Triangle calculator

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

You have entered side a, b and angle α.

Triangle has two solutions: a=54; b=62; c=11.05664267867 and a=54; b=62; c=83.933308416.

#1 Obtuse scalene triangle.

Sides: a = 54   b = 62   c = 11.05664267867

Area: T = 220.3154958523
Perimeter: p = 127.0566426787
Semiperimeter: s = 63.52882133934

Angle ∠ A = α = 40° = 0.69881317008 rad
Angle ∠ B = β = 132.4377368296° = 132°26'15″ = 2.31114681294 rad
Angle ∠ C = γ = 7.56326317037° = 7°33'45″ = 0.13219928233 rad

Height: ha = 8.16598132786
Height: hb = 7.10769341459
Height: hc = 39.85328318006

Median: ma = 35.41435890111
Median: mb = 23.62546118835
Median: mc = 57.8744336771

Inradius: r = 3.46879860609
Circumradius: R = 42.00545433252

Vertex coordinates: A[11.05664267867; 0] B[0; 0] C[-36.43883286866; 39.85328318006]
Centroid: CG[-8.46106339666; 13.28442772669]
Coordinates of the circumscribed circle: U[5.52882133934; 41.63991704605]
Coordinates of the inscribed circle: I[1.52882133934; 3.46879860609]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140° = 0.69881317008 rad
∠ B' = β' = 47.56326317037° = 47°33'45″ = 2.31114681294 rad
∠ C' = γ' = 172.4377368296° = 172°26'15″ = 0.13219928233 rad




How did we calculate this triangle?

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

a = 54 ; ; b = 62 ; ; alpha = 40° ; ;

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 54**2 = 62**2 + c**2 - 2 * 62 * c * cos(40° ) ; ; ; ; ; ; c**2 -94.99c +928 =0 ; ; a=1; b=-94.99; c=928 ; ; D = b**2 - 4ac = 94.99**2 - 4 * 1 * 928 = 5311.0071899 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 94.99 ± sqrt{ 5311.01 } }{ 2 } ; ; c_{1,2} = 47.49475547 ± 36.4383286866 ; ; c_{1} = 83.9330841566 ; ; c_{2} = 11.0564267834 ; ; ; ; (c -83.9330841566) (c -11.0564267834) = 0 ; ;
 ; ; c > 0 ; ; ; ; c = 83.933 ; ;


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 = 54 ; ; b = 62 ; ; c = 11.06 ; ;

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

p = a+b+c = 54+62+11.06 = 127.06 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 127.06 }{ 2 } = 63.53 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 63.53 * (63.53-54)(63.53-62)(63.53-11.06) } ; ; T = sqrt{ 48538.68 } = 220.31 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 220.31 }{ 54 } = 8.16 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 220.31 }{ 62 } = 7.11 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 220.31 }{ 11.06 } = 39.85 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 54**2-62**2-11.06**2 }{ 2 * 62 * 11.06 } ) = 40° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 62**2-54**2-11.06**2 }{ 2 * 54 * 11.06 } ) = 132° 26'15" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.06**2-54**2-62**2 }{ 2 * 62 * 54 } ) = 7° 33'45" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 220.31 }{ 63.53 } = 3.47 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 54 }{ 2 * sin 40° } = 42 ; ;





#2 Obtuse scalene triangle.

Sides: a = 54   b = 62   c = 83.933308416

Area: T = 1672.486554277
Perimeter: p = 199.933308416
Semiperimeter: s = 99.967654208

Angle ∠ A = α = 40° = 0.69881317008 rad
Angle ∠ B = β = 47.56326317037° = 47°33'45″ = 0.83301245241 rad
Angle ∠ C = γ = 92.43773682963° = 92°26'15″ = 1.61333364286 rad

Height: ha = 61.94439089913
Height: hb = 53.95111465408
Height: hc = 39.85328318006

Median: ma = 68.66986340938
Median: mb = 63.39985907439
Median: mc = 40.23444298561

Inradius: r = 16.73304530893
Circumradius: R = 42.00545433252

Vertex coordinates: A[83.933308416; 0] B[0; 0] C[36.43883286866; 39.85328318006]
Centroid: CG[40.12438042822; 13.28442772669]
Coordinates of the circumscribed circle: U[41.967654208; -1.78663386599]
Coordinates of the inscribed circle: I[37.967654208; 16.73304530893]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140° = 0.69881317008 rad
∠ B' = β' = 132.4377368296° = 132°26'15″ = 0.83301245241 rad
∠ C' = γ' = 87.56326317037° = 87°33'45″ = 1.61333364286 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 54 ; ; b = 62 ; ; alpha = 40° ; ; : Nr. 1

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 54**2 = 62**2 + c**2 - 2 * 62 * c * cos(40° ) ; ; ; ; ; ; c**2 -94.99c +928 =0 ; ; a=1; b=-94.99; c=928 ; ; D = b**2 - 4ac = 94.99**2 - 4 * 1 * 928 = 5311.0071899 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 94.99 ± sqrt{ 5311.01 } }{ 2 } ; ; c_{1,2} = 47.49475547 ± 36.4383286866 ; ; c_{1} = 83.9330841566 ; ; c_{2} = 11.0564267834 ; ; ; ; (c -83.9330841566) (c -11.0564267834) = 0 ; ; : Nr. 1
 ; ; c > 0 ; ; ; ; c = 83.933 ; ; : 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 = 54 ; ; b = 62 ; ; c = 83.93 ; ;

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

p = a+b+c = 54+62+83.93 = 199.93 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 199.93 }{ 2 } = 99.97 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 99.97 * (99.97-54)(99.97-62)(99.97-83.93) } ; ; T = sqrt{ 2797207.89 } = 1672.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1672.49 }{ 54 } = 61.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1672.49 }{ 62 } = 53.95 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1672.49 }{ 83.93 } = 39.85 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 54**2-62**2-83.93**2 }{ 2 * 62 * 83.93 } ) = 40° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 62**2-54**2-83.93**2 }{ 2 * 54 * 83.93 } ) = 47° 33'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 83.93**2-54**2-62**2 }{ 2 * 62 * 54 } ) = 92° 26'15" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1672.49 }{ 99.97 } = 16.73 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 54 }{ 2 * sin 40° } = 42 ; ; : Nr. 1




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