Enter the top base, bottom base and height of the trapezoid. The area is calculated with the formula "(top base + bottom base) × height ÷ 2", and the steps are shown.
Enter all three lengths in the same unit (for example, all in inches). The area comes out in that unit squared (such as in²).
Result and figure
Enter the top base, bottom base and height in the fields on the left and press "Calculate". The result will appear here.
Table of Contents
-
What you can do on this page
-
What is this calculation used for?
-
How to Use
-
Formulas and figures
-
Symbols and terms
-
Good to know before you start
-
How to calculate it in Excel
-
How to calculate it in Google Sheets
-
How to calculate it in Python
-
How to write it in LaTeX and other math languages (copy and paste)
-
How to have ChatGPT do the calculation
-
DataChef Features
-
Related Features
-
NumberChef Calculators List
What you can do on this page
- Enter the top base, the bottom base and the height, and you get the area of the trapezoid right away from the formula "(top base + bottom base) × height ÷ 2"
- The steps with your numbers plugged in are shown too, so you can use them to check your homework or a test
- Decimal lengths (for example, 2.5 in) work too
- A plain-language explanation of the formula (why you divide by 2), formulas to work back from the area to the height or a base, and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
Enter all three lengths in the same unit (for example, all in inches). The area comes out in that unit squared (such as in²).
What is this calculation used for?
Estimating the size of a lot or a field (real estate and farming)
A lot that is narrower at the street than at the back can be treated as a trapezoid to estimate its area. With 40 ft of street frontage, a back line of 60 ft and a depth of 100 ft, the area is \((40 + 60) \times 100 \div 2 = 5{,}000\) ft² (about 0.11 acre).
Lot size is the basis for the price, property taxes and, for a field, the amount of seed and fertilizer, so this is one of the area formulas you meet most often in daily life (an official survey uses more precise methods).
Finding the amount of soil or water from a ditch or levee cross section (civil engineering)
Drainage ditches and levees usually have a trapezoid-shaped cross section. A ditch 4 ft wide at the top, 2 ft wide at the bottom and 1.5 ft deep has a cross-sectional area of \((4 + 2) \times 1.5 \div 2 = 4.5\) ft². Multiply by the length of the ditch and you get a rough idea of how much soil to dig out or how much water it holds (for 100 ft of ditch, 450 ft³, or about 16.7 cubic yards).
This calculation is used every day to estimate public works such as roads, rivers and dams.
Estimating materials for a roof or wall (construction and DIY)
Trapezoid-shaped surfaces are all over a house, such as the sides of a hip roof. A roof face 10 ft long at the top, 30 ft long at the bottom and 12 ft up the slope has an area of \((10 + 30) \times 12 \div 2 = 240\) ft², or 2.4 roofing squares (1 square = 100 ft²).
How much paint, shingles or siding you need is decided by this area, so the calculation goes straight into a contractor's estimate and helps you avoid buying too much for a DIY project.
Finding the distance traveled from a speed graph (science, physics and data analysis)
For a vehicle that speeds up at a steady rate, the speed-time graph forms a trapezoid, and its area is the distance traveled. Speeding up steadily from 40 mph to 60 mph over 0.5 hours covers \((40 + 60) \times 0.5 \div 2 = 25\) miles.
Finding the area under a graph with trapezoids is a basic idea in high school physics, and it is also used in scientific computing under the name "trapezoidal rule" (numerical integration).
Formulas and figures
Formula for the area of a trapezoid
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\((\)
\(a\)
\(+\)
\(b\)
\()\)
\(\times\)
\(h\)
\(\div\)
\(2\)
In words
(symbols replaced with words)
⑤
\(S\): area of the trapezoid
\(=\)
\((\)
①
\(a\): top base
\(+\)
②
\(b\): bottom base
\()\)
\(\times\)
③
\(h\): height
\(\div\)
④
\(2\): takes half
The formula in words
①
Add the
\(a\): top base
②
and the
\(b\): bottom base
③
multiply by the
\(h\): height
④
divide by
\(2\): takes half
⑤
and you get the
\(S\): area of the trapezoid
Quick example
The area of a trapezoid with a top base of 8 in, a bottom base of 12 in and a height of 5 in is
\(S\): area of the trapezoid
\(=\)
\((\)
top base (8 in)
\(+\)
bottom base (12 in)
\()\)
\(\times\)
height (5 in)
\(\div\)
takes half (2)
\((8 + 12) \times 5 \div 2 = 20 \times 5 \div 2 = 100 \div 2 = 50\)
Key idea
Why divide by 2 at the end? Take a second copy of the same trapezoid, turn it upside down and fit it against the first one. Together they make a parallelogram whose base is (top base + bottom base) and whose height is the same. The area of a parallelogram is base × height, so that is (top base + bottom base) × height. One trapezoid is half of it, so you divide by 2.
The answer is in the length unit squared (enter inches and you get in²). It does not matter which parallel side you call the top base: only the order of the addition changes, so the area is the same.
Formula for the height (working back from the area)
Figure
Standard notation
(the usual math form)
\(h\)
\(=\)
\(S\)
\(\times\)
\(2\)
\(\div\)
\((\)
\(a\)
\(+\)
\(b\)
\()\)
In words
(symbols replaced with words)
⑤
\(h\): height
\(=\)
①
\(S\): area of the trapezoid
\(\times\)
②
\(2\): undoes the halving
\(\div\)
\((\)
③
\(a\): top base
\(+\)
④
\(b\): bottom base
\()\)
The formula in words
①
Take the
\(S\): area of the trapezoid
②
multiply it by
\(2\): undoes the halving
(back to the area of the parallelogram before dividing by 2)
③
divide by the sum of the
\(a\): top base
④
and the
\(b\): bottom base
⑤
and you get the
\(h\): height
Quick example
The height of a trapezoid with an area of 50 in², a top base of 8 in and a bottom base of 12 in is
\(h\): height
\(=\)
area (50 in²)
\(\times\)
undoes the halving (2)
\(\div\)
\((\)
top base (8 in)
\(+\)
bottom base (12 in)
\()\)
\(50 \times 2 \div (8 + 12) = 100 \div 20 = 5\)
Key idea
This works the area formula backward. Undo the final "÷ 2" with "× 2", then undo the "× height" by dividing by (top base + bottom base), and the height is what is left.
Tests often ask this kind of question: "You know the area and both bases. Find the height."
Formula for the sum of the bases (working back from the area)
Figure
Standard notation
(the usual math form)
\(a + b\)
\(=\)
\(S\)
\(\times\)
\(2\)
\(\div\)
\(h\)
In words
(symbols replaced with words)
④
\(a + b\): sum of the top and bottom bases
\(=\)
①
\(S\): area of the trapezoid
\(\times\)
②
\(2\): undoes the halving
\(\div\)
③
\(h\): height
The formula in words
①
Take the
\(S\): area of the trapezoid
②
multiply it by
\(2\): undoes the halving
③
divide by the
\(h\): height
④
and you get the
\(a + b\): sum of the top and bottom bases
Quick example
For a trapezoid with an area of 50 in² and a height of 5 in, the sum of the top and bottom bases is
\(a + b\): sum of the top and bottom bases
\(=\)
area (50 in²)
\(\times\)
undoes the halving (2)
\(\div\)
height (5 in)
\(50 \times 2 \div 5 = 100 \div 5 = 20\)
Key idea
This formula only gives you the sum of the top and bottom bases. To find the top base alone, subtract the bottom base from the sum (for example, if the sum is 20 in and the bottom base is 12 in, the top base is \(20 - 12 = 8\) in).
The trick is not to do it all in one formula, but in two steps: find the sum, then subtract.
The area of a trapezoid is "(top base + bottom base) × height ÷ 2". Two copies of the same trapezoid fit together into a parallelogram, and that is where the "÷ 2" comes from. Once you see this, working back from the area to the height or a base is just following the same formula in reverse.
Symbols and terms
Symbols
| \(S\) | ess | The area of the trapezoid you want to find. This page writes area as \(S\) (from "surface"); many US textbooks write \(A\) instead. Both stand for the same thing. |
| \(a\) | a | The length of the top base (one of the two parallel sides). (Example - if the top side is 8 in, \(a = 8\)) |
| \(b\) | bee | The length of the bottom base (the other parallel side). (Example - if the bottom side is 12 in, \(b = 12\)) Some textbooks call the two bases \(b_1\) and \(b_2\). |
| \(h\) | aitch | The height - the perpendicular distance between the top base and the bottom base. It is the first letter of "height". |
| \(\mathrm{in}^2\) | square inches | A unit of area - how many squares with 1-inch sides fit inside. If you enter lengths in feet, the area is in ft²; in centimeters, it is in cm². |
Terms
| trapezoid | A four-sided shape (quadrilateral) with one pair of opposite sides parallel. The two parallel sides are the bases, and the other two sides are the legs. In British English it is called a trapezium. |
| top base | One of the two parallel sides of a trapezoid. It usually refers to the side drawn at the top, but whichever side you call the top base, the area comes out the same. |
| bottom base | The parallel side of a trapezoid that is not the top base. It usually refers to the side drawn at the bottom. |
| height | The distance between the top base and the bottom base, measured straight across (at a right angle). Note that it is not the length of a slanted side (leg). |
| parallel | Two straight lines that never meet, however far you extend them. The perpendicular distance between two parallel lines is the same wherever you measure it. |
| parallelogram | A four-sided shape with both pairs of opposite sides parallel. Its area is base × height, and it explains why the trapezoid formula divides by 2. |
| isosceles trapezoid | A trapezoid whose two legs (slanted sides) are the same length, so it is symmetric left to right. Even for this special shape, the area comes from the same formula "(top base + bottom base) × height ÷ 2". |
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.
| What area is, and the area of a rectangle (Grade 3) |
|
| Perpendicular and parallel lines (Grade 4) |
|
| Area of parallelograms and triangles (Grade 6) |
|
| Multiplying and dividing decimals (Grades 5–6) |
|
| Finding an unknown number in an equation (Grades 3–4) |
|
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 of a trapezoid
| Top base a | 8 |
| Bottom base b | 12 |
| Height h | 5 |
| Area of the trapezoid S | =(B1+B2)*B3/2 |
Table to find the height
| Area of the trapezoid S | 50 |
| Top base a | 8 |
| Bottom base b | 12 |
| Height h | =B1*2/(B2+B3) |
Table to find the sum of the bases
| Area of the trapezoid S | 50 |
| Height h | 5 |
| Sum of the bases a+b | =B1*2/B2 |
After pasting, the upper rows are your inputs and the last row is calculated automatically.
"*" is multiplication and "/" is division. "=(B1+B2)*B3/2" is "(B1 plus B2) × B3 ÷ 2".
In the first table, for example, B4 shows 50 (for a top base of 8, a bottom base of 12 and a height of 5). Just replace the inputs with your own numbers.
"*" is multiplication and "/" is division. "=(B1+B2)*B3/2" is "(B1 plus B2) × B3 ÷ 2".
In the first table, for example, B4 shows 50 (for a top base of 8, a bottom base of 12 and a height of 5). Just replace the inputs with your own numbers.
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 of a trapezoid
| Top base a | 8 |
| Bottom base b | 12 |
| Height h | 5 |
| Area of the trapezoid S | =(B1+B2)*B3/2 |
Table to find the height
| Area of the trapezoid S | 50 |
| Top base a | 8 |
| Bottom base b | 12 |
| Height h | =B1*2/(B2+B3) |
Table to find the sum of the bases
| Area of the trapezoid S | 50 |
| Height h | 5 |
| Sum of the bases a+b | =B1*2/B2 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the inputs with the lengths of your own trapezoid.
How to calculate it in Python
top_base = 8 # top base
bottom_base = 12 # bottom base
height = 5 # height
area = (top_base + bottom_base) * height / 2 # area of the trapezoid
print(f"Area of the trapezoid: {area}")
# Working back: find the height from the area and the two bases
height_from_area = area * 2 / (top_base + bottom_base)
print(f"Height (worked back): {height_from_area}")
Runs with the standard library only. Replace the first three numbers with the lengths of your trapezoid and run it. If all lengths are in the same unit, the area comes out in that unit squared (such as in²).
How to write it in LaTeX and other math languages (copy and paste)
Formula for the area of a trapezoid
S = (a + b) × h ÷ 2
S = \frac{(a + b) \times h}{2}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>)</mo>
<mo>×</mo>
<mi>h</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</math>
S = ((a + b) xx h) / 2
(a + b)*h/2
S := (a + b)*h/2;
S = (a + b)*h/2;
S = ((a + b) × h)/2
Formula for the height (working back from the area)
h = S × 2 ÷ (a + b)
h = \frac{2S}{a + b}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow><mn>2</mn><mi>S</mi></mrow>
<mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow>
</mfrac>
</mrow>
</math>
h = (2S) / (a + b)
2*S/(a + b)
h := 2*S/(a + b);
h = 2*S/(a + b);
h = (2S)/(a + b)
Formula for the sum of the bases (working back from the area)
a + b = S × 2 ÷ h
a + b = \frac{2S}{h}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow>
<mo>=</mo>
<mfrac>
<mrow><mn>2</mn><mi>S</mi></mrow>
<mi>h</mi>
</mfrac>
</mrow>
</math>
a + b = (2S) / h
2*S/h
sumOfBases := 2*S/h;
sum_of_bases = 2*S/h;
a + b = (2S)/h
How to have ChatGPT do the calculation
You are a math calculation assistant. 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). A trapezoid has a top base of 8 in, a bottom base of 12 in and a height of 5 in. Find each of the following: 1. The area of this trapezoid (formula: (top base + bottom base) × height ÷ 2) 2. Assuming you only know the area from step 1 and the two bases, the height worked back from them (check that it matches the original height) Show the formulas you used and the numbers from the execution result.
How to Use
-
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
DataChef Features
Easy and Free
Unlimited conversions for free.
No technical knowledge required.
Intuitive and user-friendly operation.
No technical knowledge required.
Intuitive and user-friendly operation.
No Registration Required
Available immediately after access.
Can be used without registering personal information.
Can be used without registering personal information.
Safe and Secure
Fully SSL encrypted communication.
Automatic file deletion by clicking "download".
Automatic file deletion by clicking "download".
Fast
High-speed site access
and rapid file conversion.
and rapid file conversion.
No Watermark
No watermark.
No attribution required.
No attribution required.
Commercial Use Available
Free for commercial use.
No need to contact us for commercial use permission.
No need to contact us for commercial use permission.
Related Features
Triangle Area Calculator (from Three Sides with Heron's Formula)
Calculates the area of a triangle from its base and height
Area of a Rectangle Calculator (Length × Width)
Calculates the area of a rectangle from its length and width
Circle Calculator (Radius, Diameter, Circumference and Area)
Enter any one of radius, diameter, circumference or area to get all the others
Ad 
