Triangle calculator

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

You have entered side a, c and angle α.

Triangle has two solutions: a=63; b=103.0022383815; c=160 and a=63; b=210.0054848419; c=160.

#1 Obtuse scalene triangle.

Sides: a = 63   b = 103.0022383815   c = 160

Area: T = 1713.232198219
Perimeter: p = 326.0022383816
Semiperimeter: s = 163.0011191908

Angle ∠ A = α = 12° = 0.20994395102 rad
Angle ∠ B = β = 19.87224253972° = 19°52'21″ = 0.34768392535 rad
Angle ∠ C = γ = 148.1287574603° = 148°7'39″ = 2.58553138898 rad

Height: ha = 54.38883168948
Height: hb = 33.26658705308
Height: hc = 21.41553997773

Median: ma = 130.8154737457
Median: mb = 110.1465936067
Median: mc = 29.82202202513

Inradius: r = 10.51105488011
Circumradius: R = 151.5076631859

Vertex coordinates: A[160; 0] B[0; 0] C[59.2488465401; 21.41553997773]
Centroid: CG[73.08328218003; 7.13884665924]
Coordinates of the circumscribed circle: U[80; -128.6633357244]
Coordinates of the inscribed circle: I[59.99988080923; 10.51105488011]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168° = 0.20994395102 rad
∠ B' = β' = 160.1287574603° = 160°7'39″ = 0.34768392535 rad
∠ C' = γ' = 31.87224253972° = 31°52'21″ = 2.58553138898 rad




How did we calculate this triangle?

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

a = 63 ; ; c = 160 ; ; alpha = 12° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 63**2 = 160**2 + b**2 - 2 * 160 * b * cos(12° ) ; ; ; ; ; ; b**2 -313.007b +21631 =0 ; ; a=1; b=-313.007; c=21631 ; ; D = b**2 - 4ac = 313.007**2 - 4 * 1 * 21631 = 11449.5274313 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 313.01 ± sqrt{ 11449.53 } }{ 2 } ; ; b_{1,2} = 156.50361612 ± 53.5012323019 ; ; b_{1} = 210.004848422 ; ; b_{2} = 103.002383818 ; ;
 ; ; text{ Factored form: } ; ; (b -210.004848422) (b -103.002383818) = 0 ; ; ; ; b > 0 ; ; ; ; b = 210.005 ; ;


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 = 63 ; ; b = 103 ; ; c = 160 ; ;

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

p = a+b+c = 63+103+160 = 326 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 326 }{ 2 } = 163 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 163 * (163-63)(163-103)(163-160) } ; ; T = sqrt{ 2935163.82 } = 1713.23 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1713.23 }{ 63 } = 54.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1713.23 }{ 103 } = 33.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1713.23 }{ 160 } = 21.42 ; ;

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{ 103**2+160**2-63**2 }{ 2 * 103 * 160 } ) = 12° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 63**2+160**2-103**2 }{ 2 * 63 * 160 } ) = 19° 52'21" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 63**2+103**2-160**2 }{ 2 * 63 * 103 } ) = 148° 7'39" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1713.23 }{ 163 } = 10.51 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 63 }{ 2 * sin 12° } = 151.51 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 103**2+2 * 160**2 - 63**2 } }{ 2 } = 130.815 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 160**2+2 * 63**2 - 103**2 } }{ 2 } = 110.146 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 103**2+2 * 63**2 - 160**2 } }{ 2 } = 29.82 ; ;







#2 Obtuse scalene triangle.

Sides: a = 63   b = 210.0054848419   c = 160

Area: T = 3492.997704918
Perimeter: p = 433.0054848419
Semiperimeter: s = 216.502242421

Angle ∠ A = α = 12° = 0.20994395102 rad
Angle ∠ B = β = 136.1287574603° = 136°7'39″ = 2.37658743796 rad
Angle ∠ C = γ = 31.87224253972° = 31°52'21″ = 0.55662787638 rad

Height: ha = 110.8898795212
Height: hb = 33.26658705308
Height: hc = 43.66224631148

Median: ma = 184.0087522074
Median: mb = 61.3110610094
Median: mc = 132.7998788322

Inradius: r = 16.1343754908
Circumradius: R = 151.5076631859

Vertex coordinates: A[160; 0] B[0; 0] C[-45.41657386238; 43.66224631148]
Centroid: CG[38.19547537921; 14.55441543716]
Coordinates of the circumscribed circle: U[80; 128.6633357244]
Coordinates of the inscribed circle: I[6.49875757903; 16.1343754908]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 168° = 0.20994395102 rad
∠ B' = β' = 43.87224253972° = 43°52'21″ = 2.37658743796 rad
∠ C' = γ' = 148.1287574603° = 148°7'39″ = 0.55662787638 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 63 ; ; c = 160 ; ; alpha = 12° ; ; : Nr. 1

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 63**2 = 160**2 + b**2 - 2 * 160 * b * cos(12° ) ; ; ; ; ; ; b**2 -313.007b +21631 =0 ; ; a=1; b=-313.007; c=21631 ; ; D = b**2 - 4ac = 313.007**2 - 4 * 1 * 21631 = 11449.5274313 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 313.01 ± sqrt{ 11449.53 } }{ 2 } ; ; b_{1,2} = 156.50361612 ± 53.5012323019 ; ; b_{1} = 210.004848422 ; ; b_{2} = 103.002383818 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (b -210.004848422) (b -103.002383818) = 0 ; ; ; ; b > 0 ; ; ; ; b = 210.005 ; ; : 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 = 63 ; ; b = 210 ; ; c = 160 ; ;

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

p = a+b+c = 63+210+160 = 433 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 433 }{ 2 } = 216.5 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 216.5 * (216.5-63)(216.5-210)(216.5-160) } ; ; T = sqrt{ 12201028.39 } = 3493 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3493 }{ 63 } = 110.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3493 }{ 210 } = 33.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3493 }{ 160 } = 43.66 ; ;

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{ 210**2+160**2-63**2 }{ 2 * 210 * 160 } ) = 12° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 63**2+160**2-210**2 }{ 2 * 63 * 160 } ) = 136° 7'39" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 63**2+210**2-160**2 }{ 2 * 63 * 210 } ) = 31° 52'21" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3493 }{ 216.5 } = 16.13 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 63 }{ 2 * sin 12° } = 151.51 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 210**2+2 * 160**2 - 63**2 } }{ 2 } = 184.008 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 160**2+2 * 63**2 - 210**2 } }{ 2 } = 61.311 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 210**2+2 * 63**2 - 160**2 } }{ 2 } = 132.799 ; ;
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