Triangle Area Calculator — Instantly calculate the area of any triangle using base & height or three sides (Heron’s formula). Choose your method, enter your values, pick your units, and see real-time results with step-by-step formula. Mobile-optimized, SEO-rich, privacy-first, and 100% WordPress/Elementor compatible.
How to Use the Triangle Area Calculator
- Choose Calculation Method: Select “Base & Height” or “3 Sides (Heron’s formula)”.
- Enter Triangle Dimensions: Fill in the required values in your chosen units.
- Choose Units: Select cm, m, mm, in, ft, or yd. Result shown in square units.
- Calculate: Click Calculate or press Enter for instant area and explanation.
- Copy/Reset: Copy results or clear the form for new inputs.
- Mobile Ready: Fully responsive and touch-friendly for any device.
Why Use a Triangle Area Calculator?
Fast & Accurate
Instant calculation with step-by-step formula; no manual mistakes.
Multiple Methods
Base × Height or 3 Sides (Heron’s formula) for any triangle.
Any Units
Supports metric and imperial units for global usability.
Privacy First
All calculations are local; nothing is sent or stored online.
What is the Area of a Triangle?
The area of a triangle represents the total two-dimensional space enclosed by its three sides. It’s a fundamental concept in geometry, providing a measure of the size of a triangular region. The area is always expressed in square units, such as square meters (m²), square feet (ft²), or square centimeters (cm²). Calculating this area is a critical skill in fields ranging from architecture and engineering to graphic design and land surveying.
This calculator provides two of the most common and versatile methods for finding the area of any triangle:
- Using Base and Height: The classic formula is Area = ½ × base × height. This method is universal for all triangles but requires knowing the perpendicular height relative to a chosen base.
- Using Three Sides (Heron’s Formula): When you only know the lengths of the three sides (a, b, and c), you can use Heron’s formula. It is a powerful two-step method: first, calculate the semi-perimeter (s), and then apply the formula Area = √[s(s-a)(s-b)(s-c)].
Triangle Area Formula Table
Understanding the Area Formulas in Detail
To accurately find the area, it’s crucial to understand how and when to use each formula. The information you have about the triangle—whether it’s its height or the lengths of all its sides—will determine the best method.
Method 1: The Base and Height Formula (A = ½ × b × h)
This is the most widely taught formula for a triangle’s area. Its beauty lies in its simplicity, which is derived from the fact that a triangle is essentially half of a rectangle or parallelogram.
- The Base (b): The base can be any one of the triangle’s three sides. You can choose whichever side is most convenient to measure.
- The Height (h): This is the most critical component. The height is not necessarily the length of one of the other sides. It is the perpendicular distance from the chosen base to the opposite vertex (corner). This line, known as the altitude, must form a right angle (90°) with the base.
How you find the height can differ depending on the type of triangle:
- Acute Triangle: In a triangle where all angles are less than 90°, the height will fall inside the triangle.
- Right-Angled Triangle: In a right triangle, two of the sides are already perpendicular to each other. If you choose one of these sides as the base, the other automatically becomes the height.
- Obtuse Triangle: In a triangle with one angle greater than 90°, the height corresponding to one of the shorter bases will fall outside the triangle. You must extend the base line to draw the perpendicular line from the vertex.
Method 2: Heron’s Formula (Three Sides Method)
What if you don’t know the height? Measuring it can be difficult or impossible in many real-world situations. This is where Heron’s formula, named after the Greek mathematician Heron of Alexandria, becomes incredibly useful. It allows you to calculate the area using only the lengths of the three sides (a, b, and c).
The process involves two steps:
- Calculate the semi-perimeter (s): The semi-perimeter is simply half of the triangle’s total perimeter.
s = (a + b + c) / 2 - Apply Heron’s Formula: Once you have ‘s’, you plug it into the main formula.
Area = √[s(s – a)(s – b)(s – c)]
This formula works for any type of triangle—scalene, isosceles, equilateral, acute, or obtuse—as long as the three given side lengths can form a valid triangle (see the Triangle Inequality Theorem below).
Can Any Three Sides Form a Triangle? The Triangle Inequality Theorem
Before using Heron’s formula, it’s essential to verify if the given side lengths can actually form a triangle. This is determined by the Triangle Inequality Theorem, a fundamental rule in geometry.
The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For sides a, b, and c, all three of the following conditions must be true:
- a + b > c
- a + c > b
- b + c > a
Why does this matter? If this rule is not met, the sides won’t be able to connect to form a closed three-sided shape. For example, if you have sides of length 3, 4, and 8, you cannot form a triangle because 3 + 4 is not greater than 8. Our calculator automatically checks this condition when you use the “3 Sides” method. If the inequality is violated, it will notify you that the triangle is invalid, preventing an impossible calculation.
Types of Triangles and Area Calculation
Triangles can be classified by the length of their sides or the measure of their angles. Understanding these classifications can provide context for area calculations.
Classification by Sides
- Equilateral Triangle: All three sides are of equal length, and all three angles are 60°. You can use Heron’s formula or a specialized formula: Area = (s²√3)/4, where ‘s’ is the side length.
- Isosceles Triangle: Two sides are of equal length. The angles opposite the equal sides are also equal. Both the base-height method and Heron’s formula work perfectly.
- Scalene Triangle: All three sides have different lengths, and all three angles are different. This is the most general type of triangle, and both of our calculator’s methods are ideal for it.
Classification by Angles
- Right-Angled Triangle: One angle is exactly 90°. The area calculation is very straightforward: use the two perpendicular sides (legs) as the base and height. Area = ½ × leg1 × leg2.
- Acute Triangle: All three angles are less than 90°.
- Obtuse Triangle: One angle is greater than 90°.
Our calculator is designed to handle all these types. The “Base & Height” and “3 Sides” methods are universal and do not require you to pre-classify the triangle you are working with.
Advanced and Alternative Area Formulas
While our calculator uses the two most common methods, other formulas exist for finding a triangle’s area, particularly in fields like trigonometry and coordinate geometry. Understanding these can provide deeper insight.
Trigonometry Formula (Side-Angle-Side)
If you know the lengths of two sides and the angle between them (the “included angle”), you can use trigonometry to find the area. Given sides ‘a’ and ‘b’ with the included angle ‘C’, the formula is:
Area = ½ × a × b × sin(C)Here, sin(C) is the sine of angle C. This formula is extremely useful in navigation, astronomy, and physics where angles are often easier to measure than perpendicular heights.
Coordinate Geometry (The Shoelace Formula)
If you know the Cartesian coordinates (x, y) of the three vertices of a triangle, you can use the Shoelace Formula (also known as the Surveyor’s Formula). For a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃), the formula is:
Area = ½ |(x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂))|The absolute value bars (||) are used because area must always be a positive value. This method is the foundation of many computer graphics and geographic information system (GIS) calculations.
Real-World Applications of Calculating Triangle Area
The need to calculate the area of a triangle appears in a surprising number of practical, everyday situations and professional fields. It is a cornerstone of design, planning, and estimation.
Architecture & Construction
Architects and builders calculate the area of triangular gables on roofs to estimate the amount of siding or paint required. The area of triangular trusses is also analyzed for structural load calculations.
Land Surveying & Real Estate
Surveyors often divide irregularly shaped plots of land into a series of triangles. By calculating the area of each triangle (often using Heron’s formula or trigonometry) and summing them, they can determine the total acreage of the property.
Landscaping & Gardening
When designing a garden, a triangular patch of lawn, or a flower bed, knowing the area is crucial for buying the correct amount of sod, seeds, fertilizer, or mulch, which helps in managing costs and reducing waste.
Interior Design & DIY Projects
Whether you’re tiling a triangular section of a floor, painting a triangular accent wall, or cutting fabric for a craft project, calculating the area ensures you purchase the right quantity of materials.
Engineering
In civil and mechanical engineering, the cross-sectional area of triangular beams is calculated to understand their strength and resistance to bending. In aerodynamics, the surface area of triangular wings or fins is essential for calculating lift and drag.
Art & Graphic Design
Artists and designers use triangles as fundamental building blocks in compositions. Understanding their area helps in balancing a design and creating visually stable patterns and layouts.
Frequently Asked Questions
The most common formula is Area = (base × height) / 2. If you know all three sides, you can use Heron’s formula. This calculator supports both methods.
Yes! Our calculator’s “3 Sides” mode uses Heron’s formula (Area = √[s(s−a)(s−b)(s−c)]) for this exact purpose. It’s perfect when you don’t know the triangle’s height.
The height is the perpendicular distance from a base to the opposite vertex. It must form a 90-degree angle with the base, and it is not always the same as one of the side lengths.
We support cm, m, mm, in, ft, and yd. The final calculated area will be in the corresponding square units (e.g., m², ft², etc.).
Yes! You can enter decimals for the base, height, and all three side lengths as needed.
Yes! All calculations are performed instantly within your browser. No data is sent or stored on our servers. The tool is 100% free and private.
To ensure a good balance between accuracy and readability, all results are rounded to a maximum of 6 decimal places.
Yes! After calculating, click the “Copy Result” button to save the numerical answer and the step-by-step explanation to your clipboard.
Absolutely! The calculator is fully responsive and touch-optimized for a seamless experience on mobile phones, tablets, and desktops.
The calculator will show an error message. According to the Triangle Inequality Theorem, the sum of any two sides must be greater than the third. If not, the sides cannot form a closed triangle.
Yes! This calculator is a perfect tool for students, professionals, and DIY enthusiasts for any project or problem involving the area of a triangle.
Use the dropdown menu labeled “Units” to select your preferred unit of measurement before calculating. The result will be displayed in the corresponding square unit.