Triangle calculator

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

You have entered side b, c and angle α.

Right scalene triangle.

Sides: a = 3418.194426013   b = 2316   c = 2514

Area: T = 2911212
Perimeter: p = 8248.194426013
Semiperimeter: s = 4124.097713007

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 42.65325432854° = 42°39'9″ = 0.74444273147 rad
Angle ∠ C = γ = 47.34774567146° = 47°20'51″ = 0.82663690121 rad

Height: ha = 1703.363252328
Height: hb = 2514
Height: hc = 2316

Median: ma = 1709.097713007
Median: mb = 2767.88800552
Median: mc = 2635.12990291

Inradius: r = 705.9032869934
Circumradius: R = 1709.097713007

Vertex coordinates: A[2514; 0] B[0; 0] C[2514; 2316]
Centroid: CG[1676; 772]
Coordinates of the circumscribed circle: U[1257; 1158]
Coordinates of the inscribed circle: I[1808.097713007; 705.9032869934]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 137.3477456715° = 137°20'51″ = 0.74444273147 rad
∠ C' = γ' = 132.6532543285° = 132°39'9″ = 0.82663690121 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 2316 ; ; c = 2514 ; ; alpha = 90° ; ;

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

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 2316**2+2514**2 - 2 * 2316 * 2514 * cos 90° } ; ; a = 3418.19 ; ;


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 = 3418.19 ; ; b = 2316 ; ; c = 2514 ; ;

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

p = a+b+c = 3418.19+2316+2514 = 8248.19 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 8248.19 }{ 2 } = 4124.1 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 4124.1 * (4124.1-3418.19)(4124.1-2316)(4124.1-2514) } ; ; T = sqrt{ 8.475 * 10**{ 12 } } = 2911212 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2911212 }{ 3418.19 } = 1703.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2911212 }{ 2316 } = 2514 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2911212 }{ 2514 } = 2316 ; ;

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{ 2316**2+2514**2-3418.19**2 }{ 2 * 2316 * 2514 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3418.19**2+2514**2-2316**2 }{ 2 * 3418.19 * 2514 } ) = 42° 39'9" ; ; gamma = 180° - alpha - beta = 180° - 90° - 42° 39'9" = 47° 20'51" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2911212 }{ 4124.1 } = 705.9 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3418.19 }{ 2 * sin 90° } = 1709.1 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 2316**2+2 * 2514**2 - 3418.19**2 } }{ 2 } = 1709.097 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2514**2+2 * 3418.19**2 - 2316**2 } }{ 2 } = 2767.88 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 2316**2+2 * 3418.19**2 - 2514**2 } }{ 2 } = 2635.129 ; ;
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