Triangle calculator

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

You have entered side c, height hc and angle β.

Acute scalene triangle.

Sides: a = 115.4770053838   b = 136.0621843504   c = 150

Area: T = 7500
Perimeter: p = 401.5321897342
Semiperimeter: s = 200.7665948671

Angle ∠ A = α = 47.30438285023° = 47°18'14″ = 0.82656075562 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 72.69661714977° = 72°41'46″ = 1.26987875462 rad

Height: ha = 129.9043810568
Height: hb = 110.2443986218
Height: hc = 100

Median: ma = 131.0466096071
Median: mb = 115.2765584372
Median: mc = 101.4799452578

Inradius: r = 37.35769325359
Circumradius: R = 78.55553419733

Vertex coordinates: A[150; 0] B[0; 0] C[57.7355026919; 100]
Centroid: CG[69.2455008973; 33.33333333333]
Coordinates of the circumscribed circle: U[75; 23.36553964774]
Coordinates of the inscribed circle: I[64.70441051671; 37.35769325359]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.6966171498° = 132°41'46″ = 0.82656075562 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 107.3043828502° = 107°18'14″ = 1.26987875462 rad

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How did we calculate this triangle?

1. Input data entered: side c, angle β and height hc.

c = 150 ; ; beta = 60° ; ; hc = 100 ; ;

2. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 115.47**2+150**2 - 2 * 115.47 * 150 * cos(60° ) } ; ; b = 136.06 ; ;


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 = 115.47 ; ; b = 136.06 ; ; c = 150 ; ;

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

p = a+b+c = 115.47+136.06+150 = 401.53 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 401.53 }{ 2 } = 200.77 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 200.77 * (200.77-115.47)(200.77-136.06)(200.77-150) } ; ; T = sqrt{ 56250000 } = 7500 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7500 }{ 115.47 } = 129.9 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7500 }{ 136.06 } = 110.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7500 }{ 150 } = 100 ; ;

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{ 115.47**2-136.06**2-150**2 }{ 2 * 136.06 * 150 } ) = 47° 18'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 136.06**2-115.47**2-150**2 }{ 2 * 115.47 * 150 } ) = 60° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 150**2-115.47**2-136.06**2 }{ 2 * 136.06 * 115.47 } ) = 72° 41'46" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7500 }{ 200.77 } = 37.36 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 115.47 }{ 2 * sin 47° 18'14" } = 78.56 ; ;




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