# Triangle Calculator Calculate triangle area, perimeter, and semi-perimeter from three side lengths using Heron's formula with this free, step-by-step geometry calculator. --- - **Canonical URL:** https://dothecalculation.com/calculators/triangle-calculator - **Category:** Math calculators - **Publisher:** Do The Calculation (https://dothecalculation.com) - **Cost:** Free, no account or sign-up required - **Privacy:** Runs entirely in the browser; inputs are never sent to a server - **Methodology:** https://dothecalculation.com/methodology - **Reviewed by:** Dr. Elena Rostova, PhD, PhD in Pure Mathematics, Sorbonne University (https://dothecalculation.com/about/team/elena-rostova) --- ## Calculate triangle area and perimeter Enter any three side lengths to get the exact area, perimeter, and semi-perimeter using Heron's formula — no height or angle measurements required. - Exact area via Heron's formula - Perimeter and semi-perimeter - Triangle inequality validity check ## Triangle Properties and Heron’s Formula A triangle is valid only if the sum of any two sides is strictly greater than the third side. When this condition is met, Heron's formula can determine the area of the triangle without knowing its height. Heron's formula uses the semi-perimeter (half the perimeter) and side lengths: $$\text{Semi-Perimeter (s)} = \frac{a + b + c}{2}$$ $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$ It is highly useful in land measurement, construction, and geometry studies. ## The Triangle Inequality Theorem The fundamental constraint of triangle geometry is that a three-sided shape can only exist if it forms a closed loop. The mathematical statement of the **Triangle Inequality Theorem** is: $$a + b > c$$ $$a + c > b$$ $$b + c > a$$ If any single side length is equal to or greater than the sum of the other two, the sides cannot connect, and the area calculation is undefined (resulting in a flat line or disjoint segments). ## Classifying Triangles by Sides and Angles Triangles can be classified based on their side proportions and interior angles: **By Sides**: Equilateral (three equal sides), Isosceles (two equal sides), and Scalene (three unequal sides). **By Angles**: Right-angled (one 90° angle, satisfying the Pythagorean theorem \(a^2 + b^2 = c^2\)), Acute-angled (all angles under 90°), and Obtuse-angled (one angle over 90°). You can spot a right triangle from side lengths alone, without measuring any angle: if the square of the longest side equals the sum of the squares of the other two, the calculator's inputs describe a right triangle. Equilateral triangles are also worth knowing for area comparisons — for a fixed perimeter, the equilateral triangle always has the largest possible area of any triangle. ## How to Use This Calculator Enter the three side lengths (a, b, c) in any consistent unit — inches, feet, meters, or centimeters. The calculator checks the triangle inequality first, then reports the perimeter, semi-perimeter, and area via Heron's formula. The results panel returns four values: **area** (in square units), **perimeter**, **semi-perimeter**, and a **valid triangle** status. It does not compute interior angles directly — see the FAQ below for how to find each angle from the same three sides using the Law of Cosines. If the sides you enter fail the triangle inequality (for example, 2, 3, and 10), the calculator flags the triangle as invalid and area is not computed, since those three lengths cannot form a closed shape. ## Worked Example: 3-4-5 Right Triangle Using the calculator's default sides (\(a=3\), \(b=4\), \(c=5\)): $$\text{Perimeter} = 3+4+5=12, \qquad s = \frac{12}{2} = 6$$ $$\text{Area} = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6$$ Since \(3^2 + 4^2 = 5^2\) (9 + 16 = 25), this is also a right triangle. ## Worked Example: An Invalid Triangle Try sides \(a=2\), \(b=3\), \(c=10\). Checking the triangle inequality: \(2 + 3 = 5\), which is not greater than \(10\). $$a + b > c \;\rightarrow\; 5 > 10 \;\text{(false)}$$ Because one condition fails, no triangle can be formed — the two shorter sides can't reach far enough to close the shape. The calculator reports **Valid triangle: No** and does not return an area for these inputs. ## Real-World Uses for Triangle Calculations Triangle area and perimeter come up outside the classroom: estimating material for a triangular roof section or gable end, checking the square footage of a triangular lot or garden bed, calculating fabric or flooring needed for angled rooms, and verifying a support brace or truss forms a rigid (valid) triangle before cutting material. For composite shapes that combine triangles with rectangles or other polygons — a common case for room layouts and land parcels — pair this calculator with the [square footage calculator](/calculators/square-footage-calculator) to total the full area. ## Related Calculators For area across composite or irregular room and lot shapes, use the [square footage calculator](/calculators/square-footage-calculator). To work with angles and coordinates on a circle, see the [unit circle calculator](/calculators/unit-circle-calculator). For vector-based geometry in 2D or 3D, use the [vector math calculator](/calculators/vector-math-calculator). ## Frequently asked questions ### What is the triangle inequality theorem? It states that for any valid triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side. ### What is a semi-perimeter? It is half of the perimeter of the triangle, calculated as (a + b + c) / 2. ### How is Heron's formula derived? Heron's formula is derived by starting with the standard area formula (0.5 × base × height), applying the law of cosines to find the height in terms of the sides, and substituting it back. ### Can Heron's formula be used on any triangle? Yes. Heron's formula works on any valid triangle (equilateral, isosceles, or scalene) as long as you know all three side lengths. ### What is the perimeter of a triangle? The perimeter is the total boundary distance, calculated as the sum of all three side lengths (a + b + c). ### How does the calculator verify triangle validity? It checks whether the sum of any two sides is greater than the third side. For example, sides 2, 3, and 10 fail because 2 + 3 = 5 is not greater than 10, so the calculator marks the triangle invalid and does not return an area. ### What is the Pythagorean theorem? For a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). ### Does this calculator show the triangle's angles? No, it calculates area, perimeter, semi-perimeter, and validity from the three side lengths only. To find each interior angle from the same sides, apply the Law of Cosines: cos(A) = (b² + c² − a²) / (2bc), then solve for B the same way and get C as 180° − A − B. ### What is the difference between a scalene and isosceles triangle? An isosceles triangle has at least two equal side lengths and two equal angles. A scalene triangle has three unequal side lengths and three unequal angles. ### Can a triangle have two right angles? No. The sum of all interior angles in a triangle is always exactly 180°. Having two 90° angles would equal 180°, leaving 0° for the third angle, which is impossible. ## Related concepts - **Heron’s formula** — A formula for calculating the area of a triangle given the lengths of its three sides. - **Triangle inequality theorem** — The sum of any two sides of a triangle must be strictly greater than the third side, or the shape cannot close. - **Perimeter** — The total distance around the boundary of a two-dimensional shape. - **Law of Cosines** — A geometric law relating the lengths of the sides of a triangle to the cosine of one of its angles. ## Related guides - [Triangle Geometry Guide: Area, Perimeter, Heron Formula](https://dothecalculation.com/blog/math/triangle-geometry-formulas) — Learn triangle validity, perimeter, semi-perimeter, and Heron formula area with worked examples tied to the live DTC triangle calculator. - [Concrete Volume Guide: Slab Yardage, Waste Allowance, and Bag Estimates](https://dothecalculation.com/blog/utility/concrete-volume-estimation) — Learn how to calculate concrete volume for slabs and footings. 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Verify calculations manually for critical academic, professional, or scientific applications._ --- _Source: [Do The Calculation](https://dothecalculation.com/calculators/triangle-calculator). Quote freely with attribution and a link to this page._