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 = 1.81330050548   b = 7.47332514563   c = 7.25

Area: T = 6.57221433238
Perimeter: p = 16.53662565111
Semiperimeter: s = 8.26881282556

Angle ∠ A = α = 14.04° = 14°2'24″ = 0.2455044227 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 75.96° = 75°57'36″ = 1.32657520998 rad

Height: ha = 7.25
Height: hb = 1.75988444233
Height: hc = 1.81330050548

Median: ma = 7.30664524109
Median: mb = 3.73766257281
Median: mc = 4.05330991018

Inradius: r = 0.79548767993
Circumradius: R = 3.73766257281

Vertex coordinates: A[7.25; 0] B[0; 0] C[-0; 1.81330050548]
Centroid: CG[2.41766666667; 0.60443350183]
Coordinates of the circumscribed circle: U[3.625; 0.90765025274]
Coordinates of the inscribed circle: I[0.79548767993; 0.79548767993]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 165.96° = 165°57'36″ = 0.2455044227 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 104.04° = 104°2'24″ = 1.32657520998 rad

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How did we calculate this triangle?

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

c = 7.25 ; ; alpha = 14.04° ; ; beta = 90° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 14.04 ° - 90 ° = 75.96 ° ; ;

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 = 7.25 * fraction{ sin 14° 2'24" }{ sin 75° 57'36" } = 1.81 ; ;

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{ 1.81**2+7.25**2 - 2 * 1.81 * 7.25 * cos(90° ) } ; ; b = 7.47 ; ;


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 = 1.81 ; ; b = 7.47 ; ; c = 7.25 ; ;

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

p = a+b+c = 1.81+7.47+7.25 = 16.54 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.54 }{ 2 } = 8.27 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.27 * (8.27-1.81)(8.27-7.47)(8.27-7.25) } ; ; T = sqrt{ 43.19 } = 6.57 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 6.57 }{ 1.81 } = 7.25 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 6.57 }{ 7.47 } = 1.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 6.57 }{ 7.25 } = 1.81 ; ;

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{ 7.47**2+7.25**2-1.81**2 }{ 2 * 7.47 * 7.25 } ) = 14° 2'24" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1.81**2+7.25**2-7.47**2 }{ 2 * 1.81 * 7.25 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 14° 2'24" - 90° = 75° 57'36" ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 6.57 }{ 8.27 } = 0.79 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.81 }{ 2 * sin 14° 2'24" } = 3.74 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.47**2+2 * 7.25**2 - 1.81**2 } }{ 2 } = 7.306 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.25**2+2 * 1.81**2 - 7.47**2 } }{ 2 } = 3.737 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.47**2+2 * 1.81**2 - 7.25**2 } }{ 2 } = 4.053 ; ;
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