Triangle calculator

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

You have entered side a, b and angle β.

Triangle has two solutions: a=22.42; b=16.8; c=6.23774731119 and a=22.42; b=16.8; c=35.33774509269.

#1 Obtuse scalene triangle.

Sides: a = 22.42   b = 16.8   c = 6.23774731119

Area: T = 26.19332697901
Perimeter: p = 45.45774731119
Semiperimeter: s = 22.7298736556

Angle ∠ A = α = 150.0055198799° = 150°19″ = 2.61880846142 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 7.99548012007° = 7°59'41″ = 0.1439535604 rad

Height: ha = 2.3376598554
Height: hb = 3.11882464036
Height: hc = 8.39986798244

Median: ma = 5.90883784079
Median: mb = 14.15499553148
Median: mc = 19.56332738133

Inradius: r = 1.15224296445
Circumradius: R = 22.42435241655

Vertex coordinates: A[6.23774731119; 0] B[0; 0] C[20.78774620194; 8.39986798244]
Centroid: CG[9.00883117104; 2.87995599415]
Coordinates of the circumscribed circle: U[3.1198736556; 22.20655830433]
Coordinates of the inscribed circle: I[5.9298736556; 1.15224296445]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 29.99548012007° = 29°59'41″ = 2.61880846142 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 172.0055198799° = 172°19″ = 0.1439535604 rad




How did we calculate this triangle?

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

a = 22.42 ; ; b = 16.8 ; ; beta = 22° ; ;

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 16.8**2 = 22.42**2 + c**2 - 2 * 22.42 * c * cos 22° ; ; ; ; ; ; c**2 -41.575c +220.416 =0 ; ; p=1; q=-41.575; r=220.416 ; ; D = q**2 - 4pr = 41.575**2 - 4 * 1 * 220.416 = 846.80870883 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 41.57 ± sqrt{ 846.81 } }{ 2 } ; ; c_{1,2} = 20.78746202 ± 14.5499889075 ; ; c_{1} = 35.3374509275 ; ; c_{2} = 6.23747311253 ; ;
 ; ; text{ Factored form: } ; ; (c -35.3374509275) (c -6.23747311253) = 0 ; ; ; ; c > 0 ; ; ; ; c = 35.337 ; ;
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 = 22.42 ; ; b = 16.8 ; ; c = 6.24 ; ;

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

p = a+b+c = 22.42+16.8+6.24 = 45.46 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45.46 }{ 2 } = 22.73 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.73 * (22.73-22.42)(22.73-16.8)(22.73-6.24) } ; ; T = sqrt{ 686.09 } = 26.19 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 26.19 }{ 22.42 } = 2.34 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 26.19 }{ 16.8 } = 3.12 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 26.19 }{ 6.24 } = 8.4 ; ;

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{ 16.8**2+6.24**2-22.42**2 }{ 2 * 16.8 * 6.24 } ) = 150° 19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.42**2+6.24**2-16.8**2 }{ 2 * 22.42 * 6.24 } ) = 22° ; ; gamma = 180° - alpha - beta = 180° - 150° 19" - 22° = 7° 59'41" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 26.19 }{ 22.73 } = 1.15 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 22.42 }{ 2 * sin 150° 19" } = 22.42 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.8**2+2 * 6.24**2 - 22.42**2 } }{ 2 } = 5.908 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.24**2+2 * 22.42**2 - 16.8**2 } }{ 2 } = 14.15 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.8**2+2 * 22.42**2 - 6.24**2 } }{ 2 } = 19.563 ; ;







#2 Obtuse scalene triangle.

Sides: a = 22.42   b = 16.8   c = 35.33774509269

Area: T = 148.3943968072
Perimeter: p = 74.55774509269
Semiperimeter: s = 37.27987254634

Angle ∠ A = α = 29.99548012007° = 29°59'41″ = 0.52435080394 rad
Angle ∠ B = β = 22° = 0.38439724354 rad
Angle ∠ C = γ = 128.0055198799° = 128°19″ = 2.23441121787 rad

Height: ha = 13.23876421117
Height: hb = 17.666594858
Height: hc = 8.39986798244

Median: ma = 25.29547350056
Median: mb = 28.37549170748
Median: mc = 8.95990368064

Inradius: r = 3.98106609863
Circumradius: R = 22.42435241655

Vertex coordinates: A[35.33774509269; 0] B[0; 0] C[20.78774620194; 8.39986798244]
Centroid: CG[18.70883043154; 2.87995599415]
Coordinates of the circumscribed circle: U[17.66987254634; -13.80769032189]
Coordinates of the inscribed circle: I[20.47987254634; 3.98106609863]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.0055198799° = 150°19″ = 0.52435080394 rad
∠ B' = β' = 158° = 0.38439724354 rad
∠ C' = γ' = 51.99548012007° = 51°59'41″ = 2.23441121787 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 22.42 ; ; b = 16.8 ; ; beta = 22° ; ; : Nr. 1

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

b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 16.8**2 = 22.42**2 + c**2 - 2 * 22.42 * c * cos 22° ; ; ; ; ; ; c**2 -41.575c +220.416 =0 ; ; p=1; q=-41.575; r=220.416 ; ; D = q**2 - 4pr = 41.575**2 - 4 * 1 * 220.416 = 846.80870883 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 41.57 ± sqrt{ 846.81 } }{ 2 } ; ; c_{1,2} = 20.78746202 ± 14.5499889075 ; ; c_{1} = 35.3374509275 ; ; c_{2} = 6.23747311253 ; ; : Nr. 1
 ; ; text{ Factored form: } ; ; (c -35.3374509275) (c -6.23747311253) = 0 ; ; ; ; c > 0 ; ; ; ; c = 35.337 ; ; : 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 = 22.42 ; ; b = 16.8 ; ; c = 35.34 ; ;

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

p = a+b+c = 22.42+16.8+35.34 = 74.56 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74.56 }{ 2 } = 37.28 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.28 * (37.28-22.42)(37.28-16.8)(37.28-35.34) } ; ; T = sqrt{ 22020.77 } = 148.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 148.39 }{ 22.42 } = 13.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 148.39 }{ 16.8 } = 17.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 148.39 }{ 35.34 } = 8.4 ; ;

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{ 16.8**2+35.34**2-22.42**2 }{ 2 * 16.8 * 35.34 } ) = 29° 59'41" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 22.42**2+35.34**2-16.8**2 }{ 2 * 22.42 * 35.34 } ) = 22° ; ; gamma = 180° - alpha - beta = 180° - 29° 59'41" - 22° = 128° 19" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 148.39 }{ 37.28 } = 3.98 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 22.42 }{ 2 * sin 29° 59'41" } = 22.42 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.8**2+2 * 35.34**2 - 22.42**2 } }{ 2 } = 25.295 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.34**2+2 * 22.42**2 - 16.8**2 } }{ 2 } = 28.375 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 16.8**2+2 * 22.42**2 - 35.34**2 } }{ 2 } = 8.959 ; ;
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