Triangle calculator

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

You have entered side b, c and angle γ.

Triangle has two solutions: a=2.45549062585; b=33; c=32 and a=26.47875894296; b=33; c=32.

#1 Obtuse scalene triangle.

Sides: a = 2.45549062585   b = 33   c = 32

Area: T = 36.40765096341
Perimeter: p = 67.45549062585
Semiperimeter: s = 33.72774531292

Angle ∠ A = α = 3.95437798724° = 3°57'14″ = 0.06990064767 rad
Angle ∠ B = β = 112.0466220128° = 112°2'46″ = 1.95655754556 rad
Angle ∠ C = γ = 64° = 1.11770107213 rad

Height: ha = 29.66602035279
Height: hb = 2.20664551293
Height: hc = 2.27554068521

Median: ma = 32.48106613051
Median: mb = 15.58108626966
Median: mc = 17.07437600536

Inradius: r = 1.07994325173
Circumradius: R = 17.80216310476

Vertex coordinates: A[32; 0] B[0; 0] C[-0.9211459926; 2.27554068521]
Centroid: CG[10.3659513358; 0.75884689507]
Coordinates of the circumscribed circle: U[16; 7.80437214171]
Coordinates of the inscribed circle: I[0.72774531292; 1.07994325173]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 176.0466220128° = 176°2'46″ = 0.06990064767 rad
∠ B' = β' = 67.95437798724° = 67°57'14″ = 1.95655754556 rad
∠ C' = γ' = 116° = 1.11770107213 rad




How did we calculate this triangle?

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

b = 33 ; ; c = 32 ; ; gamma = 64° ; ;

2. From angle γ, side b and c we calculate side a - by using the law of cosines and quadratic equation:

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 32**2 = 33**2 + a**2 - 2 * 33 * a * cos(64° ) ; ; ; ; ; ; a**2 -28.932a +65 =0 ; ; a=1; b=-28.932; c=65 ; ; D = b**2 - 4ac = 28.932**2 - 4 * 1 * 65 = 577.089306741 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 28.93 ± sqrt{ 577.09 } }{ 2 } ; ; a_{1,2} = 14.46624784 ± 12.0113415856 ; ; a_{1} = 26.4775894256 ; ; a_{2} = 2.45490625443 ; ; ; ; text{ Factored form: } ; ;
(a -26.4775894256) (a -2.45490625443) = 0 ; ; ; ; a > 0 ; ; ; ; a = 26.478 ; ;


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 = 2.45 ; ; b = 33 ; ; c = 32 ; ;

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

p = a+b+c = 2.45+33+32 = 67.45 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67.45 }{ 2 } = 33.73 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.73 * (33.73-2.45)(33.73-33)(33.73-32) } ; ; T = sqrt{ 1325.43 } = 36.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 36.41 }{ 2.45 } = 29.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 36.41 }{ 33 } = 2.21 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 36.41 }{ 32 } = 2.28 ; ;

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{ 33**2+32**2-2.45**2 }{ 2 * 33 * 32 } ) = 3° 57'14" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 2.45**2+32**2-33**2 }{ 2 * 2.45 * 32 } ) = 112° 2'46" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 2.45**2+33**2-32**2 }{ 2 * 2.45 * 33 } ) = 64° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 36.41 }{ 33.73 } = 1.08 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.45 }{ 2 * sin 3° 57'14" } = 17.8 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 32**2 - 2.45**2 } }{ 2 } = 32.481 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 2.45**2 - 33**2 } }{ 2 } = 15.581 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 2.45**2 - 32**2 } }{ 2 } = 17.074 ; ;







#2 Acute scalene triangle.

Sides: a = 26.47875894296   b = 33   c = 32

Area: T = 392.6655345705
Perimeter: p = 91.47875894296
Semiperimeter: s = 45.73987947148

Angle ∠ A = α = 48.04662201276° = 48°2'46″ = 0.83985647344 rad
Angle ∠ B = β = 67.95437798724° = 67°57'14″ = 1.18660171979 rad
Angle ∠ C = γ = 64° = 1.11770107213 rad

Height: ha = 29.66602035279
Height: hb = 23.79878997397
Height: hc = 24.54215841065

Median: ma = 29.68655910249
Median: mb = 24.2965706843
Median: mc = 25.27990698207

Inradius: r = 8.58549517495
Circumradius: R = 17.80216310476

Vertex coordinates: A[32; 0] B[0; 0] C[9.93884803438; 24.54215841065]
Centroid: CG[13.97994934479; 8.18105280355]
Coordinates of the circumscribed circle: U[16; 7.80437214171]
Coordinates of the inscribed circle: I[12.73987947148; 8.58549517495]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.9543779872° = 131°57'14″ = 0.83985647344 rad
∠ B' = β' = 112.0466220128° = 112°2'46″ = 1.18660171979 rad
∠ C' = γ' = 116° = 1.11770107213 rad

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How did we calculate this triangle?

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

b = 33 ; ; c = 32 ; ; gamma = 64° ; ; : Nr. 1

2. From angle γ, side b and c we calculate side a - by using the law of cosines and quadratic equation:

c**2 = b**2 + a**2 - 2b a cos gamma ; ; ; ; 32**2 = 33**2 + a**2 - 2 * 33 * a * cos(64° ) ; ; ; ; ; ; a**2 -28.932a +65 =0 ; ; a=1; b=-28.932; c=65 ; ; D = b**2 - 4ac = 28.932**2 - 4 * 1 * 65 = 577.089306741 ; ; D>0 ; ; ; ; a_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 28.93 ± sqrt{ 577.09 } }{ 2 } ; ; a_{1,2} = 14.46624784 ± 12.0113415856 ; ; a_{1} = 26.4775894256 ; ; a_{2} = 2.45490625443 ; ; ; ; text{ Factored form: } ; ; : Nr. 1
(a -26.4775894256) (a -2.45490625443) = 0 ; ; ; ; a > 0 ; ; ; ; a = 26.478 ; ; : Nr. 1


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 = 26.48 ; ; b = 33 ; ; c = 32 ; ;

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

p = a+b+c = 26.48+33+32 = 91.48 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 91.48 }{ 2 } = 45.74 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 45.74 * (45.74-26.48)(45.74-33)(45.74-32) } ; ; T = sqrt{ 154186.07 } = 392.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 392.67 }{ 26.48 } = 29.66 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 392.67 }{ 33 } = 23.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 392.67 }{ 32 } = 24.54 ; ;

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{ 33**2+32**2-26.48**2 }{ 2 * 33 * 32 } ) = 48° 2'46" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 26.48**2+32**2-33**2 }{ 2 * 26.48 * 32 } ) = 67° 57'14" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 26.48**2+33**2-32**2 }{ 2 * 26.48 * 33 } ) = 64° ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 392.67 }{ 45.74 } = 8.58 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 26.48 }{ 2 * sin 48° 2'46" } = 17.8 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 32**2 - 26.48**2 } }{ 2 } = 29.686 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 32**2+2 * 26.48**2 - 33**2 } }{ 2 } = 24.296 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 33**2+2 * 26.48**2 - 32**2 } }{ 2 } = 25.279 ; ;
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