Triangle Excenter Calculator
Calculate triangle excenter instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Formula
r_A = Area / (s - a) | r_B = Area / (s - b) | r_C = Area / (s - c)
Where r_A, r_B, r_C are the exradii opposite vertices A, B, C respectively. Area is calculated using Heron formula, and s is the semi-perimeter (a+b+c)/2. Each exradius is found by dividing the area by the difference of the semi-perimeter and the opposite side.
Worked Examples
Example 1: Finding All Three Exradii
Problem: A triangle has sides a = 5, b = 6, c = 7. Calculate all three exradii and the inradius.
Solution: Semi-perimeter s = (5 + 6 + 7) / 2 = 9\nArea = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.6969\n\nInradius r = Area / s = 14.6969 / 9 = 1.6330\nExradius r_A = Area / (s - a) = 14.6969 / 4 = 3.6742\nExradius r_B = Area / (s - b) = 14.6969 / 3 = 4.8990\nExradius r_C = Area / (s - c) = 14.6969 / 2 = 7.3485\n\nVerification: 1/r = 1/3.6742 + 1/4.8990 + 1/7.3485 = 0.6124 = 1/1.6330
Result: r_A = 3.6742 | r_B = 4.8990 | r_C = 7.3485 | Inradius = 1.6330
Example 2: Excenter Coordinates for a Right Triangle
Problem: A right triangle has vertices at A(0,0), B(4,0), C(0,3). Find the excenter coordinates.
Solution: Sides: a = BC = 5, b = AC = 3, c = AB = 4\ns = (5 + 3 + 4) / 2 = 6, Area = (4 x 3)/2 = 6\n\nExcenter opposite A: I_A = (-5(0,0) + 3(4,0) + 4(0,3)) / (-5+3+4)\nI_A = (12, 12) / 2 = (6, 6)\n\nExcenter opposite B: I_B = (5(0,0) - 3(4,0) + 4(0,3)) / (5-3+4)\nI_B = (-12, 12) / 6 = (-2, 2)\n\nExcenter opposite C: I_C = (5(0,0) + 3(4,0) - 4(0,3)) / (5+3-4)\nI_C = (12, -12) / 4 = (3, -3)
Result: I_A = (6, 6) | I_B = (-2, 2) | I_C = (3, -3)
Frequently Asked Questions
What is an excenter of a triangle?
An excenter of a triangle is the center of an excircle, which is a circle that is tangent to one side of the triangle and to the extensions of the other two sides. Every triangle has exactly three excenters, one opposite each vertex. The excenter opposite vertex A (denoted I_A) is the point where the external bisector of angle B, the external bisector of angle C, and the internal bisector of angle A all meet. Each excircle lies entirely outside the triangle itself. The three excenters together with the incenter form a special quadrilateral called the excentral triangle. Excenters are important in advanced geometry, particularly in the study of triangle centers and the relationships between different circles associated with a triangle.
How do you calculate the exradius of a triangle?
The exradius is the radius of an excircle, and there is a simple formula for each of the three exradii. The exradius opposite vertex A is r_A = Area / (s - a), where s is the semi-perimeter and a is the side opposite vertex A. Similarly, r_B = Area / (s - b) and r_C = Area / (s - c). The area can be found using Heron formula: Area = sqrt(s(s-a)(s-b)(s-c)). For example, with sides 5, 6, 7: s = 9, Area = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.6969. Then r_A = 14.6969 / (9-5) = 3.6742, r_B = 14.6969 / (9-6) = 4.8990, and r_C = 14.6969 / (9-7) = 7.3485. The exradius opposite the longest side is always the largest.
How do you find the coordinates of an excenter?
The coordinates of the excenters can be found using weighted formulas based on the side lengths and vertex positions. If the vertices have coordinates A(x1,y1), B(x2,y2), C(x3,y3) and the opposite sides have lengths a, b, c respectively, then the excenter opposite A is I_A = (-a*A + b*B + c*C) / (-a + b + c). Similarly, I_B = (a*A - b*B + c*C) / (a - b + c), and I_C = (a*A + b*B - c*C) / (a + b - c). Notice the pattern: one sign is negative, corresponding to the vertex being excluded. The incenter formula uses all positive signs: I = (a*A + b*B + c*C) / (a + b + c). These formulas show that excenters are essentially signed weighted averages of the vertex positions.
What is the excentral triangle?
The excentral triangle is formed by connecting the three excenters I_A, I_B, and I_C of a triangle. It has several remarkable properties. The original triangle is the medial triangle of the excentral triangle, meaning the original vertices are the midpoints of the excentral triangle sides. The incenter of the original triangle is the orthocenter of the excentral triangle. The circumradius of the excentral triangle is 2R (twice the circumradius of the original triangle). The sides of the excentral triangle are perpendicular to the angle bisectors of the original triangle. The area of the excentral triangle is always larger than the original triangle by a factor related to the cosines of the half-angles.
What formula does Triangle Excenter Calculator use?
The formula used is described in the Formula section on this page. It is based on widely accepted standards in the relevant field. If you need a specific reference or citation, the References section provides links to authoritative sources.
How accurate are the results from Triangle Excenter Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.