8 15 21 triangle

Obtuse scalene triangle.

Sides: a = 8   b = 15   c = 21

Area: T = 46.4332747065
Perimeter: p = 44
Semiperimeter: s = 22

Angle ∠ A = α = 17.14662099989° = 17°8'46″ = 0.29992578187 rad
Angle ∠ B = β = 33.55773097619° = 33°33'26″ = 0.58656855435 rad
Angle ∠ C = γ = 129.2966480239° = 129°17'47″ = 2.25766492914 rad

Height: ha = 11.60881867662
Height: hb = 6.1911032942
Height: hc = 4.42221663871

Median: ma = 17.80444938148
Median: mb = 14.00989257261
Median: mc = 5.85223499554

Inradius: r = 2.1110579412
Circumradius: R = 13.5688010506

Vertex coordinates: A[21; 0] B[0; 0] C[6.66766666667; 4.42221663871]
Centroid: CG[9.22222222222; 1.47440554624]
Coordinates of the circumscribed circle: U[10.5; -8.59330733205]
Coordinates of the inscribed circle: I[7; 2.1110579412]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.8543790001° = 162°51'14″ = 0.29992578187 rad
∠ B' = β' = 146.4432690238° = 146°26'34″ = 0.58656855435 rad
∠ C' = γ' = 50.70435197608° = 50°42'13″ = 2.25766492914 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 = 8 ; ; b = 15 ; ; c = 21 ; ;

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

p = a+b+c = 8+15+21 = 44 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 44 }{ 2 } = 22 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22 * (22-8)(22-15)(22-21) } ; ; T = sqrt{ 2156 } = 46.43 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 46.43 }{ 8 } = 11.61 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 46.43 }{ 15 } = 6.19 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 46.43 }{ 21 } = 4.42 ; ;

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{ 8**2-15**2-21**2 }{ 2 * 15 * 21 } ) = 17° 8'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-8**2-21**2 }{ 2 * 8 * 21 } ) = 33° 33'26" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 21**2-8**2-15**2 }{ 2 * 15 * 8 } ) = 129° 17'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 46.43 }{ 22 } = 2.11 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 17° 8'46" } = 13.57 ; ;




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