7 19 20 triangle

Acute scalene triangle.

Sides: a = 7   b = 19   c = 20

Area: T = 66.45329909033
Perimeter: p = 46
Semiperimeter: s = 23

Angle ∠ A = α = 20.47221827795° = 20°28'20″ = 0.35773069946 rad
Angle ∠ B = β = 71.68223015888° = 71°40'56″ = 1.25110921781 rad
Angle ∠ C = γ = 87.84655156316° = 87°50'44″ = 1.53331934809 rad

Height: ha = 18.98765688295
Height: hb = 6.9955051674
Height: hc = 6.64552990903

Median: ma = 19.19898410624
Median: mb = 11.58766302263
Median: mc = 10.2476950766

Inradius: r = 2.88992604741
Circumradius: R = 10.00770740378

Vertex coordinates: A[20; 0] B[0; 0] C[2.2; 6.64552990903]
Centroid: CG[7.4; 2.21550996968]
Coordinates of the circumscribed circle: U[10; 0.37662057909]
Coordinates of the inscribed circle: I[4; 2.88992604741]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 159.528781722° = 159°31'40″ = 0.35773069946 rad
∠ B' = β' = 108.3187698411° = 108°19'4″ = 1.25110921781 rad
∠ C' = γ' = 92.15444843684° = 92°9'16″ = 1.53331934809 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 = 7 ; ; b = 19 ; ; c = 20 ; ;

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

p = a+b+c = 7+19+20 = 46 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 46 }{ 2 } = 23 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 23 * (23-7)(23-19)(23-20) } ; ; T = sqrt{ 4416 } = 66.45 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 66.45 }{ 7 } = 18.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 66.45 }{ 19 } = 7 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 66.45 }{ 20 } = 6.65 ; ;

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{ 7**2-19**2-20**2 }{ 2 * 19 * 20 } ) = 20° 28'20" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19**2-7**2-20**2 }{ 2 * 7 * 20 } ) = 71° 40'56" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 20**2-7**2-19**2 }{ 2 * 19 * 7 } ) = 87° 50'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 66.45 }{ 23 } = 2.89 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7 }{ 2 * sin 20° 28'20" } = 10.01 ; ;




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