Area Calculator
Calculate the area and perimeter of rectangles, circles, triangles, trapezoids, and ellipses. Instant results with formulas and step-by-step solutions.
Area Calculator
Calculator
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Formula: Rectangle: A = l × w | Circle: A = πr² | Triangle: A = ½bh
Additional inputs: Base B (bottom), Height, Semi-major Axis (a), Semi-minor Axis (b).
Worked example — Area: 180 sq ft | Perimeter: 54 ft
Formula
Rectangle: A = l × w | Circle: A = πr² | Triangle: A = ½bh
Area measures the space enclosed by a 2D shape. Each shape has its own formula: Rectangle = length × width, Circle = π × radius², Triangle = ½ × base × height, Trapezoid = ½(a+b) × h, Ellipse = π × a × b.
Worked Examples
Example 1: Room Flooring (Rectangle)
Problem:A room is 12 feet by 15 feet. How much flooring is needed?
Solution:A = l × w = 12 × 15 = 180 sq ft P = 2(12 + 15) = 54 ft (for baseboards) Diagonal = √(144 + 225) = √369 ≈ 19.21 ft
Result:Area: 180 sq ft | Perimeter: 54 ft
Example 2: Circular Garden
Problem:A circular garden has a radius of 6 meters. Find the area and circumference.
Solution:A = πr² = π × 36 ≈ 113.10 m² C = 2πr = 2 × π × 6 ≈ 37.70 m Diameter = 12 m
Result:Area: 113.10 m² | Circumference: 37.70 m
Frequently Asked Questions
What is the formula for the area of a circle?
The area of a circle is A = πr², where r is the radius. The circumference is C = 2πr or C = πd (where d is the diameter). For example, a circle with radius 7 has area = π × 49 ≈ 153.94 square units and circumference = 2π × 7 ≈ 43.98 units.
What is the difference between area and perimeter?
Area measures the space inside a 2D shape (in square units like cm², m², ft²), while perimeter measures the total distance around the outside edge (in linear units like cm, m, ft). For example, a 4×3 rectangle has area = 12 square units but perimeter = 14 units. Two shapes can have the same perimeter but different areas, and vice versa.
How do I calculate the area of an ellipse?
The area of an ellipse is A = π × a × b, where a and b are the semi-major and semi-minor axes. Unlike a circle (where a = b = r), an ellipse has two different radii. The circumference of an ellipse has no simple exact formula; Ramanujan's approximation C ≈ π(a+b)(1 + 3h/(10+√(4-3h))) where h = ((a-b)/(a+b))² gives excellent results.
How do I find the area of a trapezoid?
A trapezoid has one pair of parallel sides called the bases. Its area is A = ½ × (a + b) × h, where a and b are the two parallel bases and h is the perpendicular distance between them. In effect you average the two bases and multiply by the height, so a trapezoid with bases 6 and 10 and height 8 gives A = ½ × 16 × 8 = 64 square units. When a = b the formula collapses to the rectangle rule.
Why must the height be perpendicular to the base rather than a slanted side?
In A = ½bh for a triangle and A = ½(a+b)h for a trapezoid, h is the altitude — the shortest, perpendicular distance from the base to the opposite vertex or parallel side. A slanted side is always longer than the altitude, so substituting it inflates the area. In an obtuse triangle the altitude can even fall outside the figure, and in a right triangle the two legs are perpendicular already, so they serve directly as base and height.
What happens to the area when every dimension of a shape is doubled?
Area scales with the square of the linear scale factor. Multiply every length by k and the area is multiplied by k², while the perimeter only grows by k. Doubling a rectangle's length and width quadruples its area: a 4 × 3 rectangle covers 12 square units, but 8 × 6 covers 48. The same square law holds for circles, since A = πr² and (2r)² = 4r². This is why doubling a pizza's diameter gives four times the pizza.
How do I convert an area result from square feet to square meters?
Square the linear conversion factor rather than applying it once. Because 1 foot = 0.3048 metres exactly, 1 square foot = 0.3048² = 0.09290304 square metres, and 1 square metre ≈ 10.7639 square feet. So a 180 sq ft room is 180 × 0.09290304 ≈ 16.72 m². The same rule applies to square inches and square centimetres: convert the sides first, or convert the finished area using the squared factor.
How do I calculate the area of an irregular shape that is not on this list?
Decompose it. Split the outline into rectangles, triangles, trapezoids, and circular sectors whose areas you can compute individually, then add them. For openings such as a window in a wall or a circular hole in a plate, compute the outer area and subtract the removed region. For a curved boundary with no formula, approximate it with narrow strips of equal width and sum their areas — the trapezoidal rule — which converges on the true area as the strips get thinner.
What is the apothem, and how does it give the area of a regular polygon?
The apothem is the perpendicular distance from the centre of a regular polygon to the midpoint of any side. Once you have it, the area is A = 1/2 x perimeter x apothem, which is the same idea as slicing the polygon into n identical triangles of base s and height a. A regular hexagon of side 6 has perimeter 36 and apothem s x sqrt(3)/2 = 5.196, so A = 1/2 x 36 x 5.196 = 93.5 square units. Reach for the apothem whenever a shape is regular but is not a triangle, square, or circle.
Why does a circle enclose more space than any other shape with the same perimeter?
This is the isoperimetric inequality: for any closed plane curve, 4 x pi x A is at most P squared, and equality holds only for a circle. Corners and flat sides waste boundary length. A square of perimeter 40 has side 10 and area 100, while a circle of circumference 40 has radius 40/(2 pi) = 6.37 and area pi r squared = 127. The gap widens as a shape gets more elongated, which is why pipes, tanks, and cells that need maximum interior for minimum boundary tend toward circular cross-sections.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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