Triangle calculator

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

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

Obtuse scalene triangle.

Sides: a = 60.34333651663   b = 60.34333651663   c = 111.5

Area: T = 1287.40216402
Perimeter: p = 232.1876730333
Semiperimeter: s = 116.0933365166

Angle ∠ A = α = 22.5° = 22°30' = 0.39326990817 rad
Angle ∠ B = β = 22.5° = 22°30' = 0.39326990817 rad
Angle ∠ C = γ = 135° = 2.35661944902 rad

Height: ha = 42.66992027087
Height: hb = 42.66992027087
Height: hc = 23.09224061023

Median: ma = 84.41883358631
Median: mb = 84.41883358631
Median: mc = 23.09224061023

Inradius: r = 11.08993644814
Circumradius: R = 78.84224061023

Vertex coordinates: A[111.5; 0] B[0; 0] C[55.75; 23.09224061023]
Centroid: CG[55.75; 7.69774687008]
Coordinates of the circumscribed circle: U[55.75; -55.75]
Coordinates of the inscribed circle: I[55.75; 11.08993644814]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.5° = 157°30' = 0.39326990817 rad
∠ B' = β' = 157.5° = 157°30' = 0.39326990817 rad
∠ C' = γ' = 45° = 2.35661944902 rad

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How did we calculate this triangle?

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

c = 111.5 ; ; alpha = 22.5° ; ; beta = 22.5° ; ; gamma = 45° ; ;

2. From angle α, angle γ and side c we calculate 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 = 111.5 * fraction{ sin(22° 30') }{ sin (45° ) } = 60.34 ; ;

3. 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{ 60.34**2+111.5**2 - 2 * 60.34 * 111.5 * cos(22° 30') } ; ; b = 60.34 ; ;


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 = 60.34 ; ; b = 60.34 ; ; c = 111.5 ; ;

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

p = a+b+c = 60.34+60.34+111.5 = 232.19 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 232.19 }{ 2 } = 116.09 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 116.09 * (116.09-60.34)(116.09-60.34)(116.09-111.5) } ; ; T = sqrt{ 1657402.98 } = 1287.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1287.4 }{ 60.34 } = 42.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1287.4 }{ 60.34 } = 42.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1287.4 }{ 111.5 } = 23.09 ; ;

8. 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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 60.34**2-60.34**2-111.5**2 }{ 2 * 60.34 * 111.5 } ) = 22° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60.34**2-60.34**2-111.5**2 }{ 2 * 60.34 * 111.5 } ) = 22° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 111.5**2-60.34**2-60.34**2 }{ 2 * 60.34 * 60.34 } ) = 135° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1287.4 }{ 116.09 } = 11.09 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60.34 }{ 2 * sin 22° 30' } = 78.84 ; ;




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