13 15 20 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 15   c = 20

Area: T = 97.48884608556
Perimeter: p = 48
Semiperimeter: s = 24

Angle ∠ A = α = 40.53658021113° = 40°32'9″ = 0.70774832118 rad
Angle ∠ B = β = 48.58326895819° = 48°34'58″ = 0.84879278927 rad
Angle ∠ C = γ = 90.88215083068° = 90°52'53″ = 1.58661815491 rad

Height: ha = 14.9988224747
Height: hb = 12.99884614474
Height: hc = 9.74988460856

Median: ma = 16.43992822228
Median: mb = 15.10879449297
Median: mc = 9.84988578018

Inradius: r = 4.06220192023
Circumradius: R = 10.00111836421

Vertex coordinates: A[20; 0] B[0; 0] C[8.6; 9.74988460856]
Centroid: CG[9.53333333333; 3.25496153619]
Coordinates of the circumscribed circle: U[10; -0.15438643637]
Coordinates of the inscribed circle: I[9; 4.06220192023]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.4644197889° = 139°27'51″ = 0.70774832118 rad
∠ B' = β' = 131.4177310418° = 131°25'2″ = 0.84879278927 rad
∠ C' = γ' = 89.11884916932° = 89°7'7″ = 1.58661815491 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 = 13 ; ; b = 15 ; ; c = 20 ; ;

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

p = a+b+c = 13+15+20 = 48 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 48 }{ 2 } = 24 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 24 * (24-13)(24-15)(24-20) } ; ; T = sqrt{ 9504 } = 97.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 97.49 }{ 13 } = 15 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 97.49 }{ 15 } = 13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 97.49 }{ 20 } = 9.75 ; ;

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{ 13**2-15**2-20**2 }{ 2 * 15 * 20 } ) = 40° 32'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-13**2-20**2 }{ 2 * 13 * 20 } ) = 48° 34'58" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-13**2-15**2 }{ 2 * 15 * 13 } ) = 90° 52'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 97.49 }{ 24 } = 4.06 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 40° 32'9" } = 10 ; ;




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