Triangle calculator

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

You have entered side a, b and angle γ.

Right scalene triangle.

Sides: a = 76.4   b = 39.3   c = 85.91553653312

Area: T = 1501.26
Perimeter: p = 201.6155365331
Semiperimeter: s = 100.8087682666

Angle ∠ A = α = 62.7798805124° = 62°46'44″ = 1.09656968499 rad
Angle ∠ B = β = 27.2211194876° = 27°13'16″ = 0.47550994769 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 39.3
Height: hb = 76.4
Height: hc = 34.94774158484

Median: ma = 54.80662952588
Median: mb = 78.88765165919
Median: mc = 42.95876826656

Inradius: r = 14.89223173344
Circumradius: R = 42.95876826656

Vertex coordinates: A[85.91553653312; 0] B[0; 0] C[67.93884878071; 34.94774158484]
Centroid: CG[51.28546177128; 11.64991386161]
Coordinates of the circumscribed circle: U[42.95876826656; -0]
Coordinates of the inscribed circle: I[61.50876826656; 14.89223173344]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 117.2211194876° = 117°13'16″ = 1.09656968499 rad
∠ B' = β' = 152.7798805124° = 152°46'44″ = 0.47550994769 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle γ.

a = 76.4 ; ; b = 39.3 ; ; gamma = 90° ; ;

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

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 76.4**2+39.3**2 - 2 * 76.4 * 39.3 * cos 90° } ; ; c = 85.92 ; ;
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 = 76.4 ; ; b = 39.3 ; ; c = 85.92 ; ;

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

p = a+b+c = 76.4+39.3+85.92 = 201.62 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 201.62 }{ 2 } = 100.81 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 100.81 * (100.81-76.4)(100.81-39.3)(100.81-85.92) } ; ; T = sqrt{ 2253781.59 } = 1501.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1501.26 }{ 76.4 } = 39.3 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1501.26 }{ 39.3 } = 76.4 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1501.26 }{ 85.92 } = 34.95 ; ;

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{ 39.3**2+85.92**2-76.4**2 }{ 2 * 39.3 * 85.92 } ) = 62° 46'44" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 76.4**2+85.92**2-39.3**2 }{ 2 * 76.4 * 85.92 } ) = 27° 13'16" ; ;
 gamma = 180° - alpha - beta = 180° - 62° 46'44" - 27° 13'16" = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1501.26 }{ 100.81 } = 14.89 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 76.4 }{ 2 * sin 62° 46'44" } = 42.96 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 39.3**2+2 * 85.92**2 - 76.4**2 } }{ 2 } = 54.806 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 85.92**2+2 * 76.4**2 - 39.3**2 } }{ 2 } = 78.887 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 39.3**2+2 * 76.4**2 - 85.92**2 } }{ 2 } = 42.958 ; ;
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