Triangle calculator

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

You have entered side c, angle α and angle β.

Right scalene triangle.

Sides: a = 3.27876944458   b = 0.70994229205   c = 3.2

Area: T = 1.13550766727
Perimeter: p = 7.18771173663
Semiperimeter: s = 3.59435586831

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 12.5° = 12°30' = 0.21881661565 rad
Angle ∠ C = γ = 77.5° = 77°30' = 1.35326301703 rad

Height: ha = 0.69326067646
Height: hb = 3.2
Height: hc = 0.70994229205

Median: ma = 1.63988472229
Median: mb = 3.22195993881
Median: mc = 1.75502230944

Inradius: r = 0.31658642373
Circumradius: R = 1.63988472229

Vertex coordinates: A[3.2; 0] B[0; 0] C[3.2; 0.70994229205]
Centroid: CG[2.13333333333; 0.23664743068]
Coordinates of the circumscribed circle: U[1.6; 0.35547114602]
Coordinates of the inscribed circle: I[2.88441357627; 0.31658642373]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 167.5° = 167°30' = 0.21881661565 rad
∠ C' = γ' = 102.5° = 102°30' = 1.35326301703 rad

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How did we calculate this triangle?

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

c = 3.2 ; ; alpha = 90° ; ; beta = 12.5° ; ;

2. From angle α and angle β we calculate angle γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 90 ° - 12.5 ° = 77.5 ° ; ;

3. From angle α, angle γ and side c we calculate side a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 3.2 * fraction{ sin 90° }{ sin 77° 30' } = 3.28 ; ;

4. 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{ 3.28**2+3.2**2 - 2 * 3.28 * 3.2 * cos 12° 30' } ; ; b = 0.71 ; ;


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 = 3.28 ; ; b = 0.71 ; ; c = 3.2 ; ;

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

p = a+b+c = 3.28+0.71+3.2 = 7.19 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7.19 }{ 2 } = 3.59 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3.59 * (3.59-3.28)(3.59-0.71)(3.59-3.2) } ; ; T = sqrt{ 1.29 } = 1.14 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.14 }{ 3.28 } = 0.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.14 }{ 0.71 } = 3.2 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.14 }{ 3.2 } = 0.71 ; ;

9. 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{ 0.71**2+3.2**2-3.28**2 }{ 2 * 0.71 * 3.2 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.28**2+3.2**2-0.71**2 }{ 2 * 3.28 * 3.2 } ) = 12° 30' ; ; gamma = 180° - alpha - beta = 180° - 90° - 12° 30' = 77° 30' ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.14 }{ 3.59 } = 0.32 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.28 }{ 2 * sin 90° } = 1.64 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.71**2+2 * 3.2**2 - 3.28**2 } }{ 2 } = 1.639 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 3.2**2+2 * 3.28**2 - 0.71**2 } }{ 2 } = 3.22 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 0.71**2+2 * 3.28**2 - 3.2**2 } }{ 2 } = 1.75 ; ;
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