Perimeter of a Triangle Calculator
Solve perimeter atriangle problems step-by-step with our free calculator. See formulas, worked examples, and clear explanations.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
P = a + b + c
The perimeter P of a triangle is the sum of all three side lengths a, b, and c. The semi-perimeter s = P/2 is used in Heron's formula to calculate area and in formulas for the inradius and circumradius.
Worked Examples
Example 1: Triangular Garden Border
Problem:A triangular garden has sides of 12 feet, 15 feet, and 18 feet. Calculate the perimeter and how much fencing material is needed.
Solution:Perimeter = 12 + 15 + 18 = 45 feet\nSemi-perimeter s = 45/2 = 22.5\nArea (Heron) = sqrt(22.5 * 10.5 * 7.5 * 4.5) = sqrt(7981.875) = 89.34 sq ft\nInradius = 89.34 / 22.5 = 3.97 ft\nTriangle type: Scalene, Acute
Result:Perimeter = 45 feet of fencing needed. Area = 89.34 sq ft. The triangle is scalene and acute.
Example 2: Equilateral Triangle Perimeter
Problem:A triangular sign has all sides equal to 24 inches. Find the perimeter and all properties.
Solution:Perimeter = 24 + 24 + 24 = 72 inches = 6 feet\nArea = (sqrt(3)/4) * 24^2 = 249.42 sq inches\nAll angles = 60 degrees\nInradius = 249.42 / 36 = 6.928 inches\nCircumradius = 24 / sqrt(3) = 13.856 inches
Result:Perimeter = 72 inches. Area = 249.42 sq in. All angles 60 deg. R/r ratio = 2 (minimum possible).
Frequently Asked Questions
What is the perimeter of a triangle?
The perimeter of a triangle is the total distance around the outside of the triangle, calculated by adding the lengths of all three sides together. If a triangle has sides of length a, b, and c, then the perimeter P = a + b + c. This is one of the most fundamental measurements in geometry, applicable to any triangle regardless of its type or angle measures. The perimeter represents the total amount of fencing needed to enclose a triangular area, the length of wire needed to outline a triangular shape, or the distance walked if you travel along all three edges. Understanding perimeter is essential for construction, landscaping, material estimation, and numerous practical applications.
How is the semi-perimeter used in triangle calculations?
The semi-perimeter s = P/2 = (a + b + c) / 2 is half the perimeter and serves as a key component in several important triangle formulas. Its most famous application is in Heron's formula for calculating triangle area: Area = sqrt(s(s-a)(s-b)(s-c)). The semi-perimeter also connects to the inradius through the relationship r = Area / s, and to the exradii through ra = Area / (s-a). In coordinate geometry, the semi-perimeter appears in barycentric coordinate calculations for the incenter. The semi-perimeter provides a natural scale parameter for the triangle, and many formulas become more elegant and symmetric when expressed in terms of s rather than the full perimeter.
What is the triangle inequality theorem?
The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. This means for sides a, b, and c: a + b > c, a + c > b, and b + c > a. If any of these conditions fails, the three lengths cannot form a triangle. This theorem is fundamental because it defines which combinations of three lengths are geometrically possible. For example, sides of 3, 4, and 5 form a valid triangle because 3+4 > 5, 3+5 > 4, and 4+5 > 3. However, sides of 1, 2, and 5 cannot form a triangle because 1+2 = 3, which is not greater than 5. The theorem extends to higher dimensions and has applications in metric spaces and graph theory.
How do you find the perimeter of a triangle using coordinates?
When a triangle is defined by vertex coordinates, calculate the perimeter by finding the distance between each pair of vertices using the distance formula and adding them up. For vertices A(x1,y1), B(x2,y2), C(x3,y3): side AB = sqrt((x2-x1)^2 + (y2-y1)^2), side BC = sqrt((x3-x2)^2 + (y3-y2)^2), and side CA = sqrt((x1-x3)^2 + (y1-y3)^2). The perimeter is AB + BC + CA. For example, for vertices at (0,0), (3,0), and (0,4): AB = 3, BC = 5, CA = 4, perimeter = 12. This approach is essential in computer graphics, GIS applications, and computational geometry where shapes are defined by coordinate points rather than direct measurements.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy