Area & Perimeter Calculator (All Shapes, with Diagram)
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An area and perimeter calculator measures the inside and the outside of a shape. For a rectangle 8 cm long and 5 cm wide, the area is 8 × 5 = 40 cm² and the perimeter is 2(8 + 5) = 26 cm. A circle with a 5 cm radius has area 25π ≈ 78.54 cm² and circumference 10π ≈ 31.42 cm.
Rectangle with length 8 cm and width 5 cm: area = 40 cm², perimeter = 26 cm.
Show steps
- Area counts the unit squares that cover the rectangle: length × width = 8 × 5 = 40 cm².
- Perimeter is the distance around: two lengths plus two widths, P = 2(8 + 5) = 2 × 13 = 26 cm.
- Diagonal (Pythagorean theorem): d = √(l² + w²) = √(64 + 25) = √89 ≈ 9.43 cm.
Square units convert by the square of the length factor: 1 m = 100 cm, so 1 m² = 100 × 100 = 10,000 cm². 1 ft² = 144 in², 1 yd² = 9 ft², 1 acre = 43,560 ft².
How to use the area and perimeter calculator
- Pick a shape: square, rectangle, triangle, circle, parallelogram, trapezoid, ellipse, sector, regular polygon, rhombus, kite or ring.
- Choose the unit (mm, cm, m, km, in, ft, yd or mi) and type the measurements. Decimals and fractions such as 3/4 both work. For a square or circle you can start from the side, diagonal, radius, diameter, circumference or area.
- Read the area and the perimeter side by side. When π or a square root is involved, you also get the exact answer, such as 25π or 6√6.
- Check the formula line, which repeats the formula with your numbers, and the diagram, which is drawn to scale and labeled with your measurements.
- Open Show steps for the full working, and use the table to convert the area to other units, including acres and hectares.
Area and perimeter formulas for every shape
Area is the space inside a flat shape, measured in square units (cm², m², ft²). Perimeter is the distance around the edge, measured in ordinary length units (cm, m, ft). For a circle, the perimeter is called the circumference.
| Shape | Area | Perimeter |
|---|---|---|
| Square, side s | s² | 4s |
| Rectangle, length l, width w | l × w | 2(l + w) |
| Triangle, base b, height h | ½ × b × h | a + b + c |
| Triangle, three sides (Heron) | √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2 | a + b + c |
| Circle, radius r | πr² | 2πr = πd |
| Parallelogram, base b, height h, side s | b × h | 2(b + s) |
| Trapezoid, bases a and b, height h | ½(a + b) × h | sum of all four sides |
| Ellipse, semi-axes a and b | πab | ≈ π[3(a + b) − √((3a + b)(a + 3b))] |
| Sector, radius r, angle θ in degrees | (θ/360) × πr² | 2r + (θ/360) × 2πr |
| Regular polygon, n sides of length s | ns² / (4 tan(180°/n)) | n × s |
| Rhombus or kite, diagonals p and q | pq / 2 | 4s (rhombus), 2(a + b) (kite) |
| Ring, radii R and r | π(R² − r²) | 2π(R + r) |
The height in every formula is perpendicular to the base. For a triangle, parallelogram or trapezoid, it is usually not one of the sides.
Worked examples
Example 1: a triangle from three sides
A triangle has sides 13, 14 and 15 cm. The semi-perimeter is s = (13 + 14 + 15)/2 = 21. Heron’s formula gives A = √(21 × 8 × 7 × 6) = √7,056 = 84 cm². The perimeter is 13 + 14 + 15 = 42 cm. No height was needed.
Example 2: a circle from its diameter
A circular table is 12 ft across. The radius is 12 ÷ 2 = 6 ft. The area is π × 6² = 36π ≈ 113.10 ft², and the circumference is π × 12 = 12π ≈ 37.70 ft. Keep π exact until the last step to avoid rounding errors.
Example 3: a trapezoid with equal legs
A trapezoid has bases 12 m and 6 m and a height of 4 m. The area is ½(12 + 6) × 4 = 9 × 4 = 36 m². Each leg covers an overhang of (12 − 6)/2 = 3 m, so a leg is √(4² + 3²) = 5 m. The perimeter is 12 + 6 + 5 + 5 = 28 m.
Example 4: a slice of pizza (a sector)
A 60° slice is cut from a pizza with a 6 in radius. It is 60/360 = 1/6 of the pizza, so its area is (1/6) × π × 6² = 6π ≈ 18.85 in². The crust (arc) is (1/6) × 2π × 6 = 2π ≈ 6.28 in, and the whole edge of the slice is 6 + 6 + 2π ≈ 18.28 in.
Area of an irregular shape
Split the shape into rectangles, triangles and circle parts, find each area, then add (or subtract the pieces that are cut out). An L-shaped floor plan that is a 10 m by 8 m rectangle with a 4 m by 3 m corner missing has area 80 − 12 = 68 m². Its perimeter is still 2(10 + 8) = 36 m, because the two edges of the notch are just as long as the edges they replace.
For rooms, floors and yards with waste allowance and cost, use the square footage calculator.
Converting area units
Area units convert by the square of the length factor. Since 1 m = 100 cm, 1 m² = 100 × 100 = 10,000 cm², not 100 cm². The calculator’s table does this for you.
| From | To |
|---|---|
| 1 cm² | 100 mm² |
| 1 m² | 10,000 cm² ≈ 10.764 ft² |
| 1 ft² | 144 in² |
| 1 yd² | 9 ft² |
| 1 acre | 43,560 ft² ≈ 4,046.86 m² |
| 1 hectare | 10,000 m² ≈ 2.471 acres |
Common mistakes with area and perimeter
- Using the slanted side as the height. For a parallelogram with base 10 and slanted side 5, the area is 10 × 4 (the height), not 10 × 5.
- Mixing up radius and diameter. A circle with diameter 10 has area π × 5² = 25π, not π × 10².
- Mixing units. Convert 2 m and 50 cm to the same unit before multiplying: 200 cm × 50 cm = 10,000 cm² = 1 m².
- Forgetting square units. Area is in cm² or ft². Perimeter is in cm or ft.
- Rounding π too early. Use the exact form (25π) or the π key, then round once at the end.
- Finding a triangle’s perimeter from base and height. The perimeter needs all three sides. If you know them, Heron’s formula also gives the area.
Using the calculator to check homework
Work the problem by hand first, then compare. Estimate before you calculate: a 7.9 by 5.1 rectangle should have an area near 8 × 5 = 40. If your answer and the calculator’s differ, open the steps and find the first line where they part ways. Usually it is a radius used as a diameter, a slanted side used as a height, or a unit that was not converted. Then try one more problem of the same kind without the calculator.
Area and perimeter practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Area = length × width = 9 × 4 = 36 cm². 26 is the perimeter, 2(9 + 4), and area is measured in square units.
The radius is half the diameter, 5 cm, so A = π × 5² = 25π ≈ 78.54 cm². Using the diameter gives 100π.
s = (5 + 12 + 13)/2 = 15, so A = √(15 × 10 × 3 × 2) = √900 = 30. It is a right triangle, and ½ × 5 × 12 = 30 agrees.
A = base × height = 12 × 4 = 48 cm². Multiplying by the slanted side (12 × 5 = 60) is a common mistake.
1 ft = 12 in, so 1 ft² = 12 × 12 = 144 in². Area units convert by the square of the length factor.
6 × 3 = 18 and 2(6 + 3) = 18. For 5 by 3: 15 and 16. For 4 by 2: 8 and 12. For 8 by 1: 8 and 18.
Quiz complete
Frequently asked questions
What is area and perimeter in math?
Area is the amount of flat space inside a shape, counted in unit squares. Perimeter is the total length of the shape's boundary. A 4 by 3 rectangle covers 12 square units and has a perimeter of 4 + 3 + 4 + 3 = 14 units.
What is the area and perimeter formula of a circle?
Area = πr² and circumference (the perimeter) = 2πr, where r is the radius. For r = 5 cm, the area is 25π ≈ 78.54 cm² and the circumference is 10π ≈ 31.42 cm.
What is the area and perimeter formula of a triangle?
Area = ½ × base × height, and perimeter = a + b + c. If you only know the three sides, use Heron's formula for the area: for sides 5, 6 and 7, s = 9 and A = √(9 × 4 × 3 × 2) = 6√6 ≈ 14.70.
What is the area and perimeter formula of a trapezium?
A trapezium (called a trapezoid in the US) has area ½(a + b) × h, where a and b are the parallel sides and h is the distance between them. The perimeter is the sum of all four sides.
Can area and perimeter be the same?
The numbers can match, but the quantities cannot, because one is in square units and the other in length units. A 6 by 3 rectangle has area 18 and perimeter 18, and a 4 by 4 square has area 16 and perimeter 16. Change the unit from cm to mm and the two numbers no longer match.
How do you calculate area and perimeter in 3rd grade?
In 3rd grade, students count unit squares to find area and add side lengths to find perimeter. A rectangle 5 squares long and 2 squares wide covers 5 × 2 = 10 square units, and its perimeter is 5 + 2 + 5 + 2 = 14 units.
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