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=700; b=100.0769671742; c=800 and a=700; b=1498.956565149; c=800.

#1 Obtuse scalene triangle.

Sides: a = 700   b = 100.0769671742   c = 800

Area: T = 1396.952247159
Perimeter: p = 1600.076967174
Semiperimeter: s = 800.0354835871

Angle ∠ A = α = 2° = 0.0354906585 rad
Angle ∠ B = β = 0.28658564745° = 0°17'9″ = 0.00549891367 rad
Angle ∠ C = γ = 177.7144143526° = 177°42'51″ = 3.10216969319 rad

Height: ha = 3.9911292776
Height: hb = 27.9219597362
Height: hc = 3.4922381179

Median: ma = 450.0087743935
Median: mb = 749.9987676796
Median: mc = 300.0121615777

Inradius: r = 1.74661145552
Circumradius: R = 10028.79879218

Vertex coordinates: A[800; 0] B[0; 0] C[699.9911287998; 3.4922381179]
Centroid: CG[499.9977096; 1.16441270597]
Coordinates of the circumscribed circle: U[400; -10020.81877189]
Coordinates of the inscribed circle: I[699.9655164129; 1.74661145552]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 178° = 0.0354906585 rad
∠ B' = β' = 179.7144143526° = 179°42'51″ = 0.00549891367 rad
∠ C' = γ' = 2.28658564745° = 2°17'9″ = 3.10216969319 rad




How did we calculate this triangle?

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

a = 700 ; ; c = 800 ; ; alpha = 2° ; ;

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 700**2 = 800**2 + b**2 - 2 * 800 * b * cos 2° ; ; ; ; ; ; b**2 -1599.025b +150000 =0 ; ; p=1; q=-1599.025; r=150000 ; ; D = q**2 - 4pr = 1599.025**2 - 4 * 1 * 150000 = 1956881.98433 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 1599.03 ± sqrt{ 1956881.98 } }{ 2 } ; ; b_{1,2} = 799.51266162 ± 699.442989873 ; ; b_{1} = 1498.95565149 ; ; b_{2} = 100.069671747 ; ; ; ; text{ Factored form: } ; ; (b -1498.95565149) (b -100.069671747) = 0 ; ; ; ; b > 0 ; ; ; ; b = 1498.956 ; ;


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 = 700 ; ; b = 100.07 ; ; c = 800 ; ;

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

p = a+b+c = 700+100.07+800 = 1600.07 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1600.07 }{ 2 } = 800.03 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 800.03 * (800.03-700)(800.03-100.07)(800.03-800) } ; ; T = sqrt{ 1951476.21 } = 1396.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1396.95 }{ 700 } = 3.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1396.95 }{ 100.07 } = 27.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1396.95 }{ 800 } = 3.49 ; ;

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{ 100.07**2+800**2-700**2 }{ 2 * 100.07 * 800 } ) = 2° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 700**2+800**2-100.07**2 }{ 2 * 700 * 800 } ) = 0° 17'9" ; ; gamma = 180° - alpha - beta = 180° - 2° - 0° 17'9" = 177° 42'51" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1396.95 }{ 800.03 } = 1.75 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 700 }{ 2 * sin 2° } = 10028.8 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 100.07**2+2 * 800**2 - 700**2 } }{ 2 } = 450.008 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 800**2+2 * 700**2 - 100.07**2 } }{ 2 } = 749.998 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 100.07**2+2 * 700**2 - 800**2 } }{ 2 } = 300.012 ; ;







#2 Obtuse scalene triangle.

Sides: a = 700   b = 1498.956565149   c = 800

Area: T = 20925.11991265
Perimeter: p = 2998.956565149
Semiperimeter: s = 1499.478782574

Angle ∠ A = α = 2° = 0.0354906585 rad
Angle ∠ B = β = 175.7144143526° = 175°42'51″ = 3.06767903468 rad
Angle ∠ C = γ = 2.28658564745° = 2°17'9″ = 0.04398957217 rad

Height: ha = 59.78660546472
Height: hb = 27.9219597362
Height: hc = 52.31327978163

Median: ma = 1149.319893857
Median: mb = 57.29773709474
Median: mc = 1099.288796162

Inradius: r = 13.9554937357
Circumradius: R = 10028.79879217

Vertex coordinates: A[800; 0] B[0; 0] C[-698.0432528206; 52.31327978163]
Centroid: CG[33.98658239312; 17.43875992721]
Coordinates of the circumscribed circle: U[400; 10020.81877189]
Coordinates of the inscribed circle: I[0.52221742556; 13.9554937357]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 178° = 0.0354906585 rad
∠ B' = β' = 4.28658564745° = 4°17'9″ = 3.06767903468 rad
∠ C' = γ' = 177.7144143526° = 177°42'51″ = 0.04398957217 rad

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How did we calculate this triangle?

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

a = 700 ; ; c = 800 ; ; alpha = 2° ; ; : Nr. 1

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

a**2 = c**2 + b**2 - 2c b cos alpha ; ; ; ; 700**2 = 800**2 + b**2 - 2 * 800 * b * cos 2° ; ; ; ; ; ; b**2 -1599.025b +150000 =0 ; ; p=1; q=-1599.025; r=150000 ; ; D = q**2 - 4pr = 1599.025**2 - 4 * 1 * 150000 = 1956881.98433 ; ; D>0 ; ; ; ; b_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 1599.03 ± sqrt{ 1956881.98 } }{ 2 } ; ; b_{1,2} = 799.51266162 ± 699.442989873 ; ; b_{1} = 1498.95565149 ; ; b_{2} = 100.069671747 ; ; ; ; text{ Factored form: } ; ; (b -1498.95565149) (b -100.069671747) = 0 ; ; ; ; b > 0 ; ; ; ; b = 1498.956 ; ; : 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 = 700 ; ; b = 1498.96 ; ; c = 800 ; ;

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

p = a+b+c = 700+1498.96+800 = 2998.96 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2998.96 }{ 2 } = 1499.48 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1499.48 * (1499.48-700)(1499.48-1498.96)(1499.48-800) } ; ; T = sqrt{ 437860610.46 } = 20925.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 20925.12 }{ 700 } = 59.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 20925.12 }{ 1498.96 } = 27.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 20925.12 }{ 800 } = 52.31 ; ;

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{ 1498.96**2+800**2-700**2 }{ 2 * 1498.96 * 800 } ) = 2° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 700**2+800**2-1498.96**2 }{ 2 * 700 * 800 } ) = 175° 42'51" ; ; gamma = 180° - alpha - beta = 180° - 2° - 175° 42'51" = 2° 17'9" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 20925.12 }{ 1499.48 } = 13.95 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 700 }{ 2 * sin 2° } = 10028.8 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 1498.96**2+2 * 800**2 - 700**2 } }{ 2 } = 1149.319 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 800**2+2 * 700**2 - 1498.96**2 } }{ 2 } = 57.297 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 1498.96**2+2 * 700**2 - 800**2 } }{ 2 } = 1099.288 ; ;
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