Area Calculator
Pick a shape, enter its measurements in any length unit, and get the area as an exact value — in terms of π or a square root where that applies — alongside the decimal, the formula with your numbers substituted, a labeled figure, and the same area in other units. Free, no sign-up; everything is worked out in your browser.
Results
40 cm²
Area
Other measurements
| Measurement | Exact | Decimal |
|---|---|---|
| Perimeter | 26 cm | 26 cm |
| Diagonal | √89 cm | 9.43 cm |
The working
A = l × w
A = l × w = 8 × 5 = 40
Figure
The same area in other units
| Unit | Area |
|---|---|
| mm² | 4,000 |
| cm² | 40 |
| m² | 0.004 |
| km² | 4E-9 |
| in² | 6.200012 |
| ft² | 0.04305564 |
| yd² | 0.00478396 |
| mi² | 1.5444E-9 |
| ha | 4E-7 |
| ac | 9.8842E-7 |
How to Use the Area Calculator
- 1. Choose the shape. Square, rectangle, triangle (from a base and height, or from three sides), parallelogram, trapezoid, rhombus, regular polygon, ellipse, ring or circular sector.
- 2. Enter the measurements. Whole numbers, decimals and fractions such as 3/4 are read exactly. Pick one unit for everything, or tick the box to give each measurement its own unit — 2 ft by 30 in is fine — and choose the unit for the answer.
- 3. Read the result. The area appears in exact form first (for example 15π or 15√7/4) and as a decimal rounded to the number of places you choose. Perimeter, diagonal, apothem, arc length and other measurements that come with the shape are listed below it.
- 4. Check the working and the figure. The formula is shown with your numbers put in, and the labeled figure is drawn roughly to the proportions you typed. The last table gives the same area in other square units, each converted with the squared length factor.
Area Formulas for Common Shapes
Area measures the surface a flat figure covers, in square units. Every formula below takes the perpendicular height — measured at a right angle to the base — never the length of a slanted side.
| Shape | Area | What you need |
|---|---|---|
| Square | A = s² | Side s |
| Rectangle | A = l × w | Length l and width w |
| Triangle | A = ½ × b × h | Base b and perpendicular height h |
| Triangle, three sides | A = √(s(s − a)(s − b)(s − c)), where s = (a + b + c) / 2 | Sides a, b, c (Heron's formula) |
| Parallelogram | A = b × h | Base b and perpendicular height h |
| Trapezoid | A = ½ × (a + b) × h | Parallel sides a and b, and the height between them |
| Rhombus | A = ½ × d₁ × d₂ | Both diagonals |
| Regular polygon | A = n × s² / (4 × tan(180° / n)) | Number of sides n and side length s |
| Ellipse | A = π × a × b | Semi-axes a and b (half the width and half the height) |
| Ring (annulus) | A = π(R² − r²) | Outer radius R and inner radius r |
| Circular sector | A = ½ × r² × θ = (θ / 360°) × π × r² | Radius r and central angle θ (radians in the first form) |
For a whole circle, its circumference, or a circular segment, use the Circle Calculator; this page covers the sector, which is the slice from the center to the arc.
Worked Examples
- Trapezoid. Parallel sides 3 and 5, height 4: A = ½ × (3 + 5) × 4 = 16 square units. The midsegment, the average of the parallel sides, is 4.
- Triangle from three sides. Sides 4, 5 and 6 give s = 7.5, so A = √(7.5 × 3.5 × 2.5 × 1.5) = 15√7/4 ≈ 9.92. No height was needed.
- Regular hexagon. Side 2: A = 6 × 2² / (4 × tan 30°) = 6√3 ≈ 10.39. The calculator keeps an exact form for 3, 4, 6, 8 and 12 sides, where tan(180° / n) is a simple radical, and gives a decimal for other side counts.
- Sector. Radius 3 and a 60° angle: 60° is π/3 radians, so A = ½ × 3² × π/3 = 3π/2 ≈ 4.71, and the arc length is r × θ = π ≈ 3.14.
Area Units: Why Square Units Convert Differently
A length converts with one factor, but an area uses that factor squared. One meter is 100 centimeters, so one square meter is 100 × 100 = 10,000 square centimeters — not 100. Mixing units is where most area mistakes happen: 2 ft by 30 in is 2 ft × 2.5 ft = 5 ft², which is 5 × 144 = 720 in². The calculator converts every measurement to the answer unit first, exactly, so nothing is rounded along the way.
| Length | Area |
|---|---|
| 1 m = 100 cm | 1 m² = 10,000 cm² |
| 1 ft = 12 in | 1 ft² = 144 in² |
| 1 yd = 3 ft | 1 yd² = 9 ft² |
| 1 mi = 5,280 ft | 1 mi² = 27,878,400 ft² |
| 1 acre = 43,560 ft² | 1 hectare = 10,000 m² |
Common Mistakes
- Using the slanted side of a parallelogram or trapezoid where the formula needs the perpendicular height.
- Leaving out the ½ in the triangle, trapezoid, rhombus and sector formulas.
- Entering a sector angle in degrees where a formula expects radians — the calculator accepts either, but A = ½r²θ only works with θ in radians.
Frequently Asked Questions
- How do you find the area of a shape?
- Identify the shape, measure what its formula needs — usually a base and a perpendicular height, or two sides — and put the numbers in. A rectangle is length × width, a triangle is half base × height, and a circle-based shape involves π. Keep every measurement in the same unit before you multiply.
- What is the difference between area and perimeter?
- Perimeter is the distance around the outside of a shape, in ordinary units such as centimeters. Area is the amount of surface inside it, in square units such as square centimeters. Two shapes can share a perimeter and differ in area: a 4 by 4 square and a 1 by 7 rectangle both have perimeter 16, but their areas are 16 and 7.
- How do you convert square units, such as m² to cm²?
- Square the length conversion factor. Because 1 m = 100 cm, 1 m² = 100² = 10,000 cm². Likewise 1 ft² = 12² = 144 in². Going the other way you divide: 5,000 cm² is 0.5 m².
- How do you find the area of a triangle when you only know the three sides?
- Use Heron's formula. Find the semi-perimeter s = (a + b + c) / 2, then A = √(s(s − a)(s − b)(s − c)). The three lengths must satisfy the triangle inequality — the longest side shorter than the other two combined — or no triangle exists, and the calculator says so.
- What is the area of a regular polygon?
- A = n × s² / (4 × tan(180° / n)) for n sides of length s, which is the same as half the apothem times the perimeter. A regular hexagon with side s has area 3√3/2 × s², and an equilateral triangle has √3/4 × s².
- How do you find the area of a sector?
- Take the fraction of the full circle the angle covers and multiply by πr²: A = (θ / 360°) × π × r² for degrees. In radians this becomes A = ½ × r² × θ. A 90° sector is a quarter of the circle.
- Why is there no simple perimeter formula for an ellipse?
- The exact perimeter needs an elliptic integral with no closed form in elementary functions. The calculator shows Ramanujan's approximation, π[3(a + b) − √((3a + b)(a + 3b))], which is extremely close for ellipses that are not very elongated and is exact for a circle.