Right triangle calculator (A)

You have entered area T and angle α.

Right scalene triangle.

The lengths of the sides of the triangle:
a = 55.0000297247
b = 76.9999583856
c = 8.60223086813

Area: T = 17.5
Perimeter: p = 20.60222967917
Semiperimeter: s = 10.30111483958

Angle ∠ A = α = 35.538° = 35°32'17″ = 0.62202551096 rad
Angle ∠ B = β = 54.462° = 54°27'43″ = 0.95105412172 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Altitude (height) to the side a: h_a = 76.9999583856
Altitude (height) to the side b: h_b = 55.0000297247
Altitude (height) to the side c: h_c = 4.06986752006

Median: m_a = 7.43330001825
Median: m_b = 6.10332902273
Median: m_c = 4.30111543407

Line segment c_a = 5.6966077555
Line segment c_b = 2.90662311264

Inradius: r = 1.69988397145
Circumradius: R = 4.30111543407

Vertex coordinates: A[8.60223086813; 0] B[0; 0] C[2.90662311264; 4.06986752006]
Centroid: CG[3.83661799359; 1.35662250669]
Coordinates of the circumscribed circle: U[4.30111543407; 0]
Coordinates of the inscribed circle: I[3.30111900102; 1.69988397145]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 144.462° = 144°27'43″ = 0.62202551096 rad
∠ B' = β' = 125.538° = 125°32'17″ = 0.95105412172 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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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. Input data entered: angle α and area T

α = 35.538 ° T = 17.5

2. From the angle α, we calculate angle β:

α + β + 90 ° = 180 ° β = 90 ° - α = 90 ° - 35.538 ° = 54.462 °

3. From the area T, angle α, and angle β, we calculate hypotenuse c:

c^{2} sin α sin β = 2 T c = \frac{2 T}{sin α sin β} c = \frac{2 \cdot 1 7 . 5}{sin 3 5 ° 3 2 ^{'} 1 7 " \cdot sin 5 4 ° 2 7 ^{'} 4 3 "} = 8.602

4. From the area T and hypotenuse c, we calculate height h:

T = \frac{c \cdot h}{2} h = 2 \cdot T / c = 2 \cdot 17.5 / 8.602 = 4.069

5. From the hypotenuse c and angle α, we calculate cathetus a:

sin α = a : c a = c \cdot sin α = 8.602 \cdot sin (35.538 °) = 5

6. From the cathetus a and hypotenuse c, we calculate cathetus b - Pythagorean theorem:

c^{2} = a^{2} + b^{2} b = c^{2} - a^{2} = 8.60 2^{2} - 5^{2} = 7

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 = 5 b = 7 c = 8.6

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

p = a + b + c = 5 + 7 + 8.6 = 20.6

8. The 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{2 0 . 6}{2} = 10.3

9. The triangle area - from two legs

T = \frac{a b}{2} = \frac{5 \cdot 7}{2} = 17.5

10. Calculate the heights of the right triangle from its area.

h_{a} = b = 7 h_{b} = a = 5 T = \frac{c h _{c}}{2} h_{c} = \frac{2 T}{c} = \frac{2 \cdot 1 7 . 5}{8 . 6} = 4.07

11. Calculation of the inner angles of the triangle - basic use of sine function

sin α = \frac{a}{c} α = arcsin (\frac{a}{c}) = arcsin (\frac{5}{8 . 6}) = 35 ° 3 2^{'} 17 " sin β = \frac{b}{c} β = arcsin (\frac{b}{c}) = arcsin (\frac{7}{8 . 6}) = 54 ° 2 7^{'} 43 " γ = 90 °

12. 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 7 . 5}{1 0 . 3} = 1.7

13. 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{c}{2} = \frac{8 . 6}{2} = 4.3

14. 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}^{2} = b^{2} + (a / 2)^{2} m_{a} = b^{2} + (a / 2)^{2} = 7^{2} + (5 / 2)^{2} = 7.433 m_{b}^{2} = a^{2} + (b / 2)^{2} m_{b} = a^{2} + (b / 2)^{2} = 5^{2} + (7 / 2)^{2} = 6.103 m_{c} = R = \frac{c}{2} = \frac{8 . 6}{2} = 4.301

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See triangle basics on Wikipedia or more details on solving triangles.