Triangle calculator SSA

Triangle has two solutions with side c=293.11222965642 and with side c=13.30554806834

#1 Obtuse scalene triangle.

The lengths of the sides of the triangle:
a = 200
b = 190
c = 293.11222965642

Area: T = 18840.8955247823
Perimeter: p = 683.11222965642
Semiperimeter: s = 341.55661482821

Angle ∠ A = α = 42.58799653282° = 42°34'48″ = 0.74331605904 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 97.42200346718° = 97°25'12″ = 1.77003003624 rad

Altitude (height) to the side a : h_a = 188.40989524782
Altitude (height) to the side b : h_b = 198.3255213135
Altitude (height) to the side c : h_c = 128.55875219373

Median: m_a = 225.84881994583
Median: m_b = 232.23435229862
Median: m_c = 128.73295436204

Inradius: r = 55.16219267947
Circumradius: R = 147.79437635517

Vertex coordinates: A[293.11222965642; 0] B[0; 0] C[153.20988886238; 128.55875219373]
Centroid: CG[148.7743728396; 42.85325073124]
Coordinates of the circumscribed circle: U[146.55661482821; -19.08664335459]
Coordinates of the inscribed circle: I[151.55661482821; 55.16219267947]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 137.42200346718° = 137°25'12″ = 0.74331605904 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 82.58799653282° = 82°34'48″ = 1.77003003624 rad

How did we calculate this triangle?

The calculation of the triangle has two phases. The first phase calculates all three sides of the triangle from the input parameters. The first phase is different for the different triangles query entered. The second phase calculates other triangle characteristics, such as angles, area, perimeter, heights, the center of gravity, circle radii, etc. Some input data also results in two to three correct triangle solutions (e.g., if the specified triangle area and two sides - typically resulting in both acute and obtuse) triangle).

1. Use the Law of Cosines

a = 200 b = 190 β = 40 ° b^{2} = a^{2} + c^{2} - 2 a c cos β 19 0^{2} = 20 0^{2} + c^{2} - 2 \cdot 200 \cdot c \cdot cos 40 ° c^{2} - 306.418 c + 3900 = 0 p = 1; q = - 306.418; r = 3900 D = q^{2} - 4 p r = 306.41 8^{2} - 4 \cdot 1 \cdot 3900 = 78291.854213354 D > 0 c_{1, 2} = \frac{- q \pm D}{2 p} = \frac{3 0 6 . 4 2 \pm 7 8 2 9 1 . 8 5}{2} c_{1, 2} = 153.208889 \pm 139.903408 c_{1} = 293.112296564 c_{2} = 13.305480683 c > 0 c = 293.11

We know the lengths of all three sides of the triangle, so the triangle is uniquely specified. Next, we calculate another of its characteristics - the same procedure for calculating the triangle from the known three sides SSS.

a = 200 b = 190 c = 293.11

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

p = a + b + c = 200 + 190 + 293.11 = 683.11

3. Semiperimeter of the triangle

The semiperimeter of the triangle is half its perimeter. The semiperimeter frequently appears in formulas for triangles to be given a separate name. By the triangle inequality, the longest side length of a triangle is less than the semiperimeter.

s = \frac{p}{2} = \frac{6 8 3 . 1 1}{2} = 341.56

4. The triangle area using Heron's formula

Heron's formula gives the area of a triangle when the length of all three sides is known. There is no need to calculate angles or other distances in the triangle first. Heron's formula works equally well in all cases and types of triangles.

T = s (s - a) (s - b) (s - c) T = 341.56 (341.56 - 200) (341.56 - 190) (341.56 - 293.11) T = 354979333.74 = 18840.9

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

There are many ways to find the height of the triangle. The easiest way is from the area and base length. The triangle area is half of the product of the base's length and height. Every side of the triangle can be a base; there are three bases and three heights (altitudes). Triangle height is the perpendicular line segment from a vertex to a line containing the base.

T = \frac{a h _{a}}{2} h_{a} = \frac{2 T}{a} = \frac{2 \cdot 1 8 8 4 0 . 9}{2 0 0} = 188.41 h_{b} = \frac{2 T}{b} = \frac{2 \cdot 1 8 8 4 0 . 9}{1 9 0} = 198.33 h_{c} = \frac{2 T}{c} = \frac{2 \cdot 1 8 8 4 0 . 9}{2 9 3 . 1 1} = 128.56

6. Calculation of the inner angles of the triangle using a Law of Cosines

The Law of Cosines is useful for finding a triangle's angles when we know all three sides. The cosine rule, also known as the Law of Cosines, relates all three sides of a triangle with an angle of a triangle. The Law of Cosines extrapolates the Pythagorean theorem for any triangle. Pythagorean theorem works only in a right triangle. Pythagorean theorem is a special case of the Law of Cosines and can be derived from it because the cosine of 90° is 0. It is best to find the angle opposite the longest side first. With the Law of Cosines, there is also no problem with obtuse angles as with the Law of Sines because the cosine function is negative for obtuse angles, zero for right, and positive for acute angles. We also use an inverse cosine called arccosine to determine the angle from the cosine value.

a^{2} = b^{2} + c^{2} - 2 b c cos α α = arccos (\frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}) = arccos (\frac{1 9 0 ^{2} + 2 9 3 . 1 1 ^{2} - 2 0 0 ^{2}}{2 \cdot 1 9 0 \cdot 2 9 3 . 1 1}) = 42 ° 3 4^{'} 48 " b^{2} = a^{2} + c^{2} - 2 a c cos β β = arccos (\frac{a ^{2} + c ^{2} - b ^{2}}{2 a c}) = arccos (\frac{2 0 0 ^{2} + 2 9 3 . 1 1 ^{2} - 1 9 0 ^{2}}{2 \cdot 2 0 0 \cdot 2 9 3 . 1 1}) = 40 ° γ = 180 ° - α - β = 180 ° - 42 ° 3 4^{'} 48 " - 40 ° = 97 ° 2 5^{'} 12 "

7. Inradius

An incircle of a triangle is a tangent circle to each side. An incircle center is called an incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three-angle bisectors. The product of a triangle's inradius and semiperimeter (half the perimeter) is its area.

T = r s r = \frac{T}{s} = \frac{1 8 8 4 0 . 9}{3 4 1 . 5 6} = 55.16

8. Circumradius

The circumcircle of a triangle is a circle that passes through all of the triangle's vertices, and the circumradius of a triangle is the radius of the triangle's circumcircle. The circumcenter (center of the circumcircle) is the point where the perpendicular bisectors of a triangle intersect.

R = \frac{a b c}{4 r s} = \frac{2 0 0 \cdot 1 9 0 \cdot 2 9 3 . 1 1}{4 \cdot 5 5 . 1 6 2 \cdot 3 4 1 . 5 5 6} = 147.79

9. Calculation of medians

A median of a triangle is a line segment joining a vertex to the opposite side's midpoint. Every triangle has three medians, and they all intersect each other at the triangle's centroid. The centroid divides each median into parts in the ratio of 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. We use Apollonius's theorem to calculate a median's length from its side's lengths.

m_{a} = \frac{2 b ^{2} + 2 c ^{2} - a ^{2}}{2} = \frac{2 \cdot 1 9 0 ^{2} + 2 \cdot 2 9 3 . 1 1 ^{2} - 2 0 0 ^{2}}{2} = 225.848 m_{b} = \frac{2 c ^{2} + 2 a ^{2} - b ^{2}}{2} = \frac{2 \cdot 2 9 3 . 1 1 ^{2} + 2 \cdot 2 0 0 ^{2} - 1 9 0 ^{2}}{2} = 232.234 m_{c} = \frac{2 a ^{2} + 2 b ^{2} - c ^{2}}{2} = \frac{2 \cdot 2 0 0 ^{2} + 2 \cdot 1 9 0 ^{2} - 2 9 3 . 1 1 ^{2}}{2} = 128.73

#2 Obtuse scalene triangle.

The lengths of the sides of the triangle:
a = 200
b = 190
c = 13.30554806834

Area: T = 855.26598124209
Perimeter: p = 403.30554806834
Semiperimeter: s = 201.65327403417

Angle ∠ A = α = 137.42200346718° = 137°25'12″ = 2.39884320632 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 2.58799653282° = 2°34'48″ = 0.04550288896 rad

Altitude (height) to the side a : h_a = 8.55325981242
Altitude (height) to the side b : h_b = 9.00327348676
Altitude (height) to the side c : h_c = 128.55875219373

Median: m_a = 90.21437345869
Median: m_b = 105.18332586874
Median: m_c = 194.95106118122

Inradius: r = 4.2411250632
Circumradius: R = 147.79437635517

Vertex coordinates: A[13.30554806834; 0] B[0; 0] C[153.20988886238; 128.55875219373]
Centroid: CG[55.50547897691; 42.85325073124]
Coordinates of the circumscribed circle: U[6.65327403417; 147.64439554832]
Coordinates of the inscribed circle: I[11.65327403417; 4.2411250632]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 42.58799653282° = 42°34'48″ = 2.39884320632 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 177.42200346718° = 177°25'12″ = 0.04550288896 rad

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