What this area and perimeter calculator does
This tool computes the area and perimeter of six common two-dimensional shapes: rectangle, square, circle, triangle, trapezoid, and parallelogram. Choose a shape, enter the dimensions, and the calculator returns both measures with the exact formulas shown and the arithmetic worked out step by step. A diagram draws the shape to scale using your inputs, so you can see exactly which quantity each result refers to. The tool also reports complementary values — like the diagonal of a rectangle or the diameter of a circle — so you get the complete picture in one calculation.
Formulas for each shape
Diagonal d = √(l² + w²).
Diagonal d = s√2.
Diameter d = 2r.
For perimeter, enter the other two side lengths (or use 0 to skip).
P = a + b + c + d, the sum of all four sides.
Variable definitions
- A — area (in square units).
- P — perimeter (in linear units).
- r, d — radius and diameter of a circle.
- b, h — base and perpendicular height.
- π — pi ≈ 3.141592653589793.
Worked examples
Example 1: Rectangle 8 cm by 5 cm
- Area: A = 8 × 5 = 40 cm².
- Perimeter: P = 2(8 + 5) = 2 × 13 = 26 cm.
- Diagonal: d = √(8² + 5²) = √(64 + 25) = √89 ≈ 9.434 cm.
Example 2: Square with side 7 m
- Area: A = 7² = 49 m².
- Perimeter: P = 4 × 7 = 28 m.
- Diagonal: d = 7√2 ≈ 9.899 m.
Example 3: Circle with radius 5 cm
- Area: A = π × 5² = 25π ≈ 78.5398 cm².
- Circumference: C = 2π × 5 = 10π ≈ 31.4159 cm.
- Diameter: d = 10 cm.
Example 4: Triangle with base 12 m and height 5 m
- Area: A = ½ × 12 × 5 = 30 m².
- If the other two sides are 6 m and 8 m: perimeter = 12 + 6 + 8 = 26 m.
Example 5: Trapezoid with parallel sides 10 and 6, height 4
- Area: A = ½ × (10 + 6) × 4 = ½ × 16 × 4 = 32.
- If legs are 5 and 5: perimeter = 10 + 6 + 5 + 5 = 26.
Example 6: Parallelogram base 9, height 4, side 5
- Area: A = 9 × 4 = 36.
- Perimeter: P = 2(9 + 5) = 28.
Area vs perimeter
These two measures describe different aspects of a shape:
- Perimeter is the total length of the boundary — a 1-D measure in linear units (cm, m, km). Think of it as the fence you need to enclose a garden.
- Area is the total surface covered — a 2-D measure in square units (cm², m²). Think of it as the paint you need to cover a wall.
- Two shapes with the same area can have very different perimeters, and vice versa. A 1×16 rectangle has area 16 and perimeter 34; a 4×4 square has area 16 and perimeter 16.
Common mistakes to avoid
- Mixing up radius and diameter. The circle formula uses the radius, not the diameter. If you're given a diameter of 10 cm, use r = 5 cm in the formula.
- Using slant height instead of perpendicular height for trapezoids and parallelograms. The height h is the perpendicular distance between the parallel sides, not the length of the slanted side.
- Forgetting to square the units for area. Area is in cm², m², or km² — never just cm or m. Writing "40 cm" for an area is wrong; it should be "40 cm²".
- Adding lengths in different units. If one side is in cm and another in m, convert first. Mixing units gives nonsense.
- Using the wrong formula for the triangle area. The triangle formula is ½ × base × height, not base × height. This is because a triangle is half a parallelogram with the same base and height.
- Confusing perimeter and area. Perimeter is the length around the boundary; area is the space inside. They are not interchangeable.
FAQ
What is the difference between area and perimeter?
Perimeter is the distance around the outside of a shape — a length measured in linear units (cm, m). Area is the amount of surface the shape covers — a two-dimensional measure in square units (cm², m²). They are both geometric measures but of different kinds.
How do I find the area of a rectangle?
Multiply the length by the width: A = l × w. The perimeter is twice the sum of length and width: P = 2(l + w).
What is the formula for the area of a circle?
Area = π × r², where r is the radius. The circumference (perimeter) is 2πr, or πd where d is the diameter. If you only know the diameter, halve it to get the radius first.
Why is the area of a triangle half the base times height?
Two congruent triangles together form a parallelogram with the same base and height. The parallelogram's area is base × height, so each triangle has half that area. This is why the formula is A = ½ × b × h.
How do I find the area of a trapezoid?
Average the two parallel sides and multiply by the height: A = ½ × (a + b) × h, where a and b are the parallel sides and h is the perpendicular distance between them. The perimeter is the sum of all four side lengths.
What units should I use?
Use the same unit for every length you enter. The area will be in the square of that unit (cm² if lengths are in cm), and the perimeter will be in the original unit (cm). Mixing units (cm and m) gives wrong results.
Related tools
Related Hira Academy resources
Area and perimeter formulas appear throughout the Class 9 and 10 mathematics syllabus, especially in the geometry chapters. Revise with our free Class 9 notes and Class 10 notes.