Triangle calculator - result

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side c and aspect ratio a:b:c = 3:5:7.

Obtuse scalene triangle.

Sides: a = 7.5   b = 12.5   c = 17.5

Area: T = 40.59549408024
Perimeter: p = 37.5
Semiperimeter: s = 18.75

Angle ∠ A = α = 21.78767892983° = 21°47'12″ = 0.38802512067 rad
Angle ∠ B = β = 38.21332107017° = 38°12'48″ = 0.66769463445 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 10.82553175473
Height: hb = 6.49551905284
Height: hc = 4.6399421806

Median: ma = 14.73772826532
Median: mb = 11.92442400177
Median: mc = 5.44986236794

Inradius: r = 2.16550635095
Circumradius: R = 10.10436297108

Vertex coordinates: A[17.5; 0] B[0; 0] C[5.89328571429; 4.6399421806]
Centroid: CG[7.79876190476; 1.54664739353]
Coordinates of the circumscribed circle: U[8.75; -5.05218148554]
Coordinates of the inscribed circle: I[6.25; 2.16550635095]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 158.2133210702° = 158°12'48″ = 0.38802512067 rad
∠ B' = β' = 141.7876789298° = 141°47'12″ = 0.66769463445 rad
∠ C' = γ' = 60° = 2.09443951024 rad

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How did we calculate this triangle?

1. Input data entered: side c and aspect ratio a:b:c.

c = 17.5 ; ; a:b:c = 3:5:7 ; ;

2. From angle β, angle γ and side c we calculate b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ c } = fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = c * fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = 17.5 * fraction{ sin(38° 12'48") }{ sin (120° ) } = 12.5 ; ;


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 = 7.5 ; ; b = 12.5 ; ; c = 17.5 ; ;

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

p = a+b+c = 7.5+12.5+17.5 = 37.5 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 37.5 }{ 2 } = 18.75 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 18.75 * (18.75-7.5)(18.75-12.5)(18.75-17.5) } ; ; T = sqrt{ 1647.95 } = 40.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40.59 }{ 7.5 } = 10.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40.59 }{ 12.5 } = 6.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40.59 }{ 17.5 } = 4.64 ; ;

7. 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{ 7.5**2-12.5**2-17.5**2 }{ 2 * 12.5 * 17.5 } ) = 21° 47'12" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.5**2-7.5**2-17.5**2 }{ 2 * 7.5 * 17.5 } ) = 38° 12'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17.5**2-7.5**2-12.5**2 }{ 2 * 12.5 * 7.5 } ) = 120° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40.59 }{ 18.75 } = 2.17 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.5 }{ 2 * sin 21° 47'12" } = 10.1 ; ;




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