The area is the surface inside a shape; the perimeter is the distance around it. A rectangle 8 cm by 5 cm has an area of 8 × 5 = 40 cm² and a perimeter of 2 × (8 + 5) = 26 cm. Pick the shape, type its lengths in any unit and the calculator gives both, plus the diagonal or the hypotenuse, with each formula written out so you can follow it or copy it into homework.
The formulas
| Shape | Area | Perimeter |
|---|---|---|
| Rectangle | length × width | 2 × (length + width) |
| Square | side² | 4 × side |
| Circle | π × r² | 2 × π × r |
| Right triangle | leg a × leg b ÷ 2 | a + b + √(a² + b²) |
| Any triangle (3 sides) | √(s(s − a)(s − b)(s − c)), s = half the perimeter | a + b + c |
| Parallelogram | base × height | 2 × (base + side) |
For a parallelogram, the height is measured at right angles to the base. The slanted side changes the perimeter but not the area: push a rectangle over into a parallelogram and it keeps the same area as long as the height stays the same.
Same fence, different field
A fixed perimeter can enclose very different areas. Here is what 40 units of fence can hold:
| Shape (perimeter 40) | Area |
|---|---|
| Square 10 × 10 | 100 |
| Rectangle 12 × 8 | 96 |
| Rectangle 15 × 5 | 75 |
| Rectangle 18 × 2 | 36 |
| Rectangle 19 × 1 | 19 |
| Circle (circumference 40) | 127.32 |
Among rectangles, the square always wins; among all shapes, the circle does. That is why a long, thin plot needs much more fencing per square meter than a square one of the same size.
Checking a triangle
Three lengths only make a triangle if each one is shorter than the other two added together: 3, 4 and 8 do not (3 + 4 is less than 8). The calculator tells you when that happens. For a right triangle, the hypotenuse comes from Pythagoras: √(3² + 4²) = 5.
Everything is worked out on your device, and the calculator works offline after your first visit, so it is ready for homework on the bus.
Frequently asked questions
What is the difference between area and perimeter?
The perimeter is the distance around a shape, measured in units of length (cm, ft). The area is the surface inside it, measured in square units (cm², ft²). A fence follows the perimeter; a carpet covers the area.
How do I find the area of a triangle when I only know the three sides?
Use Heron's formula. Take half the perimeter, s = (a + b + c) ÷ 2, then area = √(s(s − a)(s − b)(s − c)). For sides 3, 4 and 5: s = 6 and the area is √(6 × 3 × 2 × 1) = 6.
What is the circumference of a circle?
Its perimeter: 2 × π × radius, or π × diameter. A circle with a 5 cm radius has a circumference of 31.42 cm and an area of 78.54 cm².
Can two shapes with the same perimeter have different areas?
Yes, very different. With 40 m of fence you can enclose 100 m² as a square but only 19 m² as a 19 × 1 m strip. A circle with the same 40 m encloses the most: 127.32 m².
Why are the areas shown in m² and ft² too?
So you can compare with a floor plan, a price per square meter or a price per square foot without converting by hand. Choose 'No unit' for school problems with plain numbers.
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