Area formulas (with real examples you can copy)

This section is intentionally detailed (and screenshot-friendly). If you’re studying, teaching, or doing a DIY project, you can use the formula blocks, example numbers, and quick checks to verify your work.

A rectangle is the easiest: multiply its two perpendicular sides. Formula: A = L × W. That’s why rectangles are the “default” shape for estimating rooms, rugs, tabletops, and screens.

• Example: L = 12 ft and W = 9 ft → A = 12 × 9 = 108 ft².
• Quick check: if you double the length, you double the area (because area scales linearly with each side).
• Real-life tip: For flooring, add 5–15% extra for cuts and waste.

A square is just a rectangle where L = W. That gives the familiar “side squared” rule. Formula: A = s².

• Example: s = 5 m → A = 5² = 25 m².
• Quick check: If the side grows by 10%, the area grows by about 21% (because area scales with the square).

A triangle can be “paired” with a copy of itself to form a rectangle (or parallelogram). That’s why the area is half of base × height. Formula: A = ½ × b × h, where h is the perpendicular height to base b.

• Example: b = 10 cm, h = 6 cm → A = ½ × 10 × 6 = 30 cm².
• Common mistake: using the slanted side as “height.” Height must be perpendicular to the base.
• Quick check: If you fold a triangle into a rectangle “box,” the triangle is exactly half the box.

Circle area is based on radius (distance from center to edge). The circle formula is one of the most reused in math and science: A = πr². If you only have diameter d, remember r = d/2, so A = π(d/2)².

• Example: r = 7 in → A = π × 7² = 49π ≈ 153.94 in².
• Quick check: Doubling r quadruples area (because of r²).
• Practical: pizza size, circular rugs, garden beds, pipes and ducts (cross-sectional area).

A parallelogram “slides” into a rectangle without changing its base or height — so the formula looks exactly like a rectangle, but the height is perpendicular to the base: A = b × h.

• Example: b = 14 m, h = 3 m → A = 14 × 3 = 42 m².
• Common mistake: using the slanted side length instead of perpendicular height.

A trapezoid has two parallel sides (often called bases) of lengths a and b, and height h. The area is the average of the parallel sides multiplied by the height: A = ½ × (a + b) × h.

• Example: a = 8 ft, b = 14 ft, h = 5 ft → A = ½ × (8+14) × 5 = ½ × 22 × 5 = 55 ft².
• Quick check: if a equals b, it becomes a rectangle: A = b × h.

An ellipse is like a stretched circle. It uses semi-major axis a and semi-minor axis b (half of the full widths). A = πab.

• Example: a = 10 cm, b = 6 cm → A = π × 10 × 6 = 60π ≈ 188.50 cm².
• Tip: if you’re given the full major/minor diameters, divide by 2 to get a and b.

A sector is a “pizza slice.” If the angle is θ degrees, it’s that fraction of a full circle: A = (θ/360) × πr². If you’re using radians, it’s A = ½r²θ.

• Example: r = 12 in, θ = 90° → A = (90/360) × π × 12² = ¼ × π × 144 = 36π ≈ 113.10 in².
• Quick check: 180° is half a circle; 90° is a quarter.

For a regular polygon (all sides and angles equal), a clean formula uses the number of sides n and side length s: A = (n × s²) / (4 × tan(π/n)). This is excellent for hexagons, octagons, and many tiling patterns. If you prefer apothem a (distance from center to midpoint of a side), another common form is A = (Perimeter × apothem)/2.

• Example (regular hexagon): n = 6, s = 4 cm → A = (6 × 16) / (4 × tan(π/6)) = 96 / (4 × 0.57735) ≈ 41.57 cm².
• Quick check: a regular hexagon is 6 equilateral triangles. If you know triangle area, you can estimate too.

• Flooring: total ft² or m² determines how much flooring to buy.
• Paint: wall area (minus windows/doors) helps estimate gallons.
• Gardens: soil, mulch, and seed coverage depend on surface area.
• Science/engineering: cross-sectional area affects flow, stress, and heat transfer.

If a dimension is off by a factor of 10 (like 3.2 instead of 32), your area will be off by 10× or 100× depending on how many dimensions are multiplied. If a rectangle’s length is 10× too big, the area becomes 10× too big. If a square’s side is 10× too big, area becomes 100× too big. When your answer looks wildly wrong, check units and decimal places first.

Frequently Asked Questions

• Why do area units have a “²” (squared)?

  Area counts how many unit-squares fit inside a shape. If the unit is a meter, the “tile” is 1 m by 1 m, which is 1 m². That’s why area is always in squared units.

• Do I need to convert units before using the calculator?

  You don’t need to convert if all inputs are in the same unit. If one measurement is in inches and another is in feet, convert one of them first so they match.

• What if my shape is irregular?

  Break it into smaller familiar shapes (rectangles, triangles, circles), compute each area, then add them. For curved/complex boundaries, approximation or CAD tools may be needed.

• What’s the difference between perimeter and area?

  Perimeter is the distance around the boundary (linear units like ft or m). Area is the space inside (squared units like ft² or m²). A big perimeter doesn’t always mean a big area.

• Why does doubling radius make circle area 4× bigger?

  Because the formula is A = πr². If r becomes 2r, then r² becomes (2r)² = 4r², so the area becomes 4 times larger.

• Is this calculator accurate?

  The math is exact for the formulas shown. Accuracy in real projects depends on measurement accuracy and whether the chosen shape matches the real object.