Triangle calculator

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

You have entered side a, height ha and height hb.

Triangle has two solutions: a=6.6; b=4.4; c=9.33435952167 and a=6.6; b=4.4; c=6.22328611049.

#1 Obtuse scalene triangle.

Sides: a = 6.6   b = 4.4   c = 9.33435952167

Area: T = 13.2
Perimeter: p = 20.33435952167
Semiperimeter: s = 10.16767976084

Angle ∠ A = α = 40.00438517017° = 40°14″ = 0.69881989257 rad
Angle ∠ B = β = 25.37661709696° = 25°22'34″ = 0.4432897735 rad
Angle ∠ C = γ = 114.6219977329° = 114°37'12″ = 22.0004959929 rad

Height: ha = 4
Height: hb = 6
Height: hc = 2.82884920641

Median: ma = 6.50875340825
Median: mb = 7.77880460165
Median: mc = 3.11114305524

Inradius: r = 1.29883439337
Circumradius: R = 5.13334773692

Vertex coordinates: A[9.33435952167; 0] B[0; 0] C[5.96331898044; 2.82884920641]
Centroid: CG[5.09989283404; 0.9432830688]
Coordinates of the circumscribed circle: U[4.66767976084; -2.13985953293]
Coordinates of the inscribed circle: I[5.76767976084; 1.29883439337]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.9966148298° = 139°59'46″ = 0.69881989257 rad
∠ B' = β' = 154.624382903° = 154°37'26″ = 0.4432897735 rad
∠ C' = γ' = 65.38800226713° = 65°22'48″ = 22.0004959929 rad




How did we calculate this triangle?

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

a = 6.6 ; ; ha = 4 ; ; hb = 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 = 6.6 ; ; b = 4.4 ; ; c = 9.33 ; ;

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

p = a+b+c = 6.6+4.4+9.33 = 20.33 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 20.33 }{ 2 } = 10.17 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 10.17 * (10.17-6.6)(10.17-4.4)(10.17-9.33) } ; ; T = sqrt{ 174.24 } = 13.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.2 }{ 6.6 } = 4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.2 }{ 4.4 } = 6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.2 }{ 9.33 } = 2.83 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 4.4**2+9.33**2-6.6**2 }{ 2 * 4.4 * 9.33 } ) = 40° 14" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.6**2+9.33**2-4.4**2 }{ 2 * 6.6 * 9.33 } ) = 25° 22'34" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 6.6**2+4.4**2-9.33**2 }{ 2 * 6.6 * 4.4 } ) = 114° 37'12" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.2 }{ 10.17 } = 1.3 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.6 }{ 2 * sin 40° 14" } = 5.13 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.4**2+2 * 9.33**2 - 6.6**2 } }{ 2 } = 6.508 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 9.33**2+2 * 6.6**2 - 4.4**2 } }{ 2 } = 7.778 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.4**2+2 * 6.6**2 - 9.33**2 } }{ 2 } = 3.111 ; ;







#2 Acute scalene triangle.

Sides: a = 6.6   b = 4.4   c = 6.22328611049

Area: T = 13.2
Perimeter: p = 17.22328611049
Semiperimeter: s = 8.61114305524

Angle ∠ A = α = 74.62197130844° = 74°37'11″ = 1.30223596802 rad
Angle ∠ B = β = 400.0002642442° = 40°1″ = 0.69881363127 rad
Angle ∠ C = γ = 65.38800226713° = 65°22'48″ = 1.14110966606 rad

Height: ha = 4
Height: hb = 6
Height: hc = 4.24224215413

Median: ma = 4.261051642
Median: mb = 6.02551141205
Median: mc = 4.66767976084

Inradius: r = 1.53328463627
Circumradius: R = 3.42325736077

Vertex coordinates: A[6.22328611049; 0] B[0; 0] C[5.05658737589; 4.24224215413]
Centroid: CG[3.76595782879; 1.41441405138]
Coordinates of the circumscribed circle: U[3.11114305524; 1.42658366026]
Coordinates of the inscribed circle: I[4.21114305524; 1.53328463627]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105.3880286916° = 105°22'49″ = 1.30223596802 rad
∠ B' = β' = 1409.999735756° = 139°59'59″ = 0.69881363127 rad
∠ C' = γ' = 114.6219977329° = 114°37'12″ = 1.14110966606 rad

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How did we calculate this triangle?

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

a = 6.6 ; ; ha = 4 ; ; hb = 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 = 6.6 ; ; b = 4.4 ; ; c = 6.22 ; ;

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

p = a+b+c = 6.6+4.4+6.22 = 17.22 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 17.22 }{ 2 } = 8.61 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.61 * (8.61-6.6)(8.61-4.4)(8.61-6.22) } ; ; T = sqrt{ 174.24 } = 13.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 13.2 }{ 6.6 } = 4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 13.2 }{ 4.4 } = 6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 13.2 }{ 6.22 } = 4.24 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 4.4**2+6.22**2-6.6**2 }{ 2 * 4.4 * 6.22 } ) = 74° 37'11" ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 6.6**2+6.22**2-4.4**2 }{ 2 * 6.6 * 6.22 } ) = 40° 1" ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 6.6**2+4.4**2-6.22**2 }{ 2 * 6.6 * 4.4 } ) = 65° 22'48" ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 13.2 }{ 8.61 } = 1.53 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6.6 }{ 2 * sin 74° 37'11" } = 3.42 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.4**2+2 * 6.22**2 - 6.6**2 } }{ 2 } = 4.261 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 6.22**2+2 * 6.6**2 - 4.4**2 } }{ 2 } = 6.025 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 4.4**2+2 * 6.6**2 - 6.22**2 } }{ 2 } = 4.667 ; ;
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