

Area: T = 77.12661142545
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 27.32880163439° = 27°19'41″ = 0.47769638632 rad
Angle ∠ B = β = 47.26878899574° = 47°16'4″ = 0.82549803102 rad
Angle ∠ C = γ = 105.4044093699° = 105°24'15″ = 1.84396484801 rad

Height: ha = 15.42552228509
Height: hb = 9.64107642818
Height: hc = 7.34553442147

Median: ma = 17.98661057486
Median: mb = 14.37701078632
Median: mc = 8.23110388166

Inradius: r = 3.28219623087
Circumradius: R = 10.89112526985

Vertex coordinates: A[21; 0] B[0; 0] C[6.78657142857; 7.34553442147]
Centroid: CG[9.26219047619; 2.44884480716]
Coordinates of the circumscribed circle: U[10.5; -2.8932988998]
Coordinates of the inscribed circle: I[7.5; 3.28219623087]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.6721983656° = 152°40'19″ = 0.47769638632 rad
∠ B' = β' = 132.7322110043° = 132°43'56″ = 0.82549803102 rad
∠ C' = γ' = 74.59659063013° = 74°35'45″ = 1.84396484801 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 = 16 ; ; c = 21 ; ;

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

p = a+b+c = 10+16+21 = 47 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 47 }{ 2 } = 23.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23.5 * (23.5-10)(23.5-16)(23.5-21) } ; ; T = sqrt{ 5948.44 } = 77.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 77.13 }{ 10 } = 15.43 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 77.13 }{ 16 } = 9.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 77.13 }{ 21 } = 7.35 ; ;

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**2-16**2-21**2 }{ 2 * 16 * 21 } ) = 27° 19'41" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 16**2-10**2-21**2 }{ 2 * 10 * 21 } ) = 47° 16'4" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-10**2-16**2 }{ 2 * 16 * 10 } ) = 105° 24'15" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 77.13 }{ 23.5 } = 3.28 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 27° 19'41" } = 10.89 ; ;




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