Triangle calculator

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

You have entered side a, b and height ha.

Triangle has two solutions: a=7.5; b=5.3; c=11.12877733506 and a=7.5; b=5.3; c=6.69772128723.

#1 Obtuse scalene triangle.

Sides: a = 7.5   b = 5.3   c = 11.12877733506

Area: T = 17.25
Perimeter: p = 23.92877733506
Semiperimeter: s = 11.96438866753

Angle ∠ A = α = 35.80110000919° = 35°48'4″ = 0.62548453271 rad
Angle ∠ B = β = 24.41773446738° = 24°25'2″ = 0.42661630592 rad
Angle ∠ C = γ = 119.7821655234° = 119°46'54″ = 2.09105842673 rad

Height: ha = 4.6
Height: hb = 6.50994339623
Height: hc = 3.11003507092

Median: ma = 7.86774118916
Median: mb = 9.11113209729
Median: mc = 3.34986064361

Inradius: r = 1.44218391337
Circumradius: R = 6.41105650824

Vertex coordinates: A[11.12877733506; 0] B[0; 0] C[6.82991892256; 3.11003507092]
Centroid: CG[5.98656541921; 1.03334502364]
Coordinates of the circumscribed circle: U[5.56438866753; -3.18441026586]
Coordinates of the inscribed circle: I[6.66438866753; 1.44218391337]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144.1998999908° = 144°11'56″ = 0.62548453271 rad
∠ B' = β' = 155.5832655326° = 155°34'58″ = 0.42661630592 rad
∠ C' = γ' = 60.21883447657° = 60°13'6″ = 2.09105842673 rad




How did we calculate this triangle?

1. Input data entered: side a, b and height ha.

a = 7.5 ; ; b = 5.3 ; ; ha = 4.6 ; ;


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 = 7.5 ; ; b = 5.3 ; ; c = 11.13 ; ;

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

p = a+b+c = 7.5+5.3+11.13 = 23.93 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 23.93 }{ 2 } = 11.96 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 11.96 * (11.96-7.5)(11.96-5.3)(11.96-11.13) } ; ; T = sqrt{ 297.56 } = 17.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.25 }{ 7.5 } = 4.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.25 }{ 5.3 } = 6.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.25 }{ 11.13 } = 3.1 ; ;

6. 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{ 7.5**2-5.3**2-11.13**2 }{ 2 * 5.3 * 11.13 } ) = 35° 48'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.3**2-7.5**2-11.13**2 }{ 2 * 7.5 * 11.13 } ) = 24° 25'2" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 11.13**2-7.5**2-5.3**2 }{ 2 * 5.3 * 7.5 } ) = 119° 46'54" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.25 }{ 11.96 } = 1.44 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.5 }{ 2 * sin 35° 48'4" } = 6.41 ; ;





#2 Acute scalene triangle.

Sides: a = 7.5   b = 5.3   c = 6.69772128723

Area: T = 17.25
Perimeter: p = 19.49772128723
Semiperimeter: s = 9.74986064361

Angle ∠ A = α = 76.44001541856° = 76°24'1″ = 1.33334342396 rad
Angle ∠ B = β = 43.38215010487° = 43°22'53″ = 0.75771500278 rad
Angle ∠ C = γ = 60.21883447657° = 60°13'6″ = 1.05110083863 rad

Height: ha = 4.6
Height: hb = 6.50994339623
Height: hc = 5.15113966568

Median: ma = 4.73437965871
Median: mb = 6.59876382235
Median: mc = 5.56438866753

Inradius: r = 1.76994836809
Circumradius: R = 3.85881769808

Vertex coordinates: A[6.69772128723; 0] B[0; 0] C[5.45109735355; 5.15113966568]
Centroid: CG[4.04993954693; 1.71771322189]
Coordinates of the circumscribed circle: U[3.34986064361; 1.91663414494]
Coordinates of the inscribed circle: I[4.44986064361; 1.76994836809]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 103.6599845814° = 103°35'59″ = 1.33334342396 rad
∠ B' = β' = 136.6188498951° = 136°37'7″ = 0.75771500278 rad
∠ C' = γ' = 119.7821655234° = 119°46'54″ = 1.05110083863 rad

Calculate another triangle

How did we calculate this triangle?

1. Input data entered: side a, b and height ha.

a = 7.5 ; ; b = 5.3 ; ; ha = 4.6 ; ; : 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 = 7.5 ; ; b = 5.3 ; ; c = 6.7 ; ;

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

p = a+b+c = 7.5+5.3+6.7 = 19.5 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 19.5 }{ 2 } = 9.75 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 9.75 * (9.75-7.5)(9.75-5.3)(9.75-6.7) } ; ; T = sqrt{ 297.56 } = 17.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 17.25 }{ 7.5 } = 4.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 17.25 }{ 5.3 } = 6.51 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 17.25 }{ 6.7 } = 5.15 ; ;

6. 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{ 7.5**2-5.3**2-6.7**2 }{ 2 * 5.3 * 6.7 } ) = 76° 24'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.3**2-7.5**2-6.7**2 }{ 2 * 7.5 * 6.7 } ) = 43° 22'53" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.7**2-7.5**2-5.3**2 }{ 2 * 5.3 * 7.5 } ) = 60° 13'6" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 17.25 }{ 9.75 } = 1.77 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.5 }{ 2 * sin 76° 24'1" } = 3.86 ; ;




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