What is the Area of a Triangle?
The area of a triangle is the measure of the two-dimensional region enclosed by its three sides. In geometric terms, it represents the total space contained within the triangle's boundaries, expressed in square units such as square centimeters (cm²), square meters (m²), square inches (in²), or square feet (ft²).
Mathematicians developed several formulas to calculate triangle area based on whichever measurements are available. The calculator above supports all four primary methods: base and height, three sides (Heron's formula), two sides and included angle (SAS), and two angles and included side (ASA). Each method works for any triangle type, whether equilateral, isosceles, right, scalene, or obtuse.
Area of Triangle Formulas and Step-by-Step Examples
Below are the four universal methods for calculating the area of any triangle. Each section includes the formula, a worked example showing how to find the area, and a reverse calculation showing how to find a missing parameter when you already know the area. Our calculator handles all these calculations automatically when you select the appropriate method.
Base and Height Method
The most fundamental formula for triangle area uses the base and the perpendicular height:
where $b$ is the length of the base and $h$ is the perpendicular height, the distance from the base to the opposite vertex, measured at a 90° angle. This height is sometimes called the altitude of the triangle.
Example: Finding the Area
Given: Base $b = 10$ cm, Height $h = 6$ cm
- Identify the known values: $b = 10$, $h = 6$
- Substitute into the formula: $A = \frac{1}{2} \times 10 \times 6$
- Calculate: $A = 30$ cm²
Reverse Example: Finding the Height from Area
If you know the area and the base, rearrange the formula to solve for height:
Given: Area $A = 45$ cm², Base $b = 9$ cm
- Substitute: $h = \frac{2 \times 45}{9}$
- Calculate: $h = \frac{90}{9} = 10$ cm
Three Sides Method (Heron's Formula)
When you know all three side lengths but not the height, Heron's formula provides an elegant solution. Named after Heron of Alexandria, a Greek mathematician and engineer who documented it around 60 AD, this formula has been used for over two thousand years.
First, calculate the semi-perimeter $s$ (half the perimeter):
Then, the area is:
Example: Finding the Area from Three Sides
Given: Sides $a = 3$ cm, $b = 4$ cm, $c = 5$ cm
- Calculate the semi-perimeter: $s = \frac{3+4+5}{2} = 6$
- Apply Heron's formula: $A = \sqrt{6(6-3)(6-4)(6-5)}$
- Simplify: $A = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6$ cm²
Reverse Example: Finding a Missing Side from Area
If you know the area and two sides, finding the third side requires a multi-step process. First, use the SAS area formula to find the sine of the included angle, then use the Law of Cosines to find the missing side.
Given: Area $A = 6$ cm², sides $a = 3$ cm, $b = 4$ cm. Find side $c$.
- Use $A = \frac{1}{2}ab\sin(C)$ to find $\sin(C)$: $$\sin(C) = \frac{2A}{ab} = \frac{2 \times 6}{3 \times 4} = \frac{12}{12} = 1$$
- Find angle $C$: $C = \arcsin(1) = 90°$
- Use the Law of Cosines: $c = \sqrt{a^2 + b^2 - 2ab\cos(C)}$ $$c = \sqrt{9 + 16 - 24\cos(90°)} = \sqrt{25 - 0} = 5 \text{ cm}$$
Two Sides and Included Angle (SAS)
When you know two sides and the angle between them, the trigonometric area formula is:
where $a$ and $b$ are the two known sides, and $C$ is the included angle (the angle formed between sides $a$ and $b$). This method works for any triangle, including obtuse triangles where the height falls outside the triangle.
Example: Finding the Area (SAS)
Given: $a = 8$ cm, $b = 10$ cm, included angle $\gamma = 45°$
- Substitute: $A = \frac{1}{2} \times 8 \times 10 \times \sin(45°)$
- Calculate: $A = 40 \times 0.7071 \approx 28.28$ cm²
Reverse Example 1: Finding the Included Angle
If you know the area and two sides, rearrange to find the angle:
Given: Area $A = 25$ cm², $a = 10$ cm, $b = 8$ cm
- Calculate $\sin(C) = \frac{2 \times 25}{10 \times 8} = \frac{50}{80} = 0.625$
- Find the angle: $C = \arcsin(0.625) \approx 38.68°$
Reverse Example 2: Finding a Missing Side
If you know the area, one side, and the included angle, solve for the other side:
Given: Area $A = 30$ cm², $b = 12$ cm, $\gamma = 30°$
- Substitute: $a = \frac{2 \times 30}{12 \times \sin(30°)}$
- Calculate: $a = \frac{60}{12 \times 0.5} = \frac{60}{6} = 10$ cm
Two Angles and Included Side (ASA)
When you know two angles and the side between them, use this advanced formula:
where $\alpha$ and $\beta$ are the two known angles, and $c$ is the included side. This formula relies on the angle sum property of triangles: the three interior angles always add up to 180°.
Example: Finding the Area (ASA)
Given: $\alpha = 60°$, $\beta = 60°$, included side $c = 10$ cm
- Note: $\alpha + \beta = 120°$
- Substitute: $A = \frac{100 \times \sin(60°) \times \sin(60°)}{2 \times \sin(120°)}$
- Calculate: $A = \frac{100 \times 0.8660 \times 0.8660}{2 \times 0.8660} = \frac{75}{1.732} \approx 43.30$ cm²
Note: This is an equilateral triangle with side 10 cm, and the result matches the specialized formula $A = \frac{\sqrt{3}}{4} \times 10^2$.
Reverse Example: Finding the Included Side
Rearrange the formula to solve for $c$:
Given: Area $A = 50$ cm², $\alpha = 45°$, $\beta = 60°$
- Note: $\alpha + \beta = 105°$
- Substitute: $c = \sqrt{\frac{2 \times 50 \times \sin(105°)}{\sin(45°) \times \sin(60°)}}$
- Calculate: $c = \sqrt{\frac{100 \times 0.9659}{0.7071 \times 0.8660}} = \sqrt{\frac{96.59}{0.6124}} \approx \sqrt{157.74} \approx 12.56$ cm
How to Calculate Area of Different Triangle Types
While the four universal methods above work for any triangle, certain triangle types have specialized formulas that simplify the calculation. The calculator above automatically applies the appropriate method when you enter measurements for these specific triangle types.
Area of Equilateral Triangle
An equilateral triangle has all three sides equal ($a = b = c$) and all three angles equal to 60°. The area formula simplifies to:
Derivation: Drop a perpendicular from one vertex to the opposite side, creating two 30-60-90 right triangles. The height is $h = \frac{\sqrt{3}}{2}a$. Substituting into $A = \frac{1}{2}bh$ gives $A = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{4}a^2$.
Example
An equilateral triangle with side length 8 cm:
- $A = \frac{\sqrt{3}}{4} \times 8^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3}$
- $A \approx 16 \times 1.732 \approx 27.71$ cm²
Area of Isosceles Triangle
An isosceles triangle has two equal sides. If the equal sides have length $a$ and the base has length $b$:
Alternative method: Split the isosceles triangle down its axis of symmetry to create two congruent right triangles. Each has hypotenuse $a$, base $\frac{b}{2}$, and height $h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2}$. Then use $A = \frac{1}{2}bh$.
Example
An isosceles triangle with equal sides of 10 cm and base 12 cm:
- Find the height: $h = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$ cm
- Calculate area: $A = \frac{1}{2} \times 12 \times 8 = 48$ cm²
Area of Right Triangle
A right triangle has one 90° angle. The two sides forming the right angle (called legs) naturally serve as base and height:
If you know the hypotenuse $c$ and one leg, use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the other leg first.
Example
A right triangle with legs 6 cm and 8 cm:
- $A = \frac{1}{2} \times 6 \times 8 = 24$ cm²
Note: The hypotenuse would be $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$ cm, the classic 3-4-5 right triangle scaled by 2.
Area of Scalene Triangle
A scalene triangle has all three sides and all three angles different. Since the height is rarely known directly, Heron's formula (the Three Sides method) is typically the most practical approach. Simply measure or obtain all three side lengths and apply the formula from the SSS section above.
Area of Obtuse Triangle
An obtuse triangle has one angle greater than 90°. A common point of confusion is that the perpendicular height from the opposite vertex falls outside the triangle, beyond the extended base. However, all the standard formulas still apply perfectly. Just ensure you measure the height correctly as the perpendicular distance from the vertex to the line containing the base, not just to the base segment itself. The SAS, ASA, and Heron's formulas handle obtuse triangles automatically without any special adjustments.
Your Triangle Area Questions, Answered
Which calculation method should I use?
Use this practical decision guide:
- If you know the base and perpendicular height: Use the Base and Height method (simplest)
- If you know all three side lengths: Use Heron's Formula (SSS)
- If you know two sides and the angle between them: Use the SAS method
- If you know two angles and any side: Use the ASA method
When in doubt, Heron's formula is the most versatile because it only requires side lengths, which are often the easiest measurements to obtain in real-world situations.
Why do I need at least one side length if I have all three angles?
Three angles define the shape of a triangle but not its size. Triangles with the same angles but different side lengths are called similar triangles. They have the same proportions but different scales. Without at least one side length, you cannot determine the actual size or area.
Think of it this way: knowing a triangle has angles 60°, 60°, 60° tells you it's equilateral, but it could be tiny (1 cm sides) or enormous (100 m sides). The angles alone don't tell you which.
How do I handle unit conversions correctly?
Always convert all measurements to the same unit before calculating. For example, if one side is in meters and another in centimeters, convert both to the same unit first. A common mistake is mixing units and getting wildly incorrect results.
For area conversions, remember that units are squared: $1 \text{ m}^2 = 10{,}000 \text{ cm}^2$ (not 100). This is because $1 \text{ m} \times 1 \text{ m} = 100 \text{ cm} \times 100 \text{ cm} = 10{,}000 \text{ cm}^2$. Our calculator handles these conversions automatically, but understanding the principle helps you verify results.
What if my triangle has an angle greater than 90 degrees?
All the formulas work perfectly for obtuse triangles. The only consideration is that when using the base-height method, the perpendicular height extends outside the triangle. You must measure from the vertex to the line containing the base, not just to the base itself. The SAS, ASA, and Heron's formulas handle obtuse triangles automatically without any special adjustments.
Why does my calculator show slightly different results than my manual calculation?
Manual calculations often involve rounding intermediate steps (e.g., using $\pi \approx 3.14$ or $\sin(45°) \approx 0.71$). Our calculator uses high-precision floating-point arithmetic internally, carrying many more decimal places through each step. Small differences in the final result, typically in the third or fourth decimal place, are normal and indicate greater precision, not error.
For most practical purposes, rounding to 2–4 decimal places is sufficient. If you need exact values, keep intermediate results in symbolic form (e.g., $16\sqrt{3}$ instead of 27.71).
How do I find the area of a triangle given sides?
Use Heron's formula (the Three Sides method). Enter all three side lengths into the calculator, and it will instantly compute the area. Heron's formula is the most reliable method when height is unknown because it requires only the side lengths.
Remember the triangle inequality: the sum of any two sides must be greater than the third side. If this condition isn't met, the three lengths cannot form a valid triangle.
How do I find the area of a triangle given angles?
You cannot find the area with angles alone. You need at least one side length. If you have two angles and any side, use the ASA method. If you have all three angles and one side, you can still use the ASA method (the third angle is found by subtracting the known angles from 180°).
For example, if you know angles are 50°, 60°, and 70°, and one side is 15 cm, you can calculate the area using the ASA formula with any pair of angles and their included side.
References and Mathematical Sources
The formulas and methods presented on this page are based on established geometric principles documented in standard mathematical references, including:
- Euclid's Elements (c. 300 BC): foundational propositions on triangle geometry
- Heron of Alexandria's Metrica (c. 60 AD): original documentation of Heron's formula
- Modern Trigonometry: standard treatments of the Law of Sines, Law of Cosines, and trigonometric area formulas
- NIST Digital Library of Mathematical Functions: verification of trigonometric identities and constants
All calculations performed by this calculator have been verified against established mathematical identities and theorems. The implementation uses IEEE 754 double-precision floating-point arithmetic to ensure numerical accuracy across all supported unit conversions.