10 17 20 triangle

Obtuse scalene triangle.

Sides: a = 10   b = 17   c = 20

Area: T = 84.9565503059
Perimeter: p = 47
Semiperimeter: s = 23.5

Angle ∠ A = α = 29.98326844873° = 29°58'58″ = 0.52332965629 rad
Angle ∠ B = β = 58.16333049964° = 58°9'48″ = 1.0155141176 rad
Angle ∠ C = γ = 91.85440105163° = 91°51'14″ = 1.60331549147 rad

Height: ha = 16.99111006118
Height: hb = 9.99547650658
Height: hc = 8.49655503059

Median: ma = 17.87545629317
Median: mb = 13.3322291626
Median: mc = 9.72111110476

Inradius: r = 3.61551277897
Circumradius: R = 10.00552376761

Vertex coordinates: A[20; 0] B[0; 0] C[5.275; 8.49655503059]
Centroid: CG[8.425; 2.8321850102]
Coordinates of the circumscribed circle: U[10; -0.3243698866]
Coordinates of the inscribed circle: I[6.5; 3.61551277897]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0177315513° = 150°1'2″ = 0.52332965629 rad
∠ B' = β' = 121.8376695004° = 121°50'12″ = 1.0155141176 rad
∠ C' = γ' = 88.14659894837° = 88°8'46″ = 1.60331549147 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 = 17 ; ; c = 20 ; ;

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

p = a+b+c = 10+17+20 = 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)(23.5-17)(23.5-20) } ; ; T = sqrt{ 7217.44 } = 84.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 84.96 }{ 10 } = 16.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 84.96 }{ 17 } = 9.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 84.96 }{ 20 } = 8.5 ; ;

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**2-17**2-20**2 }{ 2 * 17 * 20 } ) = 29° 58'58" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-10**2-20**2 }{ 2 * 10 * 20 } ) = 58° 9'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-10**2-17**2 }{ 2 * 17 * 10 } ) = 91° 51'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 84.96 }{ 23.5 } = 3.62 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 10 }{ 2 * sin 29° 58'58" } = 10.01 ; ;




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