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=8.3; b=10.7; c=4.076593068 and a=8.3; b=10.7; c=11.18876289319.

#1 Obtuse scalene triangle.

Sides: a = 8.3   b = 10.7   c = 4.076593068

Area: T = 15.28441880223
Perimeter: p = 23.076593068
Semiperimeter: s = 11.538796534

Angle ∠ A = α = 44.5° = 44°30' = 0.77766715171 rad
Angle ∠ B = β = 115.3677049386° = 115°22'1″ = 2.01435348601 rad
Angle ∠ C = γ = 20.13329506144° = 20°7'59″ = 0.35113862764 rad

Height: ha = 3.68329368728
Height: hb = 2.85768575743
Height: hc = 7.5499729128

Median: ma = 6.95219137979
Median: mb = 3.75988702364
Median: mc = 9.35661048131

Inradius: r = 1.32546865952
Circumradius: R = 5.92108805068

Vertex coordinates: A[4.076593068; 0] B[0; 0] C[-3.55658491259; 7.5499729128]
Centroid: CG[0.1733360518; 2.54999097093]
Coordinates of the circumscribed circle: U[2.038796534; 5.55990937434]
Coordinates of the inscribed circle: I[0.838796534; 1.32546865952]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5° = 135°30' = 0.77766715171 rad
∠ B' = β' = 64.63329506144° = 64°37'59″ = 2.01435348601 rad
∠ C' = γ' = 159.8677049386° = 159°52'1″ = 0.35113862764 rad




How did we calculate this triangle?

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

a = 8.3 ; ; b = 10.7 ; ; alpha = 44.5° ; ;

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 8.3**2 = 10.7**2 + c**2 - 2 * 10.7 * c * cos 44° 30' ; ; ; ; ; ; c**2 -15.264c +45.6 =0 ; ; p=1; q=-15.264; r=45.6 ; ; D = q**2 - 4pr = 15.264**2 - 4 * 1 * 45.6 = 50.576252026 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.26 ± sqrt{ 50.58 } }{ 2 } ; ; c_{1,2} = 7.63177981 ± 3.55584912595 ; ; c_{1} = 11.1876289359 ; ; c_{2} = 4.07593068405 ; ; ; ; text{ Factored form: } ; ; (c -11.1876289359) (c -4.07593068405) = 0 ; ; ; ; c > 0 ; ; ; ; c = 11.188 ; ;


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 = 8.3 ; ; b = 10.7 ; ; c = 4.08 ; ;

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

p = a+b+c = 8.3+10.7+4.08 = 23.08 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.08 }{ 2 } = 11.54 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.54 * (11.54-8.3)(11.54-10.7)(11.54-4.08) } ; ; T = sqrt{ 233.61 } = 15.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15.28 }{ 8.3 } = 3.68 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15.28 }{ 10.7 } = 2.86 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15.28 }{ 4.08 } = 7.5 ; ;

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{ 10.7**2+4.08**2-8.3**2 }{ 2 * 10.7 * 4.08 } ) = 44° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.3**2+4.08**2-10.7**2 }{ 2 * 8.3 * 4.08 } ) = 115° 22'1" ; ; gamma = 180° - alpha - beta = 180° - 44° 30' - 115° 22'1" = 20° 7'59" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15.28 }{ 11.54 } = 1.32 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.3 }{ 2 * sin 44° 30' } = 5.92 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.7**2+2 * 4.08**2 - 8.3**2 } }{ 2 } = 6.952 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.08**2+2 * 8.3**2 - 10.7**2 } }{ 2 } = 3.759 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.7**2+2 * 8.3**2 - 4.08**2 } }{ 2 } = 9.356 ; ;







#2 Acute scalene triangle.

Sides: a = 8.3   b = 10.7   c = 11.18876289319

Area: T = 41.9522093287
Perimeter: p = 30.18876289319
Semiperimeter: s = 15.09438144659

Angle ∠ A = α = 44.5° = 44°30' = 0.77766715171 rad
Angle ∠ B = β = 64.63329506144° = 64°37'59″ = 1.12880577935 rad
Angle ∠ C = γ = 70.86770493856° = 70°52'1″ = 1.2376863343 rad

Height: ha = 10.10989381414
Height: hb = 7.84215127639
Height: hc = 7.5499729128

Median: ma = 10.12993642722
Median: mb = 8.2710672316
Median: mc = 7.77216947779

Inradius: r = 2.77994228809
Circumradius: R = 5.92108805068

Vertex coordinates: A[11.18876289319; 0] B[0; 0] C[3.55658491259; 7.5499729128]
Centroid: CG[4.91444926859; 2.54999097093]
Coordinates of the circumscribed circle: U[5.59438144659; 1.94106353846]
Coordinates of the inscribed circle: I[4.39438144659; 2.77994228809]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135.5° = 135°30' = 0.77766715171 rad
∠ B' = β' = 115.3677049386° = 115°22'1″ = 1.12880577935 rad
∠ C' = γ' = 109.1332950614° = 109°7'59″ = 1.2376863343 rad

Calculate another triangle

How did we calculate this triangle?

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

a = 8.3 ; ; b = 10.7 ; ; alpha = 44.5° ; ; : Nr. 1

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

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 8.3**2 = 10.7**2 + c**2 - 2 * 10.7 * c * cos 44° 30' ; ; ; ; ; ; c**2 -15.264c +45.6 =0 ; ; p=1; q=-15.264; r=45.6 ; ; D = q**2 - 4pr = 15.264**2 - 4 * 1 * 45.6 = 50.576252026 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 15.26 ± sqrt{ 50.58 } }{ 2 } ; ; c_{1,2} = 7.63177981 ± 3.55584912595 ; ; c_{1} = 11.1876289359 ; ; c_{2} = 4.07593068405 ; ; ; ; text{ Factored form: } ; ; (c -11.1876289359) (c -4.07593068405) = 0 ; ; ; ; c > 0 ; ; ; ; c = 11.188 ; ; : 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 = 8.3 ; ; b = 10.7 ; ; c = 11.19 ; ;

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

p = a+b+c = 8.3+10.7+11.19 = 30.19 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.19 }{ 2 } = 15.09 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.09 * (15.09-8.3)(15.09-10.7)(15.09-11.19) } ; ; T = sqrt{ 1759.98 } = 41.95 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 41.95 }{ 8.3 } = 10.11 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 41.95 }{ 10.7 } = 7.84 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 41.95 }{ 11.19 } = 7.5 ; ;

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{ 10.7**2+11.19**2-8.3**2 }{ 2 * 10.7 * 11.19 } ) = 44° 30' ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 8.3**2+11.19**2-10.7**2 }{ 2 * 8.3 * 11.19 } ) = 64° 37'59" ; ; gamma = 180° - alpha - beta = 180° - 44° 30' - 64° 37'59" = 70° 52'1" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 41.95 }{ 15.09 } = 2.78 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 8.3 }{ 2 * sin 44° 30' } = 5.92 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.7**2+2 * 11.19**2 - 8.3**2 } }{ 2 } = 10.129 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 11.19**2+2 * 8.3**2 - 10.7**2 } }{ 2 } = 8.271 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 10.7**2+2 * 8.3**2 - 11.19**2 } }{ 2 } = 7.772 ; ;
Calculate another triangle

Look also our friend's collection of math examples and problems:

See more informations about triangles or more information about solving triangles.