Area Formulas for Common Shapes
By the Calculator.hair team · Updated · 8 min read
Short answer
Area measures the flat space inside a shape, in square units. The main formulas are: square = s², rectangle = l × w, parallelogram = b × h, triangle = ½ × b × h, trapezoid = ½ × (a + b) × h, and circle = π × r². For example, a rectangle 12 feet long and 8 feet wide covers 12 × 8 = 96 square feet.
Key takeaways
- Almost every area formula is a version of base times height, with a factor added for shapes that are not rectangles.
- The height is always measured at a right angle to the base, never along a slanted side.
- A circle's formula uses the radius, so halve the diameter first: a 10-unit diameter gives about 78.54 square units.
- Convert all lengths to one unit before multiplying, and remember that 1 square yard is 9 square feet, not 3.
- Split an irregular shape into rectangles and triangles, find each area, and add them.
What are the area formulas for common shapes?
The table lists the area formula for each common flat shape, along with one worked example. Here, area means the amount of surface a shape covers, and it is always stated in square units such as square feet (ft²) or square meters (m²). The examples use feet and come from our area calculator, which covers every shape here except the rhombus (hand arithmetic). Its triangle form takes three sides, so the triangle row uses sides of 13, 14 and 15, whose 14-unit base has a height of 12.
| Shape | Formula | Example inputs | Area |
|---|---|---|---|
| Square | A = s² | s = 5 | 25 ft² |
| Rectangle | A = l × w | l = 12, w = 8 | 96 ft² |
| Parallelogram | A = b × h | b = 10, h = 6 | 60 ft² |
| Triangle | A = ½ × b × h | b = 14, h = 12 | 84 ft² |
| Trapezoid | A = ½ × (a + b) × h | a = 6, b = 10, h = 4 | 32 ft² |
| Rhombus | A = ½ × d₁ × d₂ | d₁ = 8, d₂ = 6 | 24 ft² |
| Circle | A = π × r² | r = 6 | 113.097336 ft² |
| Sector | A = (θ ÷ 360) × π × r² | r = 6, θ = 90° | 28.274334 ft² |
| Ellipse | A = π × a × b | a = 5, b = 3 (semi-axes) | 47.12389 ft² |
What is the area formula of a square?
The area of a square is the side length squared: A = s². Because all four sides are equal, you only need one measurement. A square with 5-foot sides covers 5 × 5 = 25 square feet, which our calculator also reports as 2.322576 square meters.
To work backward from area to side, take the square root. A square patio of 25 square feet has sides of √25 = 5 feet. To use the calculator, enter the same number for length and width.
What is the area formula of a rectangle?
The area of a rectangle is length times width: A = l × w. A room 12 feet long and 8 feet wide has an area of 12 × 8 = 96 square feet, or 8.918692 square meters.
Picture a grid: 12 rows of 8 one-foot tiles makes 96 tiles. Our guide on how to calculate square footage applies this to rooms and whole houses.
What is the area of a parallelogram formula?
The area of a parallelogram is base times height: A = b × h. A parallelogram is a four-sided shape whose opposite sides are parallel, and its height is the perpendicular distance between the two bases. With a base of 10 and a height of 6, the area is 10 × 6 = 60 square feet.
The formula works because you can slice a right triangle off one end, slide it to the other end, and form a rectangle with the same base and height. The slanted side plays no part in the result. If that parallelogram has a slanted side of 7, multiplying 10 × 7 gives 70, which overstates the area by 10 square feet. A rhombus is a parallelogram with four equal sides, so b × h works for it too, and when you know the diagonals instead, A = ½ × d₁ × d₂ gives the same answer.
What is the area of a triangle formula?
The area of a triangle is half the base times the height: A = ½ × b × h. Any triangle is half of a parallelogram with the same base and height, which is where the ½ comes from. A triangle with a base of 14 and a height of 12 has an area of ½ × 14 × 12 = 84 square units.
If you know the three sides but not the height, use Heron's formula: A = √(s × (s − a) × (s − b) × (s − c)), where s is the semiperimeter, or half the perimeter. For sides of 13, 14 and 15, s = 21, so A = √(21 × 8 × 7 × 6) = √7,056 = 84. Our triangle calculator confirms this and shows that the height to the 14-unit side is 12, which ties the two methods together. A right triangle is the easy case, because its two legs already meet at a right angle: legs of 3 and 4 give ½ × 3 × 4 = 6.
For angle rules and special triangles, see triangle formulas.
What is the area of a circle formula?
The area of a circle is pi times the radius squared: A = π × r². The radius is the distance from the center to the edge, and π is about 3.14159. A circle with a 6-foot radius has an area of π × 6² = π × 36 ≈ 113.097 square feet.
If you measured the diameter, the distance straight across, halve it first. A circle 10 units across has a radius of 5, so its area is π × 25 ≈ 78.54 square units. The circle calculator accepts any one of radius, diameter, circumference or area and returns the other three. For instance, a circle with an area of 50 has a radius of about 3.9894 and a circumference of about 25.0663.
A sector, or pie slice, is a fraction of the full circle. A 90° sector is one quarter of a circle, so a 6-foot radius gives 113.097336 ÷ 4 = 28.274334 square feet. An ellipse stretches the circle formula by using its two semi-axes, the longest and shortest distances from center to edge, in place of r × r: A = π × a × b.
What is the area of a trapezoid formula?
The area of a trapezoid is the average of the two parallel sides times the height: A = ½ × (a + b) × h. A trapezoid with parallel sides of 6 and 10 and a height of 4 has an area of ½ × (6 + 10) × 4 = 8 × 4 = 32 square feet.
Averaging the parallel sides gives an equal-area rectangle, here 8 wide and 4 tall. The slanted sides are not part of the formula.
How do you find the area of an irregular shape?
Split the shape into rectangles, triangles and circle parts, find each area, and add them. Subtract any cutouts. Take an L-shaped room made of a 12 × 8 foot main section and a 5 × 5 foot alcove. The total is 96 + 25 = 121 square feet.
For a shape with a hole, subtract. A 12 × 8 foot lawn with a circular flower bed of 2-foot radius has 96 − (π × 4) ≈ 96 − 12.57 = 83.43 square feet of grass (hand arithmetic). Once you have an area, you can turn it into material amounts. Our guide to how much mulch you need shows that next step for garden beds.
How do you convert square units?
Square the length conversion factor, then multiply. There are 3 feet in a yard, so there are 3 × 3 = 9 square feet in a square yard. Using the linear factor by mistake is one of the most common area errors. Also convert before you multiply: a space 12 feet by 96 inches is 12 feet by 8 feet, or 96 square feet, not 12 × 96. The calculator takes a separate unit for each side and handles this for you.
| Unit | Equals | How to convert |
|---|---|---|
| 1 square foot | 144 square inches | 12 × 12 |
| 1 square yard | 9 square feet | 3 × 3 |
| 1 square yard | 0.836127 square meters | 0.9144 × 0.9144 |
| 1 square foot | 0.092903 square meters | 0.3048 × 0.3048 |
| 1 square meter | about 10.7639 square feet | 1 ÷ 0.092903 |
| 1 acre | 43,560 square feet | by definition |
What mistakes should you avoid when calculating area?
Most wrong answers come from using the wrong measurement rather than the wrong formula.
- Using a slanted side as the height of a parallelogram, triangle or trapezoid.
- Putting the diameter into π × r². A diameter of 10 treated as a radius gives about 314.16, four times the correct 78.54.
- Forgetting the ½ in the triangle and trapezoid formulas.
- Multiplying lengths in different units, such as feet by inches.
- Confusing area with perimeter. Perimeter is the distance around a shape, in plain units, while area is the space inside, in square units. A 12 × 8 rectangle has a perimeter of 40 feet and an area of 96 square feet.
Calculators in this guide
Frequently asked questions
What is the formula for the area of a square?
Multiply the side by itself: A = s². A square with 5-foot sides has an area of 25 square feet.
How do you find the area of a circle from the diameter?
Divide the diameter by 2 to get the radius, then use A = π × r². A diameter of 10 gives a radius of 5 and an area of about 78.54 square units.
How do you find the area of a parallelogram without the height?
If you know two adjacent sides and the angle between them, use A = a × b × sin(θ). Sides of 10 and 8 at 30° give 10 × 8 × 0.5 = 40 square units.
Why is area measured in square units?
Area counts how many one-unit squares fit inside a shape. A 96 square foot room could be covered by 96 tiles that are each one foot on a side.
How do you find the area of a triangle with only three sides?
Use Heron's formula. For sides of 13, 14 and 15, the semiperimeter is 21 and the area is √(21 × 8 × 7 × 6) = 84 square units.