Exradii Calculator
Calculate the three exradii (excircle radii) of a triangle
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Understanding Exradii
An exradius (or excircle radius) is the radius of an excircle of a triangle. An excircle is a circle that lies outside the triangle and is tangent to one side and to the extensions of the other two sides.
Exradii Formulas
Key Properties
- Each triangle has exactly three excircles and three exradii
- The exradius is always larger than the inradius
- Relationship: 1/r = 1/r_A + 1/r_B + 1/r_C (where r is the inradius)
- The center of an excircle is called the excenter
- Alternative formula: r_A = A·tan(A/2) where A is the angle at vertex A
Comparison with Inradius
Inradius (r): Radius of inscribed circle (inside triangle)
Formula: r = A / s
Exradius (r_x): Radius of escribed circle (outside triangle)
Formula: r_x = A / (s - x), where x is the side opposite to the excircle
Frequently Asked Questions
What is an excircle?
An excircle (or escribed circle) of a triangle is a circle that lies outside the triangle and is tangent to one side of the triangle and to the extensions of the other two sides. Each triangle has three excircles, one opposite each vertex.
How many exradii does a triangle have?
Every triangle has exactly three exradii, one for each excircle. The excircle opposite vertex A has radius r_A, opposite B has r_B, and opposite C has r_C.
What is the difference between inradius and exradius?
The inradius is the radius of the incircle (inscribed circle) that fits inside the triangle and is tangent to all three sides. An exradius is the radius of an excircle that lies outside the triangle. Exradii are always larger than the inradius.
How do you find exradii without area?
You can use the formula r_A = s·tan(A/2), where s is the semiperimeter and A is the angle at vertex A. However, calculating the area first using Heron's formula is often more straightforward.
What is the relationship between all radii?
The inradius and exradii satisfy the equation: 1/r = 1/r_A + 1/r_B + 1/r_C. Also, r_A + r_B + r_C - r = 4R, where R is the circumradius.
Where is the excenter located?
The excenter (center of an excircle) is located at the intersection of one internal angle bisector and two external angle bisectors. For example, the excenter opposite vertex A is where the internal bisector of angle A meets the external bisectors of angles B and C.
What are practical applications of exradii?
Exradii are used in advanced geometry problems, triangle center theory, and in some engineering applications involving tangent circles. They also appear in competition mathematics and geometric proofs.
Can exradii be used to find triangle area?
Yes! If you know the exradii and sides, you can verify or calculate the area using the formula A = r_A(s - a) = r_B(s - b) = r_C(s - c). This provides an alternative method for area calculation.