Triangle calculator

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

You have entered side a, c and angle γ.

Right scalene triangle.

Sides: a = 828   b = 2974.922184771   c = 3088

Area: T = 1231617.645495
Perimeter: p = 6890.922184771
Semiperimeter: s = 3445.461092386

Angle ∠ A = α = 15.55333016278° = 15°33'12″ = 0.2711456323 rad
Angle ∠ B = β = 74.44766983722° = 74°26'48″ = 1.29993400038 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 2974.922184771
Height: hb = 828
Height: hc = 797.6879821861

Median: ma = 3003.591051803
Median: mb = 1702.388773492
Median: mc = 1544

Inradius: r = 357.4610923857
Circumradius: R = 1544

Vertex coordinates: A[3088; 0] B[0; 0] C[222.0165544041; 797.6879821861]
Centroid: CG[1103.339851468; 265.8933273954]
Coordinates of the circumscribed circle: U[1544; -0]
Coordinates of the inscribed circle: I[470.5399076143; 357.4610923857]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 164.4476698372° = 164°26'48″ = 0.2711456323 rad
∠ B' = β' = 105.5533301628° = 105°33'12″ = 1.29993400038 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

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

a = 828 ; ; c = 3088 ; ; gamma = 90° ; ;

2. From angle γ, side a and side c we calculate side b - by using the law of cosines and quadratic equation:

c**2 = a**2 + b**2 - 2a b cos gamma ; ; ; ; 3088**2 = 828**2 + b**2 - 2 * 828 * b * cos 90° ; ; ; ; ; ; b**2 -8850160 =0 ; ; p=1; q=-0; r=-8850160 ; ; D = q**2 - 4pr = 0**2 - 4 * 1 * (-8850160) = 35400640 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ ± sqrt{ 35400640 } }{ 2 } = fraction{ ± 8 sqrt{ 553135 } }{ 2 } ; ; b_{1,2} = ± 4 sqrt{ 553135} = ± 2974.92184771 ; ; b_{1} = 4 sqrt{ 553135} = 2974.92184771 ; ;
b_{2} = - 4 sqrt{ 553135} = -2974.92184771 ; ; ; ; text{ Factored form: } ; ; (b -2974.92184771) (b +2974.92184771) = 0 ; ; ; ; b > 0 ; ; ; ; b = 2974.922 ; ;
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 = 828 ; ; b = 2974.92 ; ; c = 3088 ; ;

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

p = a+b+c = 828+2974.92+3088 = 6890.92 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 6890.92 }{ 2 } = 3445.46 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3445.46 * (3445.46-828)(3445.46-2974.92)(3445.46-3088) } ; ; T = sqrt{ 1.517 * 10**{ 12 } } = 1231617.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1231617.64 }{ 828 } = 2974.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1231617.64 }{ 2974.92 } = 828 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1231617.64 }{ 3088 } = 797.68 ; ;

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{ 2974.92**2+3088**2-828**2 }{ 2 * 2974.92 * 3088 } ) = 15° 33'12" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 828**2+3088**2-2974.92**2 }{ 2 * 828 * 3088 } ) = 74° 26'48" ; ; gamma = 180° - alpha - beta = 180° - 15° 33'12" - 74° 26'48" = 90° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1231617.64 }{ 3445.46 } = 357.46 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 828 }{ 2 * sin 15° 33'12" } = 1544 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2974.92**2+2 * 3088**2 - 828**2 } }{ 2 } = 3003.591 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3088**2+2 * 828**2 - 2974.92**2 } }{ 2 } = 1702.388 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2974.92**2+2 * 828**2 - 3088**2 } }{ 2 } = 1544 ; ;
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