Right triangle calculator (a,c)

You have entered cathetus a and hypotenuse c.

Right scalene triangle.

The lengths of the sides of the triangle:
a = 1000
b = 1732.05108075689
c = 2000

Area: T = 866025.404378444
Perimeter: p = 4732.05108075689
Semiperimeter: s = 2366.02554037844

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Altitude (height) to the side a: h_a = 1732.05108075689
Altitude (height) to the side b: h_b = 1000
Altitude (height) to the side c: h_c = 866.02554037844

Median: m_a = 1802.7765637732
Median: m_b = 1322.87656555323
Median: m_c = 1000

Line segment c_a = 1500
Line segment c_b = 500

Inradius: r = 366.02554037844
Circumradius: R = 1000

Vertex coordinates: A[2000; 0] B[0; 0] C[500; 866.02554037844]
Centroid: CG[833.33333333333; 288.67551345948]
Coordinates of the circumscribed circle: U[1000; -0]
Coordinates of the inscribed circle: I[633.97545962156; 366.02554037844]

Exterior (or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 120° = 1.04771975512 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: cathetus a and hypotenuse c

a = 1000 c = 2000

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

c^{2} = a^{2} + b^{2} b = c^{2} - a^{2} = 200 0^{2} - 100 0^{2} = 1732.051

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 = 1000 b = 1732.05 c = 2000

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

4. 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.

5. The triangle area - from two legs

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

7. 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{1 0 0 0}{2 0 0 0}) = 30 ° sin β = \frac{b}{c} β = arcsin (\frac{b}{c}) = arcsin (\frac{1 7 3 2 . 0 5}{2 0 0 0}) = 60 ° γ = 90 °

8. 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.

9. 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{2 0 0 0}{2} = 1000

10. 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.

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See more information about triangles or more details on solving triangles.