Area Calculator
Area + perimeter for 12 shapes (circle, square, rectangle, triangle, parallelogram, trapezoid, ellipse, regular polygon, annulus, sector, rhombus, kite) with diagrams, animated derivations, and step-by-step work.
Area Calculator — every common 2D shape, with an animated how-it-works view
An area calculator computes how much flat 2D space a shape covers (square units) and, where defined, the perimeter (linear units). This one supports twelve common shapes — circle, square, rectangle, triangle (base-height + Heron), parallelogram, trapezoid, ellipse, regular polygon (pentagon, hexagon, octagon, …), annulus (ring), sector of a circle, rhombus, and kite.
Beyond the answer, every shape includes a “How this works” tab: a short animation that shows where each formula comes from — a circle as wedges fanning into a rectangle, a triangle as half a parallelogram, a regular polygon as n triangles from the centre. A numbered step-by-step derivation runs alongside, so you understand the formula instead of memorising it.
The “How this works” view — formulas built up from one simple idea
Pick any shape, switch to the How this works tab, and the animation will play (looping). The numbered steps below the animation walk you through the derivation:
- Rectangle & square: a line of length a swept through a distance b traces out the area. Multiplication was invented for exactly this — b copies of a is just
a × b. - Triangle: two identical triangles snap into a parallelogram of base × height. One triangle is half:
½ · b · h. - Circle: slice the disc into wedges, fan them into a near-rectangle of width r and length πr. Multiply:
π·r². - Parallelogram & trapezoid: slice and slide / average the bases — both reduce to a rectangle.
- Regular polygon: n identical triangles from the centre, each ½·s·apothem. Add them up.
- Sector: a fraction θ/(2π) of the full disc's area πr².
- Rhombus & kite: both fit inside a rectangle whose sides are the perpendicular diagonals — and fill exactly half:
½·d₁·d₂.
Area formula reference — all 12 shapes
| Shape | Area formula | Common use |
|---|---|---|
| Circle | A = π · r² | Pizza, wheel, plate |
| Square | A = side² | Tiles, postage stamps |
| Rectangle | A = w · h | Rooms, screens, paper |
| Triangle (b·h) | A = ½ · b · h | Roof slope, sail |
| Triangle (Heron) | A = √(s(s−a)(s−b)(s−c)) | Three measured sides |
| Parallelogram | A = b · h | Diamond tiles, banners |
| Trapezoid | A = ½ · (a + b) · h | Bucket, dam wall |
| Ellipse | A = π · a · b | Athletics tracks, eggs |
| Regular polygon | A = (n · s²) / (4 · tan(π/n)) | Hexagon tiles, stop signs |
| Annulus | A = π · (R² − r²) | Washers, pipe wall, CDs |
| Sector | A = ½ · r² · θ | Pie slice, sprinkler arc |
| Rhombus | A = ½ · d₁ · d₂ | Diamond cards, argyle |
| Kite | A = ½ · d₁ · d₂ | Flying kites, sails |
Twelve formulas, four families
Once you spot the patterns, almost everything is variation:
- base × height family: rectangle (full), parallelogram (full), triangle (half), trapezoid (half × averaged base). Drop or apply the ½, and the base × height idea covers four shapes.
- π · radius² family: circle (πr²), ellipse (πab — a stretched circle), sector (½r²θ — fraction of a disc), annulus (π(R²−r²) — disc minus disc).
- half × diagonal product: rhombus and kite both use ½·d₁·d₂ because both fit half of their bounding rectangle.
- n triangles from a centre: the regular polygon is just n equal triangles meeting at the centroid.
Perpendicular height vs. slant side
Triangles, parallelograms, and trapezoids all use perpendicular height in their area formulas — the straight-line distance from the base to the opposite side. Using the slanted side instead gives a number bigger than the true area.
For a right triangle, the two legs are perpendicular, so the leg IS the height when the other leg is the base. For acute or obtuse triangles, drop a perpendicular from the apex (or use Heron's formula instead — it doesn't need a height).
Common area mistakes
- Forgetting to square units. A 3 cm × 4 cm rectangle has area 12 cm², not 12 cm. Area is always in square units.
- Using slant as height. For triangles, parallelograms, and trapezoids: perpendicular height only.
- Mixing π and 2π. Circle area uses π·r². Circumference uses 2·π·r. They're easy to swap when tired.
- Heron with invalid sides. Three lengths can only form a triangle if each pair sums to more than the third. The calculator validates this.
- Sector angle in degrees inside the radian formula. A = ½·r²·θ ONLY if θ is in radians. For degrees, use (θ°/360)·πr² — or convert first (× π/180).
- Annulus with inner ≥ outer. The hole can't be bigger than the disc. The calculator errors out; by hand you'd get a negative area.
Frequently asked questions
What 2D shapes does this area calculator support?
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What 2D shapes does this area calculator support?
▾Twelve shapes — circle, square, rectangle, triangle (base-height + Heron), parallelogram, trapezoid, ellipse, regular polygon (pentagon, hexagon, octagon, n-gon), annulus (ring), sector of a circle, rhombus, and kite. Each one returns area, perimeter (where defined), the relevant extras (apothem, arc length, side length, diagonals), and a step-by-step worked solution.
What is the "How this works" view?
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What is the "How this works" view?
▾A short animation that shows where each formula comes from. The circle is built by slicing a disc into wedges and fanning them into a near-rectangle of width r and length πr. The triangle is two copies snapped into a parallelogram. Each shape gets a numbered step-by-step derivation alongside the animation, so you understand the formula instead of memorising it.
Why is a triangle's area exactly half a parallelogram?
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Why is a triangle's area exactly half a parallelogram?
▾Take any triangle and put a copy of it upside-down next to itself. The two triangles snap together into a parallelogram of the same base and height. Since two triangles make one parallelogram, one triangle is exactly half: A = ½·base·height.
When should I use Heron's formula vs. ½·base·height?
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When should I use Heron's formula vs. ½·base·height?
▾Use ½·base·height when you can identify or measure the perpendicular height. Use Heron when you only know the three side lengths (e.g., a measured triangular plot of land). Both give the same answer for the same triangle — Heron is just the route that avoids needing the height.
What is the formula for a regular polygon (pentagon, hexagon, octagon)?
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What is the formula for a regular polygon (pentagon, hexagon, octagon)?
▾A = (n · s²) / (4 · tan(π/n)), where n is the number of sides and s is one side's length. The intuition: split the polygon into n identical triangles meeting at the centre — each triangle has base s and height equal to the apothem. Add up n of them and the formula falls out.
How does the sector formula handle degrees vs radians?
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How does the sector formula handle degrees vs radians?
▾A = ½·r²·θ when θ is in radians, or (θ°/360)·π·r² in degrees. The calculator has a toggle so you can use whichever your inputs are in — internally it always converts to radians before computing. The arc length follows the same fraction trick: r·θ in radians.
Why do rhombus and kite use the same formula?
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Why do rhombus and kite use the same formula?
▾Both shapes have two perpendicular diagonals that form a bounding rectangle of width d₁ and height d₂. The rhombus or kite is exactly half of that bounding rectangle — you can verify by counting the four corner triangles outside the shape. So both use A = ½·d₁·d₂.
Can I compute area in square feet / m² / cm² here?
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Can I compute area in square feet / m² / cm² here?
▾Yes — the calculator is unit-agnostic. Put in 3 (meters) × 4 (meters) and you get 12 (square meters). Put in 3 feet × 4 feet and you get 12 sq ft. The math is the same; the units are up to you.
Why doesn't the parallelogram show a perimeter?
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Why doesn't the parallelogram show a perimeter?
▾Computing the perimeter requires the slant-side length, which this calculator doesn't collect (it asks only for base + perpendicular height, enough for area). Same for the kite — without knowing where the diagonals cross, the four sides aren't uniquely determined.
How accurate is the ellipse perimeter?
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How accurate is the ellipse perimeter?
▾Ellipse perimeter has no closed-form expression in elementary functions — the exact integral is an elliptic integral. The calculator uses Ramanujan's 1914 approximation, accurate to about 5 × 10⁻⁵ for normal ellipses (better than the precision you'll ever need in practice).
Want the full reference?
The Area Guide walks through every formula with worked examples, the perpendicular-height trap, the diagonal-product family, and interactive embeds you can play with inline.
Read the Area Guide →