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 = 490   b = 960   c = 1077.822187768

Area: T = 235200
Perimeter: p = 2527.822187768
Semiperimeter: s = 1263.911093884

Angle ∠ A = α = 27.04105239095° = 27°2'26″ = 0.47219461737 rad
Angle ∠ B = β = 62.95994760905° = 62°57'34″ = 1.09988501531 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 960
Height: hb = 490
Height: hc = 436.4365750417

Median: ma = 990.7769902652
Median: mb = 685.9330025586
Median: mc = 538.9110938839

Inradius: r = 186.0899061161
Circumradius: R = 538.9110938839

Vertex coordinates: A[1077.822187768; 0] B[0; 0] C[222.7644080942; 436.4365750417]
Centroid: CG[433.5298652873; 145.4798583472]
Coordinates of the circumscribed circle: U[538.9110938839; 0]
Coordinates of the inscribed circle: I[303.9110938839; 186.0899061161]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.959947609° = 152°57'34″ = 0.47219461737 rad
∠ B' = β' = 117.041052391° = 117°2'26″ = 1.09988501531 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 = 490 ; ; b = 960 ; ; 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{ 490**2+960**2 - 2 * 490 * 960 * cos 90° } ; ; c = 1077.82 ; ;
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 = 490 ; ; b = 960 ; ; c = 1077.82 ; ;

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

p = a+b+c = 490+960+1077.82 = 2527.82 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2527.82 }{ 2 } = 1263.91 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1263.91 * (1263.91-490)(1263.91-960)(1263.91-1077.82) } ; ; T = sqrt{ 55319040000 } = 235200 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 235200 }{ 490 } = 960 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 235200 }{ 960 } = 490 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 235200 }{ 1077.82 } = 436.44 ; ;

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{ 960**2+1077.82**2-490**2 }{ 2 * 960 * 1077.82 } ) = 27° 2'26" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 490**2+1077.82**2-960**2 }{ 2 * 490 * 1077.82 } ) = 62° 57'34" ; ;
gamma = 180° - alpha - beta = 180° - 27° 2'26" - 62° 57'34" = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 235200 }{ 1263.91 } = 186.09 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 490 }{ 2 * sin 27° 2'26" } = 538.91 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 960**2+2 * 1077.82**2 - 490**2 } }{ 2 } = 990.77 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1077.82**2+2 * 490**2 - 960**2 } }{ 2 } = 685.93 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 960**2+2 * 490**2 - 1077.82**2 } }{ 2 } = 538.911 ; ;
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