Enter the lengths of the three sides of the triangle. Even without the height, Heron's formula gives the area. The semiperimeter s is shown too.
Enter all three sides in the same unit (for example, all in inches). The area is in that unit squared (square inches for inches). The sum of any two sides must be greater than the third, or no triangle can be made.
Result and figure
Enter the lengths of the three sides of the triangle in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the lengths of the three sides of a triangle and get its area on the spot (no need to measure the height)
- The calculation uses Heron's formula \(S = \sqrt{s(s-a)(s-b)(s-c)}\) and also shows the semiperimeter \(s\) found along the way
- Also covers the familiar "base × height ÷ 2" and the area of an equilateral triangle, \(S = \frac{\sqrt{3}}{4}a^2\), with plain-language explanations
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter all three sides in the same unit (for example, all in inches). The area is in that unit squared (square inches for inches). Also, if the three sides break the triangle inequality ("the sum of any two sides is greater than the third side"), no triangle can be made, so the area cannot be calculated.
What is this calculation used for?
Measuring the area of land (surveying and real estate)
Splitting a plot of land into triangles and adding up their areas has long been a basic way to find the area of land. When the height of each triangle is hard to measure on site, measuring the three sides and using Heron's formula works instead.
For example, a triangular lot with sides of 100 feet, 150 feet and 170 feet has an area of about 7,446 square feet (about 0.17 acre). Being able to check for yourself that the area in a listing or a deed makes sense is reassuring knowledge when buying land, one of the biggest purchases in life.
Estimating paint and materials for roofs and walls (construction and DIY)
Buildings have many triangular surfaces, such as the gable end of a house (the triangle under a pitched roof) and the edges of sloped roofs. A triangular gable wall with a base of 24 feet and a height of 8 feet is \(24 \times 8 \div 2 = 96\) square feet.
A paint can says how many square feet it covers (a gallon often covers about 350 to 400 square feet), so once you know the area, you can estimate the number of cans and the cost yourself.
Estimating materials for crafts and sewing
Crafts often use triangular pieces, such as pennant flags and quilt patches. One triangular pennant with a base of 8 inches and a height of 12 inches has an area of \(8 \times 12 \div 2 = 48\) square inches.
Add up the area for the number of pieces you need, and you can decide how much fabric to buy without waste.
3D graphics and games are made of triangles
3D models in games and movies are made by combining a huge number of small triangles (polygons). Even a character's face that looks smoothly curved is a collection of triangles when you zoom in.
For a computer, Heron's formula, which finds the area from the distances between three vertices (the three side lengths), is easy to handle, and it is still used as a basic part of geometry-processing programs. It is one of the pieces of math behind game programmers and CG designers.
Formulas and figures
Heron's formula (area from the three side lengths)
Figure
Standard notation
(the usual math form)
\(s\)
\(=\)
\((a+b+c)\)
\(\div\)
\(2\)
\(S\)
\(=\)
\(\sqrt{s(s-a)(s-b)(s-c)}\)
In words
(symbols replaced with words)
③
\(s\): semiperimeter
\(=\)
①
\((a+b+c)\): sum of the three sides
\(\div\)
②
2 (to take half)
⑤
\(S\): area of the triangle
\(=\)
④
square root of \(s\), \(s-a\), \(s-b\) and \(s-c\) multiplied together
The formula in words
①
Divide the
sum of the three sides \((a+b+c)\)
②
by
2 (to take half)
③
to get the
semiperimeter \(s\)
④
Then take the
square root of the four numbers \(s\), \(s-a\), \(s-b\) and \(s-c\) multiplied together
⑤
and you get the
area of the triangle \(S\)
Quick example
For a triangle with sides 3 ft, 4 ft and 5 ft, first find the semiperimeter, then
semiperimeter \(s\)
\(=\)
sum of the three sides (3 + 4 + 5)
\(\div\)
2
area of the triangle \(S\)
\(=\)
\(\sqrt{6 \times 3 \times 2 \times 1}\)
\(s = (3 + 4 + 5) \div 2 = 6\)
\(S = \sqrt{6 \times (6-3) \times (6-4) \times (6-5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6\)
Key idea
The key point of Heron's formula is that it finds the area from the three side lengths alone, without measuring the height. It is named after Heron of Alexandria, a mathematician of about the first century (he is also called Hero, so it is sometimes called Hero's formula). In the US, it often appears in Geometry or Precalculus.
When the three sides cannot form a triangle (when they break the triangle inequality in formula 4), the number under the square root becomes 0 or less, and no area can be calculated. The formula itself tells you whether the three sides can form a triangle.
Base × height ÷ 2 (the basic formula from elementary school)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(b\)
\(\times\)
\(h\)
\(\div\)
\(2\)
In words
(symbols replaced with words)
④
\(S\): area of the triangle
\(=\)
①
\(b\): base
\(\times\)
②
\(h\): height
\(\div\)
③
2 (to take half)
The formula in words
①
Multiply the
base \(b\)
②
by the
height \(h\)
③
divide by
2 (to take half)
④
and you get the
area of the triangle \(S\)
Quick example
The area of a triangle with a base of 6 in and a height of 4 in is
area of the triangle \(S\)
\(=\)
base (6 in)
\(\times\)
height (4 in)
\(\div\)
2
\(6 \times 4 \div 2 = 12\)
Key idea
Why divide by 2? Take a second copy of the same triangle, flip it and attach it, and you get a parallelogram of "base × height" exactly. The triangle is half of it, so you divide by 2. Any side can be the base. The height is the distance from the opposite vertex to the base (or the base extended), measured at a right angle to it.
If you know the height, this formula is the quickest way. For example, the triangle with sides 3, 4 and 5 is a right triangle, so with base 3 and height 4, \(3 \times 4 \div 2 = 6\). This matches the area 6 found with Heron's formula.
Area of an equilateral triangle (from one side length)
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(\dfrac{\sqrt{3}}{4}\)
\(\times\)
\(a^2\)
In words
(symbols replaced with words)
③
\(S\): area of the equilateral triangle
\(=\)
②
\(\dfrac{\sqrt{3}}{4}\): constant (about 0.433)
\(\times\)
①
\(a^2\): the side length \(a\), squared
The formula in words
①
Multiply the
side length \(a\), squared
②
by the
constant \(\dfrac{\sqrt{3}}{4}\) (about 0.433)
③
and you get the
area of the equilateral triangle \(S\)
Quick example
The area of an equilateral triangle with 10-inch sides is
area of the equilateral triangle \(S\)
\(=\)
constant (about 0.433)
\(\times\)
side squared (10 × 10 = 100)
\(S = \dfrac{\sqrt{3}}{4} \times 10^{2} \approx 0.433 \times 100 = 43.3\)
Key idea
All three sides of an equilateral triangle are equal, so putting \(a = b = c\) into Heron's formula simplifies it all the way to this form (\(s = \frac{3a}{2}\) and \(s - a = \frac{a}{2}\), so \(S = \sqrt{\frac{3a}{2} \times (\frac{a}{2})^3} = \sqrt{\frac{3a^4}{16}} = \frac{\sqrt{3}}{4}a^2\)).
Since \(\sqrt{3} \approx 1.732\), \(\frac{\sqrt{3}}{4} \approx 0.433\). Remember it as "about 43% of the side squared", and you can use it for quick mental estimates.
Triangle inequality (can the three sides form a triangle?)
Figure
Standard notation
(the usual math form)
\(a + b\)
\(>\)
\(c\)
\(b + c\)
\(>\)
\(a\)
\(c + a\)
\(>\)
\(b\)
In words
(symbols replaced with words)
①
\(a + b\): sum of sides \(a\) and \(b\)
\(>\)
②
\(c\): the remaining side
\(b + c\): sum of sides \(b\) and \(c\)
\(>\)
\(a\): the remaining side
\(c + a\): sum of sides \(c\) and \(a\)
\(>\)
\(b\): the remaining side
The formula in words
①
Only when the
sum of two sides
②
is greater than the
remaining side
for all three combinations can the three sides form a triangle
Quick example
With sides 1, 2 and 10, the other two sides together (1 + 2 = 3) cannot reach across the longest side, 10, so no triangle can be made
sum of two sides (1 + 2 = 3)
\(<\)
remaining side (10)
\(1 + 2 = 3 < 10\)
Key idea
It also fails when the sum of two sides is exactly equal to the third side (for example, 1, 2, 3). The three sides collapse flat onto one straight line (area 0), which does not count as a triangle. This calculator also treats the equal case as an error.
The link to Heron's formula is neat too: when the sides break the triangle inequality, the number under the square root, \(s(s-a)(s-b)(s-c)\), is 0 or less, and no area can be calculated.
If you know only the three sides, find the semiperimeter s and use Heron's formula. If you know the height, use base × height ÷ 2. Either way, you reach the same area. Before calculating, just check that the three sides can form a triangle (the triangle inequality).
Symbols and terms
Symbols
| \(a\), \(b\), \(c\) | a, b, c | The lengths of the three sides of the triangle. Use the same unit for all three (if one is in inches, all are in inches). |
| \(s\) | s (lowercase) | The semiperimeter, half the sum of the three sides (the perimeter): \(s = (a+b+c) \div 2\). It is a step along the way in Heron's formula. |
| \(S\) | S (uppercase) | The area of the triangle you want to find (US textbooks often use \(A\) or \(K\) for area instead). Note that it is different from the lowercase \(s\) (the semiperimeter). |
| \(\sqrt{\phantom{a}}\) | square root (radical sign) | The sign for the square root, the number that gives the original number when squared. Example: \(\sqrt{36} = 6\) (6 squared is 36). |
| \(b\), \(h\) (in formula 2) | b, h | In the formula "base × height ÷ 2", \(b\) stands for the base and \(h\) for the height. Note that this \(b\) has a different meaning from side \(b\) in Heron's formula. |
Terms
| Heron's formula | A formula that gives the area of a triangle from the three side lengths alone. It is named after Heron of Alexandria, a mathematician of about the first century (also called Hero, so it is sometimes called Hero's formula). |
| semiperimeter | Half the perimeter (the sum of the three sides). It is used along the way in Heron's formula, and its symbol is the lowercase \(s\). |
| triangle inequality | The condition for three lengths to form a triangle - the sum of any two sides must be greater than the third side. If two sides add up to exactly the third side, the shape collapses onto a straight line and is not a triangle. |
| base | The side chosen as the reference when finding the area. Any side can be chosen, and the height depends on which side is the base. |
| height | The length of the perpendicular segment from the vertex opposite the base down to the base (or the base extended). It changes depending on which side is the base. |
| equilateral triangle | A triangle whose three sides are all equal. The side length \(a\) alone decides the area: \(S = \frac{\sqrt{3}}{4}a^2\). |
| square root | A number that gives the original number when squared. The sign is \(\sqrt{\phantom{a}}\) (the radical sign). Example: \(\sqrt{144} = 12\). It is taught in Grade 8. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to these topics is the fastest way forward.
| Area of a triangle (Grade 6) |
|
| The triangle inequality (Grade 7) |
|
| Expressions and substitution (Grades 6–7) |
|
| Square roots (Grade 8) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the area with Heron's formula
| Side a | 3 |
| Side b | 4 |
| Side c | 5 |
| Semiperimeter s | =(B1+B2+B3)/2 |
| Area of the triangle S | =SQRT(B4*(B4-B1)*(B4-B2)*(B4-B3)) |
Table to find the area with base × height ÷ 2
| Base | 6 |
| Height | 4 |
| Area of the triangle S | =B1*B2/2 |
Table to find the area of an equilateral triangle
| Side length a | 10 |
| Area of the equilateral triangle S | =SQRT(3)/4*B1^2 |
In the first table, B1 to B3 are the inputs for the three sides, and B4 (the semiperimeter) and B5 (the area) are calculated automatically. "SQRT" is Excel's square root (√) function, and "^2" means squared.
For example, the first table shows 6 in B4 and 6 in B5 (a triangle with sides 3, 4 and 5 has area 6). The second table shows 12 in B3, and the third shows about 43.3 in B2. Just replace the input numbers with the values for your own triangle.
For example, the first table shows 6 in B4 and 6 in B5 (a triangle with sides 3, 4 and 5 has area 6). The second table shows 12 in B3, and the third shows about 43.3 in B2. Just replace the input numbers with the values for your own triangle.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the area with Heron's formula
| Side a | 3 |
| Side b | 4 |
| Side c | 5 |
| Semiperimeter s | =(B1+B2+B3)/2 |
| Area of the triangle S | =SQRT(B4*(B4-B1)*(B4-B2)*(B4-B3)) |
Table to find the area with base × height ÷ 2
| Base | 6 |
| Height | 4 |
| Area of the triangle S | =B1*B2/2 |
Table to find the area of an equilateral triangle
| Side length a | 10 |
| Area of the equilateral triangle S | =SQRT(3)/4*B1^2 |
The same formulas and function (SQRT) as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the input numbers with the values for your own triangle.
How to calculate it in Python
import math
edge_a = 3.0 # length of side a
edge_b = 4.0 # length of side b
edge_c = 5.0 # length of side c
# First check the triangle inequality (the sum of any two sides is greater than the third)
if edge_a + edge_b > edge_c and edge_b + edge_c > edge_a and edge_c + edge_a > edge_b:
half_perimeter = (edge_a + edge_b + edge_c) / 2 # semiperimeter s
area = math.sqrt(half_perimeter * (half_perimeter - edge_a)
* (half_perimeter - edge_b) * (half_perimeter - edge_c))
print(f"Semiperimeter s: {half_perimeter}")
print(f"Area of the triangle S: {area}")
else:
print("These three sides cannot form a triangle")
Runs with just math from the standard library. math.sqrt(...) calculates the square root (√). Replace the three side lengths at the top with your own values and run it.
How to write it in LaTeX and other math languages (copy and paste)
Heron's formula (area from the three side lengths)
S = √(s(s − a)(s − b)(s − c)), s = (a + b + c) / 2
S = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \dfrac{a+b+c}{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<msqrt>
<mrow>
<mi>s</mi>
<mo>(</mo><mi>s</mi><mo>−</mo><mi>a</mi><mo>)</mo>
<mo>(</mo><mi>s</mi><mo>−</mo><mi>b</mi><mo>)</mo>
<mo>(</mo><mi>s</mi><mo>−</mo><mi>c</mi><mo>)</mo>
</mrow>
</msqrt>
<mo>,</mo>
<mspace width="1em"/>
<mi>s</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow>
<mn>2</mn>
</mfrac>
</mrow>
</math>
S = sqrt(s(s-a)(s-b)(s-c)), s = (a+b+c)/2
s = (a + b + c)/2; Sqrt[s (s - a) (s - b) (s - c)]
s := (a + b + c)/2; S := sqrt(s*(s - a)*(s - b)*(s - c));
s = (a + b + c)/2; S = sqrt(s*(s - a)*(s - b)*(s - c));
S = √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2
Base × height ÷ 2 (the basic formula from elementary school)
S = b × h ÷ 2
S = \dfrac{1}{2} b h
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mfrac><mn>1</mn><mn>2</mn></mfrac>
<mi>b</mi>
<mi>h</mi>
</mrow>
</math>
S = 1/2 b h
b*h/2
S := b*h/2;
S = b*h/2;
S = bh/2
Area of an equilateral triangle (from one side length)
S = (√3 / 4) × a²
S = \dfrac{\sqrt{3}}{4} a^{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mfrac>
<msqrt><mn>3</mn></msqrt>
<mn>4</mn>
</mfrac>
<msup><mi>a</mi><mn>2</mn></msup>
</mrow>
</math>
S = (sqrt(3)/4) a^2
Sqrt[3]/4*a^2
S := sqrt(3)/4*a^2;
S = sqrt(3)/4*a^2;
S = (√3/4)a^2
Triangle inequality (can the three sides form a triangle?)
a + b > c, b + c > a, c + a > b
a + b > c, \quad b + c > a, \quad c + a > b
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>a</mi><mo>+</mo><mi>b</mi><mo>></mo><mi>c</mi>
<mo>,</mo>
<mspace width="1em"/>
<mi>b</mi><mo>+</mo><mi>c</mi><mo>></mo><mi>a</mi>
<mo>,</mo>
<mspace width="1em"/>
<mi>c</mi><mo>+</mo><mi>a</mi><mo>></mo><mi>b</mi>
</mrow>
</math>
a + b > c, b + c > a, c + a > b
a + b > c && b + c > a && c + a > b
is(a + b > c and b + c > a and c + a > b);
(a + b > c) && (b + c > a) && (c + a > b)
a + b > c, b + c > a, c + a > b
How to have ChatGPT do the calculation
You are a calculation assistant for geometry. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). The three sides of a triangle are 7.5, 10.2 and 12.3 (in feet). 1. Check whether these three sides can form a triangle (the triangle inequality). 2. Find the semiperimeter s = (a + b + c) ÷ 2. 3. Find the area of the triangle with Heron's formula S = √(s(s−a)(s−b)(s−c)). Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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