Area and Perimeter Calculator
Pick the shape, enter the measurements and see the area, the perimeter and the formulas applied, all explained, in the unit you want.
Educational use: results are estimates made for study and reference. Checking the answer and deciding what to do with it is the sole responsibility of whoever uses the tool. Read the legal notice.
On this page
- How to calculate here
- The shapes and their measurements
- The formulas for each shape
- Solved examples
- Common mistakes in plane geometry
- Handy area conversions
- Geometry: measuring the earth
- What it is used for in practice
- Frequently asked questions about area and perimeter
- Before you buy material or cut anything
- Want to know more?
- Plane geometry glossary
- Tools that go well with this one
How to calculate here
- Pick the shape. The form shows only the fields that shape needs.
- Enter the measurements in the same unit. If one is in meters and another in centimeters, convert first (100 cm = 1 m).
- Select the unit you used, so the result comes with the right name (m², cm² and so on).
- Calculate and see the area highlighted, the perimeter right below it and the step-by-step work applying the formulas to your numbers.
For the triangle, there are two ways: with base and height (gives only the area) or with all three sides (gives area and perimeter, using Heron's formula). For the trapezoid and the parallelogram, the slanted sides are optional and are only used to get the perimeter.
Area or perimeter? Area is how much surface the shape covers, like the floor of a room or the paint on a wall. Perimeter is the outline, like the baseboard that runs around the room or the fence around a lot.
The shapes and their measurements
This drawing shows which measurements go into each calculation. Pay attention to the height of the triangle: it is always perpendicular to the base, even when the triangle is slanted.
The formulas for each shape
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | l × w | 2 × (l + w) |
| Square | side² | 4 × side |
| Triangle (base and height) | b × h ÷ 2 | sum of the three sides |
| Triangle (three sides) | √[s(s−a)(s−b)(s−c)] | a + b + c |
| Circle | π × r² | 2 × π × r (circumference) |
| Trapezoid | (B + b) × h ÷ 2 | sum of the four sides |
| Parallelogram | base × height | 2 × (base + side) |
| Rhombus | D × d ÷ 2 | 4 × side |
Heron's formula, for when you only have the sides
Area = √[ s · (s − a) · (s − b) · (s − c) ]s is the semiperimeter
It is very handy: you do not need to know the height or the angles, only the lengths of the three sides. But keep in mind that the triangle has to exist, which means each side must be shorter than the sum of the other two.
Where does the ÷ 2 for the triangle come from?
Any triangle fits exactly into half of a parallelogram (or rectangle) with the same base and the same height. That is why the area of a triangle is half of base × height. The trapezoid follows the same reasoning: two identical trapezoids, one flipped upside down, form a parallelogram with base B + b.
Solved examples
1. The floor of a room
A room measures 4.5 m by 3.2 m. How many square meters of flooring are needed? And how much baseboard?
That is 14.4 m² of flooring. The baseboard runs 15.4 m, minus the width of the doors.
2. The pizza
A pizza is 30 cm in diameter. What is its area?
A useful fact: a 40 cm pizza has an area of 1,256.6 cm², almost double that of the 30 cm one (1.78 times, to be exact). Doubling the diameter quadruples the area.
3. A trapezoid-shaped lot
A lot has bases of 20 m and 12 m, and a height of 8 m. What is the area?
4. A triangle with only the sides
Sides of 13, 14 and 15 cm.
This set of measurements is famous because the area comes out as a whole number.
5. Rhombus
Diagonals of 10 cm and 6 cm.
Common mistakes in plane geometry
Mixing units
Multiplying 4 m by 80 cm without converting gives a meaningless result. Convert everything to the same unit first. Remember that 1 m² equals 10,000 cm², not 100 cm².
Using the slanted side as the height
In the triangle, the trapezoid and the parallelogram, the height is the perpendicular distance between the bases. The slanted side is always longer than the height. Using it in the formula overestimates the area.
Confusing radius with diameter
The circle formula uses the radius. If you measured the diameter (from edge to edge, through the center), divide it by 2 first. Forgetting this multiplies the area by 4.
Forgetting that area is "squared"
An area result always comes in square units (m², cm²). If a result in m² shows up as just "m", something is wrong.
Forgetting waste on building jobs
For floors and wall coverings, the calculated area is the net area. In practice, it is common to add a margin for cuts and breakage. How much depends on the material and the layout of the installation.
Handy area conversions
| Unit | Equals |
|---|---|
| 1 m² | 10,000 cm² |
| 1 cm² | 100 mm² |
| 1 hectare (ha) | 10,000 m² (a square of 100 m × 100 m) |
| 1 km² | 1,000,000 m² = 100 ha |
| 1 square foot | 0.0929 m² |
| 1 square inch | 6.4516 cm² |
| 1 acre | approximately 4,047 m² |
If you need other conversions, the site's Unit Converter covers area, length, volume and much more.
Geometry: measuring the earth
The word "geometry" comes from Greek and literally means "earth measurement." The Egyptians were already measuring plots of land along the Nile, and the yearly floods wiped out the boundaries, forcing surveyors to redo the markings every year. That is where practical rules for the areas of rectangles, triangles and circles were born.
The Greeks turned these rules into proofs. Archimedes, in the 3rd century BC, bounded the value of π between 3 + 10/71 and 3 + 1/7 using inscribed and circumscribed polygons of up to 96 sides. The formula for the area of a triangle from its sides is attributed to Heron of Alexandria, around the 1st century.
What it is used for in practice
- Renovations: flooring, tile, paint, carpet, grass and ceilings all depend on area.
- Fences and finishing: the perimeter tells you how much wire, wall, baseboard or tape you will need.
- Packaging and design: calculating surfaces of paper, fabric and sheet metal.
- Gardening and farming: estimating the area of garden beds and planting fields.
- Maps and floor plans: comparing the sizes of lots, rooms and regions.
- School and exams: plane geometry is guaranteed to show up on tests.
Frequently asked questions about area and perimeter
What is the difference between area and perimeter?
Area is the measure of the surface (how much it covers), in square units. Perimeter is the measure of the outline (how long it is around), in linear units.
How do I calculate the area of a rectangle?
Multiply the length by the width. A rectangle of 5 m by 3 m has 15 m².
How do I calculate the area of a circle if I only have the diameter?
Divide the diameter by 2 to get the radius and apply π × radius². For a 30 cm diameter, the radius is 15 cm and the area is about 706.86 cm².
What is the circumference?
It is the perimeter of the circle, that is, the length of the line that surrounds it. It is calculated as 2 × π × radius.
How do I calculate the area of a triangle without knowing the height?
Use Heron's formula with the three sides. The tool does this in the "Triangle (three sides)" option.
Why does the tool say my sides do not form a triangle?
Because each side of a triangle must be shorter than the sum of the other two. With sides 1, 2 and 10, for example, the two short sides could never meet.
How do I calculate the area of a trapezoid?
Add the two bases, multiply by the height and divide by 2: (B + b) × h ÷ 2.
How many square meters are in a hectare?
10,000 m², the equivalent of a square 100 m on each side.
Can I mix different units in the fields?
No. Convert everything to the same unit first, otherwise the result is meaningless.
The perimeter of the trapezoid does not show up. Why?
Because you need to know both slanted sides, and they are not used in the area calculation. Fill in the optional fields to get the perimeter.
Before you buy material or cut anything
The results are mathematical and exact for perfect shapes. Real lots and rooms rarely are. Walls are not perfectly straight, corners are not exactly 90°, and measurements with a tape measure have errors. When buying material, measure carefully, add a safety margin suited to the material and, if the cost is significant, ask for guidance from whoever will do the work.
The display is limited to six decimal places. Round the result according to your use, and remember that the perimeter of the trapezoid and the parallelogram is only calculated when you enter the slanted sides.
Bottom line: this tool is educational. Results do not replace your own checking, and how you use them is entirely your responsibility. More details in the legal notice.
Want to know more?
- The circle is the champion. Among all shapes with the same perimeter, the circle has the largest area. That is why soap bubbles are spherical, and the same amount of fence encloses more land in a circle than in any other shape.
- π is irrational. Its decimal places neither repeat nor end. For everyday construction, 3.1416 is more than enough.
- Same perimeter, different areas. A 10 × 2 rectangle and a 6 × 6 one both have a perimeter of 24, but their areas are 20 and 36. The square is the rectangle that "yields" the most for a given perimeter.
Plane geometry glossary
- Area
- The measure of the surface occupied by a plane shape, in square units.
- Perimeter
- The sum of the lengths of all the sides of a shape; the length of its outline.
- Height
- The perpendicular distance between the base and the opposite side or vertex.
- Radius
- The distance from the center of a circle to its edge. Half the diameter.
- Diameter
- A segment that crosses the circle through the center. Equal to twice the radius.
- Circumference
- The line that surrounds the circle; it is its perimeter.
- Semiperimeter
- Half the perimeter. It is used in Heron's formula.
- Diagonal
- A segment that joins two non-adjacent vertices of a shape.