Triangle calculator

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

You have entered area S and aspect ratio a:b:c = 9:15:16.

Acute scalene triangle.

Sides: a = 25.86884030848   b = 43.11440051413   c = 45.98882721507

Area: T = 548
Perimeter: p = 114.9710680377
Semiperimeter: s = 57.48553401884

Angle ∠ A = α = 33.55773097619° = 33°33'26″ = 0.58656855435 rad
Angle ∠ B = β = 67.11546195238° = 67°6'53″ = 1.17113710869 rad
Angle ∠ C = γ = 79.32880707142° = 79°19'41″ = 1.38545360232 rad

Height: ha = 42.36882898557
Height: hb = 25.42109739134
Height: hc = 23.83221630438

Median: ma = 42.65664852962
Median: mb = 30.45223129984
Median: mc = 27.11657807353

Inradius: r = 9.53328652175
Circumradius: R = 23.39988509885

Vertex coordinates: A[45.98882721507; 0] B[0; 0] C[10.0659934533; 23.83221630438]
Centroid: CG[18.68327355612; 7.94440543479]
Coordinates of the circumscribed circle: U[22.99441360753; 4.33331205534]
Coordinates of the inscribed circle: I[14.37113350471; 9.53328652175]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 146.4432690238° = 146°26'34″ = 0.58656855435 rad
∠ B' = β' = 112.8855380476° = 112°53'7″ = 1.17113710869 rad
∠ C' = γ' = 100.6721929286° = 100°40'19″ = 1.38545360232 rad

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How did we calculate this triangle?

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

S = 548 ; ; a:b:c = 9:15:16 ; ;

2. From side a and angle β we calculate height hc:

h_c = a * sin beta = 25.868 * sin 67° 6'53" = 23.832 ; ;


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 = 25.87 ; ; b = 43.11 ; ; c = 45.99 ; ;

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

p = a+b+c = 25.87+43.11+45.99 = 114.97 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 114.97 }{ 2 } = 57.49 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.49 * (57.49-25.87)(57.49-43.11)(57.49-45.99) } ; ; T = sqrt{ 300304 } = 548 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 548 }{ 25.87 } = 42.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 548 }{ 43.11 } = 25.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 548 }{ 45.99 } = 23.83 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 43.11**2+45.99**2-25.87**2 }{ 2 * 43.11 * 45.99 } ) = 33° 33'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 25.87**2+45.99**2-43.11**2 }{ 2 * 25.87 * 45.99 } ) = 67° 6'53" ; ; gamma = 180° - alpha - beta = 180° - 33° 33'26" - 67° 6'53" = 79° 19'41" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 548 }{ 57.49 } = 9.53 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 25.87 }{ 2 * sin 33° 33'26" } = 23.4 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.11**2+2 * 45.99**2 - 25.87**2 } }{ 2 } = 42.656 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 45.99**2+2 * 25.87**2 - 43.11**2 } }{ 2 } = 30.452 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 43.11**2+2 * 25.87**2 - 45.99**2 } }{ 2 } = 27.116 ; ;
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