math

Triangle Area Calculator

Calculate the area of a triangle from its base and height using the formula A = (1/2) × base × height.

Reviewed by Christopher FloiedPublished April 8, 2026Updated May 21, 2026

This free online triangle area calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Base

Height

Minimum: 0

Results

Area

How to Use This Calculator

Enter your input values

Fill in all required input fields for the Triangle Area Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

Read the results

The Triangle Area Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Triangle Area Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

•Use the Triangle Area Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
•Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
•Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
•Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Triangle Area Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the area of a triangle from its base and height using the formula A = (1/2) × base × height. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Triangle Area Calculator

The Triangle Area Calculator finds the area of any triangle given its base and perpendicular height. This classic formula A = (1/2) × b × h has been known since ancient times and is one of the most frequently used formulas in geometry. The key insight is that any triangle has exactly HALF the area of a rectangle with the same base and height — which is why the (1/2) appears in the formula. Whether you're calculating the area of a lot, a roof section, a sail, a piece of land, or a geometric figure in a math problem, this simple formula is indispensable. The calculator works for any triangle as long as you have the base and the perpendicular height.

The Math Behind It

The area of a triangle is one of the most fundamental formulas in geometry. Despite the many shapes of triangles, the area calculation is surprisingly simple. **The Basic Formula**: A = (1/2) × base × height Where: - A = Area - base = Any side of the triangle - height = Perpendicular distance from the opposite vertex to the base **The 'Half Rectangle' Insight**: Any triangle occupies exactly HALF the area of a rectangle with the same base and height. This is why the formula has a (1/2) factor. Proof idea: Any triangle can be split by a line from a vertex perpendicular to the opposite side, creating two right triangles. Each right triangle is half of a rectangle. Adding them together gives half the original rectangle. **Finding the Height**: The 'height' must be perpendicular to the chosen base: - **Right triangle**: One leg is the height, the other is the base - **Acute triangle**: Drop a perpendicular from any vertex to the opposite side - **Obtuse triangle**: May need to extend the base to meet the perpendicular **Multiple Bases**: Every triangle has three sides, so three possible bases. You can use any side as the base, with the corresponding height: A = (1/2) × a × h_a = (1/2) × b × h_b = (1/2) × c × h_c All give the same area. **Special Triangles**: **Right Triangle**: - Two legs are perpendicular - A = (1/2) × leg₁ × leg₂ **Equilateral Triangle**: - All sides equal length s - Height = (s√3)/2 - Area = (s²√3)/4 For side = 10: Height = 5√3 ≈ 8.66 Area = 25√3 ≈ 43.30 **Isosceles Triangle**: - Two sides equal - Symmetric around axis through apex - Use regular formula **Heron's Formula**: When you only know the three side lengths (no height): A = √[s(s-a)(s-b)(s-c)] Where s = (a+b+c)/2 (semi-perimeter) Example: Sides 3, 4, 5 s = (3+4+5)/2 = 6 A = √[6×3×2×1] = √36 = 6 Verify with base × height formula: (1/2) × 3 × 4 = 6 ✓ **Triangle Classification by Sides**: - **Equilateral**: All three sides equal - **Isosceles**: Two sides equal - **Scalene**: All sides different **Triangle Classification by Angles**: - **Right**: One 90° angle - **Acute**: All angles less than 90° - **Obtuse**: One angle greater than 90° **Triangle Inequality**: For three lengths to form a triangle: The sum of any two sides must be greater than the third side. Example: 3, 4, 8 3 + 4 = 7 < 8, so these CAN'T form a triangle. **The 30-60-90 Triangle**: Special right triangle with angles 30°, 60°, 90°. Sides are in ratio: 1 : √3 : 2 Examples: - 1 : 1.732 : 2 - 5 : 8.66 : 10 - 10 : 17.32 : 20 If hypotenuse = 10, the sides are 5 and 5√3. **The 45-45-90 Triangle**: Isosceles right triangle. Sides in ratio: 1 : 1 : √2 Examples: - 1 : 1 : 1.414 - 5 : 5 : 7.07 - 10 : 10 : 14.14 **Common Triangle Problems**: **Example 1**: Triangular lot A triangular plot of land has base 100 ft and height 80 ft. Area = (1/2) × 100 × 80 = 4,000 sq ft **Example 2**: Sail A sailboat's triangular sail has base 8 ft and height 15 ft. Area = (1/2) × 8 × 15 = 60 sq ft **Example 3**: Gable end House gable has 20 ft base and 8 ft height. Area = (1/2) × 20 × 8 = 80 sq ft **Using Sine**: When you know two sides and the angle between them: A = (1/2) × a × b × sin(C) Where C is the angle between sides a and b. Example: Sides 10 and 15, angle 30° A = (1/2) × 10 × 15 × sin(30°) A = 75 × 0.5 = 37.5 **Coordinate Geometry**: For triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): A = (1/2) × |x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| 'Shoelace formula' useful for coordinate problems. **Real-World Applications**: **Architecture**: - Triangular facades - Gables and pediments - Truss calculations **Engineering**: - Structural triangulation - Roof loading - Bridge design **Art and Design**: - Composition ratios - Perspective drawing - Logos and layouts **Navigation**: - Triangulation (GPS, surveying) - Sailing (wind angles) - Triangle inequality in routing **Sports**: - Triangular offense setups - Defensive triangles - Soccer positioning **Nature**: - Many crystal formations - Rock climbing routes - Mountain peaks **Triangle Types in Different Cultures**: - **Egyptian**: Used right triangles for pyramids - **Greek**: Pythagoras and geometric proofs - **Roman**: Surveying and engineering - **Medieval**: Gothic architecture - **Modern**: Trigonometry and physics **Pythagorean Triples** (right triangles with integer sides): Classic examples: - 3-4-5 - 5-12-13 - 8-15-17 - 7-24-25 Areas: - 3-4-5: Area = 6 - 5-12-13: Area = 30 - 8-15-17: Area = 60 - 7-24-25: Area = 84 **Formulas Summary**: 1. **Base × height**: A = (1/2)bh 2. **Heron's formula**: A = √[s(s-a)(s-b)(s-c)] 3. **Two sides + angle**: A = (1/2)ab sin(C) 4. **Three vertices**: A = (1/2)|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| 5. **Equilateral (side s)**: A = (s²√3)/4 Choose the formula that matches your known quantities. **Common Mistakes**: 1. **Forgetting the 1/2**: It's half the base × height 2. **Using slant height**: Must be PERPENDICULAR height 3. **Confusing base and side**: Any side can be the base 4. **Negative area**: Take absolute value if formula gives negative 5. **Mixing units**: Base and height must use same units

Formula Reference

Triangle Area

A = (1/2) × b × h

Variables: b = base, h = perpendicular height

Heron's Formula

A = √[s(s-a)(s-b)(s-c)]

Variables: s = semi-perimeter = (a+b+c)/2

Worked Examples

Example 1: Standard Triangle

Find the area of a triangle with base 10 cm and height 6 cm.

Step 1:A = (1/2) × base × height

Step 2:A = (1/2) × 10 × 6

Step 3:A = (1/2) × 60

Step 4:A = 30 cm²

Area = 30 cm². This triangle has exactly half the area of a 10 × 6 rectangle (which has area 60).

Example 2: Right Triangle

A right triangle has legs of 5 and 12.

Step 1:Both legs are perpendicular, so either can be base/height

Step 2:A = (1/2) × 5 × 12

Step 3:A = (1/2) × 60

Step 4:A = 30

Area = 30 square units. For right triangles, the two legs ARE the base and height, making calculation especially easy.

Common Mistakes & Tips

!Forgetting the (1/2) factor. It's half the base × height.
!Using slant height instead of perpendicular height. Height must be perpendicular to the base.
!Mixing units. Base and height must be in the same units.
!Confused about which side is the base. Any side can be the base — just use the corresponding perpendicular height.

Related Concepts

Circle Area

Another fundamental area formula.

Pythagorean Theorem

For right triangles.

Square Footage

For rectangular areas.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Cube Volume Calculator

Frequently Asked Questions

Why is the triangle area formula (1/2)bh?

Because every triangle is exactly half the area of a rectangle with the same base and height. You can see this visually by drawing a rectangle around any triangle — the triangle occupies exactly half of the rectangle's area. The (1/2) in the formula represents this 'half rectangle' relationship. It works for any triangle, not just right triangles.

What's the perpendicular height of a triangle?

It's the shortest distance from a vertex to the opposite side (the base), measured perpendicularly. For right triangles, one leg is the height relative to the other leg as base. For acute triangles, drop a perpendicular from any vertex to the opposite side. For obtuse triangles, you may need to extend the base line to meet the perpendicular from the obtuse-angle vertex.

How do I find the area if I only know the three sides?

Use Heron's formula: A = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2. This works for any triangle given just the three side lengths. Example: sides 3, 4, 5. s = 6. A = √[6 × 3 × 2 × 1] = √36 = 6. Pythagorean triples like 3-4-5 always give integer areas, but Heron's formula works for any valid triangle.

Can a triangle have more than one base?

Yes! Any of the three sides can be called the 'base.' Each base has its own corresponding height (perpendicular distance from the opposite vertex). The area is the same regardless of which side you choose as the base. This is useful when one height is easier to calculate than another — choose whichever base gives you the easiest calculation.