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.

Obtuse scalene triangle.

Sides: a = 5.74765599714   b = 5.74765599714   c = 10.8

Area: T = 10.61333720312
Perimeter: p = 22.29331199427
Semiperimeter: s = 11.14765599714

Angle ∠ A = α = 20° = 0.34990658504 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 140° = 2.44334609528 rad

Height: ha = 3.69438175479
Height: hb = 3.69438175479
Height: hc = 1.9655439265

Median: ma = 8.15993956808
Median: mb = 8.15993956808
Median: mc = 1.9655439265

Inradius: r = 0.95221656958
Circumradius: R = 8.4010908665

Vertex coordinates: A[10.8; 0] B[0; 0] C[5.4; 1.9655439265]
Centroid: CG[5.4; 0.65551464217]
Coordinates of the circumscribed circle: U[5.4; -6.43554694]
Coordinates of the inscribed circle: I[5.4; 0.95221656958]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160° = 0.34990658504 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 40° = 2.44334609528 rad

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How did we calculate this triangle?

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

c = 10.8 ; ; alpha = 20° ; ; beta = 20° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 20 ° - 20 ° = 140 ° ; ;

3. 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 = 10.8 * fraction{ sin(20° ) }{ sin (140° ) } = 5.75 ; ;

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{ 5.75**2+10.8**2 - 2 * 5.75 * 10.8 * cos(20° ) } ; ; b = 5.75 ; ;


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 = 5.75 ; ; b = 5.75 ; ; c = 10.8 ; ;

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

p = a+b+c = 5.75+5.75+10.8 = 22.29 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 22.29 }{ 2 } = 11.15 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.15 * (11.15-5.75)(11.15-5.75)(11.15-10.8) } ; ; T = sqrt{ 112.64 } = 10.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 10.61 }{ 5.75 } = 3.69 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 10.61 }{ 5.75 } = 3.69 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 10.61 }{ 10.8 } = 1.97 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 5.75**2-5.75**2-10.8**2 }{ 2 * 5.75 * 10.8 } ) = 20° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.75**2-5.75**2-10.8**2 }{ 2 * 5.75 * 10.8 } ) = 20° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 10.8**2-5.75**2-5.75**2 }{ 2 * 5.75 * 5.75 } ) = 140° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 10.61 }{ 11.15 } = 0.95 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.75 }{ 2 * sin 20° } = 8.4 ; ;




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