Triangle calculator

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

You have entered side a, b and angle γ.

Acute scalene triangle.

Sides: a = 5.64   b = 4.917   c = 4.91766841664

Area: T = 11.3588313097
Perimeter: p = 15.47436841664
Semiperimeter: s = 7.73768420832

Angle ∠ A = α = 69.99547433355° = 69°59'41″ = 1.22216387303 rad
Angle ∠ B = β = 55.00552566645° = 55°19″ = 0.96600228347 rad
Angle ∠ C = γ = 55° = 0.96599310886 rad

Height: ha = 4.02877706018
Height: hb = 4.62200175298
Height: hc = 4.62203143063

Median: ma = 4.02877706112
Median: mb = 4.68548126266
Median: mc = 4.68550612271

Inradius: r = 1.46880812888
Circumradius: R = 3.00110815457

Vertex coordinates: A[4.91766841664; 0] B[0; 0] C[3.23545472188; 4.62203143063]
Centroid: CG[2.71770771284; 1.54401047688]
Coordinates of the circumscribed circle: U[2.45883420832; 1.72113496582]
Coordinates of the inscribed circle: I[2.82198420832; 1.46880812888]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 110.0055256665° = 110°19″ = 1.22216387303 rad
∠ B' = β' = 124.9954743335° = 124°59'41″ = 0.96600228347 rad
∠ C' = γ' = 125° = 0.96599310886 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle γ.

a = 5.64 ; ; b = 4.917 ; ; gamma = 55° ; ;

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

c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 5.64**2+4.92**2 - 2 * 5.64 * 4.92 * cos 55° } ; ; c = 4.92 ; ;
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 = 5.64 ; ; b = 4.92 ; ; c = 4.92 ; ;

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

p = a+b+c = 5.64+4.92+4.92 = 15.47 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 15.47 }{ 2 } = 7.74 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 7.74 * (7.74-5.64)(7.74-4.92)(7.74-4.92) } ; ; T = sqrt{ 129.01 } = 11.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11.36 }{ 5.64 } = 4.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11.36 }{ 4.92 } = 4.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11.36 }{ 4.92 } = 4.62 ; ;

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{ 4.92**2+4.92**2-5.64**2 }{ 2 * 4.92 * 4.92 } ) = 69° 59'41" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 5.64**2+4.92**2-4.92**2 }{ 2 * 5.64 * 4.92 } ) = 55° 19" ; ;
 gamma = 180° - alpha - beta = 180° - 69° 59'41" - 55° 19" = 55° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11.36 }{ 7.74 } = 1.47 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 5.64 }{ 2 * sin 69° 59'41" } = 3 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.92**2+2 * 4.92**2 - 5.64**2 } }{ 2 } = 4.028 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.92**2+2 * 5.64**2 - 4.92**2 } }{ 2 } = 4.685 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.92**2+2 * 5.64**2 - 4.92**2 } }{ 2 } = 4.685 ; ;
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