You're staring at a floor plan, a garden bed, or a homework problem, and you need the area of some shape. Maybe it's a circle. Maybe it's a trapezoid you haven't thought about since eighth grade. Either way, you don't want to dig through a textbook — you just want the answer.
Here's every area and perimeter formula you're likely to need, along with tips for using them without making mistakes.
Rectangle
The rectangle is the one everyone remembers:
- Area = width × height
- Perimeter = 2 × (width + height)
If your room is 12 feet wide and 15 feet long, the area is 180 square feet and the perimeter is 54 feet. Straightforward. The only gotcha? Make sure both measurements use the same unit. Mixing inches and feet is a surprisingly common way to end up with nonsense numbers.
Square
A square is just a rectangle where all sides are equal, so the formula gets even simpler:
- Area = side²
- Perimeter = 4 × side
A 10-meter square has an area of 100 m² and a perimeter of 40 m. If you know the area and need the side length, take the square root — √100 = 10.
Circle
Circles bring π into the picture. You'll use 3.14159 (or just let a calculator handle it):
- Area = π × r²
- Circumference = 2 × π × r
Where r is the radius — half the diameter. Got a circular patio that's 8 feet across? The radius is 4 feet, so the area is π × 16 ≈ 50.27 square feet.
One common mistake: confusing radius and diameter. If someone gives you the diameter, divide by two before plugging it in. Skip that step and your area comes out four times too large.
Triangle
The basic triangle formula works when you know the base and the height (the perpendicular distance from the base to the opposite vertex):
- Area = ½ × base × height
A triangle with a 10 cm base and 7 cm height has an area of 35 cm². Note that the height has to be perpendicular to the base — not the length of one of the slanted sides.
What if you don't know the height? If you have all three side lengths (a, b, c), use Heron's formula:
- Calculate the semi-perimeter: s = (a + b + c) / 2
- Area = √[s × (s − a) × (s − b) × (s − c)]
It looks intimidating, but it's just plugging in numbers.
Trapezoid
A trapezoid has one pair of parallel sides (the bases) and two non-parallel sides (the legs). The area formula averages the two bases:
- Area = ½ × (top base + bottom base) × height
Say you're calculating the area of a garden bed shaped like a trapezoid. The top edge is 6 feet, the bottom edge is 10 feet, and the distance between them is 4 feet. That's ½ × (6 + 10) × 4 = 32 square feet.
The height here is the perpendicular distance between the two parallel sides, not the length of the slanted legs.
Parallelogram
A parallelogram looks like a tilted rectangle. Both pairs of opposite sides are parallel:
- Area = base × height
- Perimeter = 2 × (side₁ + side₂)
The height is perpendicular to the base, not the length of the slanted side. This trips people up — if you use the slanted side length instead of the actual height, your answer will be too large.
Ellipse
Think of an ellipse as a squished circle. Instead of one radius, you have two: a semi-major axis (a) and a semi-minor axis (b):
- Area = π × a × b
If the semi-major axis is 6 units and the semi-minor axis is 4 units, the area is π × 6 × 4 ≈ 75.40 square units.
The perimeter of an ellipse doesn't have a clean exact formula. There are approximations (Ramanujan's is popular), but for quick work, just let a calculator handle it.
Sector (pie slice of a circle)
A sector is a wedge cut from a circle, defined by a central angle:
- Area = (θ / 360) × π × r²
Where θ is the angle in degrees. A 90° sector of a circle with radius 5 gives you (90/360) × π × 25 ≈ 19.63 square units. That's exactly one quarter of the full circle's area, which makes intuitive sense.
Rhombus
A rhombus has four equal sides, but it's tilted so opposite angles aren't 90°. The easiest area formula uses the diagonals:
- Area = ½ × d₁ × d₂
Where d₁ and d₂ are the two diagonals. If the diagonals are 8 and 6 units, the area is ½ × 8 × 6 = 24 square units.
You can also use base × height if you know the perpendicular height, just like a parallelogram.
Picking the right units
Area is always in square units. If you measure in feet, your area is in square feet (ft²). In meters, square meters (m²). This matters when you're buying materials — flooring is sold by the square foot, paint coverage is listed per gallon in square feet, and tile comes in specific sizes you'll need to convert.
Quick conversions worth remembering:
- 1 square meter = 10.764 square feet
- 1 acre = 43,560 square feet
- 1 hectare = 10,000 square meters
When do you actually need these formulas?
More often than you'd think. Calculating how much paint to buy, figuring out how many tiles cover a bathroom floor, sizing a garden bed, estimating how much mulch to order, measuring a lot size — all of these come back to area.
Even if math isn't your thing, you can skip the manual calculation entirely. Open the Area Calculator, pick your shape from the dropdown, enter your measurements, and get both area and perimeter instantly. It supports all nine shapes covered here — rectangle, square, circle, triangle, trapezoid, parallelogram, ellipse, sector, and rhombus — and shows you the formula and step-by-step work so you can verify the result.
Common mistakes to watch for
Using the wrong height. For triangles, trapezoids, and parallelograms, the height must be perpendicular to the base. The slanted side is not the height.
Forgetting to halve the diameter. Circle formulas use the radius. If you're given a diameter of 10, your radius is 5. Plugging in 10 instead of 5 gives you an area that's four times too big.
Mixing units. If your width is in inches and your length is in feet, convert one before multiplying. A 3-foot by 24-inch rectangle isn't 72 square anything — you need both in feet (3 × 2 = 6 ft²) or both in inches (36 × 24 = 864 in²).
Rounding too early. When a formula involves π or square roots, keep all the decimal places until the final answer. Rounding intermediate steps can throw off your result by a surprising amount, especially for larger measurements.
FAQ
How do I calculate the area of an irregular shape?
Break it into simpler shapes you know the formulas for. A room with an alcove? That's a rectangle plus a smaller rectangle. An L-shaped garden? Two rectangles. Calculate each piece separately and add them up. For curved irregular shapes, try approximating with a grid or using the Area Calculator for each regular section.
What's the difference between area and perimeter?
Area measures the surface inside a shape (in square units). Perimeter measures the total distance around the outside edge (in linear units). You need area to figure out how much paint covers a wall. You need perimeter to figure out how much trim goes around it.
Can I calculate area from just the perimeter?
Not without additional information. Many different shapes can share the same perimeter but have wildly different areas. A long, skinny rectangle and a nearly-square rectangle can both have a perimeter of 20, but their areas will be very different. You need at least two independent measurements for most shapes.
Geometry doesn't have to mean flashbacks to pop quizzes. Pick your shape, grab the formula, punch in the numbers — or just let the Area Calculator do it for you.