Area Calculator
Area and perimeter of 22 flat shapes, with the formula, every step and a labelled diagram.
Result
This is the last result worked out. Fix the input above to update it.
Step by step
Area in other units
| Unit | Area |
|---|
About the Area Calculator
Choose a shape, type its measurements and get the area and the perimeter at once, with the formula, your numbers substituted step by step, and a diagram drawn to the proportions you entered so you can see which length is which. It covers 22 shapes: squares and rectangles; triangles from the base and height, from three sides (Heron’s formula), from two sides and the angle between them, equilateral and right triangles; circles, semicircles, quarter circles, sectors, segments, rings (annulus) and ellipses; trapezoids, parallelograms, rhombuses and kites; regular polygons from the side, circumradius or apothem; and any polygon from the coordinates of its corners, by the shoelace formula.
When the inputs are whole numbers, fractions or roots, the answer is also given exactly — 25π cm², 3√15/4 m² — next to the decimal. You can set π to 22/7 or 3.14 to match a textbook, and the area is converted to every common unit, from mm² to hectares and acres.
How to use it
- Choose the shape and the unit your lengths are in (mm, cm, m, km, inches, feet, yards or miles).
- Type the measurements labelled on the diagram. Decimals, fractions such as 7/2 or 1 1/2, and roots such as 2√3 all work. For “Any polygon”, list the corner coordinates in order around the outline, one “x, y” pair per line.
- Read the area and the perimeter (or circumference). The steps show each formula with your numbers, and exact forms such as 25π appear when your inputs allow.
- Set π to 22/7 or 3.14 if your textbook asks for it, choose the decimal places, and copy or download the full working. The table converts the area to other units.
Examples
r = 7 cm
A = 154 cm², C = 44 cm
A = (22/7) × 7² = 154 and C = 2 × (22/7) × 7 = 44. With π itself: A = 49π ≈ 153.938 cm².
a = 13, b = 14, c = 15
A = 84
s = (13 + 14 + 15)/2 = 21, so A = √(21 × 8 × 7 × 6) = √7056 = 84.
r = 6 cm, θ = 60°
A = 6π ≈ 18.8496 cm²
A = (60/360) × π × 6². The arc is (60/360) × 2π × 6 = 2π ≈ 6.2832 cm.
(0, 0), (4, 0), (4, 3)
A = 6
Shoelace sum: (0×0 − 4×0) + (4×3 − 4×0) + (4×0 − 0×3) = 12, and 12 ÷ 2 = 6.
side s = 1
A = 3√3/2 ≈ 2.5981
Apothem a = 1/(2 tan 30°) ≈ 0.8660, perimeter 6, so A = ½ × 6 × 0.8660.
a = 5, b = 3
A = 15π ≈ 47.1239, P ≈ 25.5270
Ramanujan’s perimeter 25.52699886 agrees with the exact value 25.52699886 to eight decimal places.
Common uses
- Checking school geometry homework, including NCERT problems that take π = 22/7.
- Working out the floor, wall or garden area of a room or plot made of rectangles, triangles and curves.
- Finding the area of an irregular plot or a CAD outline from its corner coordinates.
- Converting a measured area between cm², m², square feet, square yards, acres and hectares.
The formulas
- Rectangle: A = l × w, P = 2(l + w). Square: A = a², P = 4a.
- Triangle: A = ½ × base × height; from three sides, Heron’s formula A = √(s(s − a)(s − b)(s − c)) with s = (a + b + c)/2; from two sides and the included angle, A = ½ab sin C. Equilateral: A = (√3/4)a².
- Circle: A = πr², C = 2πr. Sector of angle θ: A = (θ/360°)πr², arc = (θ/360°)2πr — or, with θ in radians, A = ½r²θ and arc = rθ. Segment: A = ½r²(θ − sin θ) with θ in radians. Ring: A = π(R² − r²).
- Ellipse: A = πab. Trapezoid: A = ½(a + b)h. Parallelogram: A = bh = ab sin θ. Rhombus and kite: A = ½ × p × q (the diagonals).
- Regular polygon with n sides of length s: apothem a = s / (2 tan(180°/n)), A = ½ × perimeter × apothem.
The perimeter of an ellipse
An ellipse has no simple formula for its perimeter: the exact value is an elliptic integral. The calculator shows Srinivasa Ramanujan’s second approximation from 1914 (Quarterly Journal of Mathematics 45, 350–372), P ≈ π(a + b)(1 + 3h/(10 + √(4 − 3h))) with h = (a − b)²/(a + b)², and also the exact perimeter computed with the arithmetic–geometric mean. For everyday ellipses the two agree to many decimal places; even for an ellipse 100 times longer than it is wide, Ramanujan’s value is only 0.024 % short.
Area from coordinates: the shoelace formula
For vertices (x₁, y₁) … (xₙ, yₙ) listed in order around a polygon, twice the area is the sum of xᵢyᵢ₊₁ − xᵢ₊₁yᵢ over every edge, wrapping from the last vertex back to the first (Gauss’s area formula). The sum is positive when the vertices go anticlockwise and negative when they go clockwise; the area is half its absolute value. The formula works for any simple polygon, convex or not. If the edges cross each other — a bow-tie shape — the result is a net signed area in which loops that turn the other way subtract, so the calculator warns you and you should list the corners in order around the outline.
Exact answers, π and units
Lengths you type as whole numbers, decimals, fractions or square roots are carried exactly, so a circle of radius 5 gives 25π, an equilateral triangle of side 6 gives 9√3, and a square of side 1/3 gives 1/9. The decimal uses the value of π you choose: π itself, 3.14, or 22/7 as many Indian textbooks specify. Unit conversions use exact factors — 1 inch = 2.54 cm, 1 foot = 0.3048 m, 1 acre = 43,560 square feet = 4,046.8564224 m², 1 hectare = 10,000 m².
Limitations
- The shapes are ideal flat figures. For land, the result is only as accurate as the measurements, and a plot on a slope or with curved edges needs to be surveyed.
- Calculations use double-precision numbers (about 15 significant digits); results are rounded to the decimal places you choose, and the unit table keeps at least six significant digits. Lengths must be positive and at most 10¹² in the chosen unit.
- For a trapezoid or kite, side lengths that do not fit the other measurements are flagged but not corrected — the perimeter simply adds the sides you typed.
- A polygon whose edges cross itself gives a net signed area, not the area enclosed by its outline.
Privacy
Everything happens in your browser. What you enter or open here is not uploaded or stored by MySmartCoPilot.
Frequently asked questions
How do I find the area of a triangle when I only know the three sides?
Use Heron’s formula. Add the sides and halve the total to get s, then A = √(s(s − a)(s − b)(s − c)). For sides 13, 14 and 15, s = 21 and A = √(21 × 8 × 7 × 6) = 84. Choose “Triangle (three sides — Heron)” and the steps show this for your numbers; the sides must satisfy the triangle inequality (each shorter than the other two together).
Should I use π, 3.14 or 22/7?
Use the value your question asks for. NCERT and many other textbook problems say “take π = 22/7”, which is convenient when the radius is a multiple of 7 (r = 7 gives an area of exactly 154). 22/7 is about 0.04 % larger than π, and 3.14 about 0.05 % smaller. For real measurements use π itself; the exact answer in terms of π (such as 49π) is shown either way.
What is the difference between area and perimeter?
Area is the amount of surface inside a shape, measured in square units such as cm² or m². Perimeter is the length of its boundary, measured in plain length units such as cm or m. For a circle the perimeter is called the circumference.
Why is the perimeter “not determined” for some shapes?
Some measurements fix the area but not the outline. A triangle’s base and height give its area, but many triangles share them, with different slanted sides. The same happens for a parallelogram without its slanted side and a kite known only by its diagonals. Enter the extra side lengths, or use the “three sides” or “two sides and an angle” options.
How do I find the area of an irregular shape?
If you know the corners, choose “Any polygon (vertex coordinates)” and list them in order — measure them from a corner on a sketch or plan. Otherwise split the shape into rectangles, triangles and circle parts, work out each one here, and add (or subtract) the areas.
How do I convert cm² to m²?
Divide by 10,000, because 1 m = 100 cm and so 1 m² = 100 × 100 cm². In the same way 1 ft² = 144 in² and 1 m² ≈ 10.7639 ft². The table under each result lists the area in every common unit.