How do you calculate the area of a circle?

The area of a circle is A = π × r², where r is the radius (not diameter). For diameter d, use A = π × d²/4. For a circle with 10 cm radius: A = π × 100 ≈ 314 cm². For 20 cm diameter (= 10 cm radius): same answer.

What is the formula for the area of a trapezoid?

The area of a trapezoid is A = (1/2) × (base₁ + base₂) × height, where base₁ and base₂ are the two parallel sides and height is the perpendicular distance between them. The two bases need not be equal; if they are, the shape is a rectangle.

How do I find the area of a triangle?

Simplest: A = (1/2) × base × height. Height must be perpendicular to the base. For triangle with sides only (no height): use Heron's formula A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. For sides a, b and included angle C: A = (1/2) × a × b × sin(C).

What units is area measured in?

Square units of length: m², ft², in², cm², mi², km². For land: acres (4,047 m²), hectares (10,000 m²). 1 m² ≈ 10.76 ft². 1 acre ≈ 0.405 hectares. Choose units matching the scale of your problem.

How do I find the area of an irregular shape?

Three approaches: (1) Divide into rectangles, triangles, and circular sectors; calculate each and sum. (2) For polygons with coordinates: use Shoelace formula. (3) For arbitrary curves: numerical integration (trapezoidal or Simpson's rule). CAD and GIS software automate this for any closed boundary.

How much paint do I need for a room?

Calculate wall + ceiling area, subtract doors/windows. Typical paint coverage: 1 gallon ≈ 400 ft² (1 L ≈ 10 m²) one coat. For two coats, double. 12 × 14 ft room with 8 ft walls: 416 ft² walls + 168 ft² ceiling = ~584 ft² → ~1.5 gallons per coat.

How do I calculate area of L-shaped room?

Divide into two rectangles, calculate each area, then add. Example: L-shape 5×8 with 3×3 cutout = (5×8) - (3×3) = 31 sq units. Or: split the L into two rectangles whose areas sum to total. Both methods give same answer.

What's the difference between area and perimeter?

Area: amount of 2D space inside (m², ft²). Perimeter: total length of boundary (m, ft). For 10×5 rectangle: area = 50 m², perimeter = 30 m. Two different quantities measured in different units (squared vs linear). Easy to confuse but important to distinguish.

Area Calculator

Find the area of common geometric shapes. Select a shape and enter the required dimensions to calculate its area. Supports circles, rectangles, triangles, and trapezoids.

Area is the measure of two-dimensional space enclosed by a shape. Whether calculating how much paint to buy for a wall, how much lawn fertilizer to spread, how much fabric to make a dress, or how many square feet a house has, you're working with area. The calculation depends on the shape — circles, rectangles, triangles, trapezoids, and irregular shapes each have their own formulas.

The most familiar formula is rectangle area: length × width. From this, every other shape's area can be derived. A triangle is half a rectangle: ½ × base × height. A circle, derived through integration, gives πr². A trapezoid combines triangles and rectangles. These foundational formulas are the building blocks for measuring all 2D geometry.

Units of area are square units: square meters (m²), square feet (ft²), square miles, acres, hectares. Conversions span enormous ranges: 1 m² = 10.76 ft²; 1 acre = 4,047 m²; 1 hectare = 10,000 m² = 2.47 acres; 1 km² = 100 hectares = 247 acres.

For irregular shapes, area is computed via integration or numerical approximation (Simpson's rule, trapezoidal rule, Monte Carlo). In practice, many tools — CAD software, GIS systems, mapping apps — calculate area of any closed shape automatically using algorithms like the Shoelace formula or vector cross products.

Common applications: construction (paint, flooring, roofing), landscaping (lawn, mulch, fertilizer), real estate (property size, room dimensions), agriculture (field acreage), sewing/fabric, packaging, surveying, and any analysis requiring a 2D extent measurement.

Inputs

Shape

Length / Radius / Base

Width / Height

Top Base (trapezoid only)

Results

Area

50 sq units

Perimeter

30 units

Formula Used

A = length x width = 10 x 5

Last updated: May 29, 2026

Formula

**Common area formulas:** | Shape | Formula | |---|---| | Square | s² (where s = side length) | | Rectangle | l × w | | Triangle | ½ × b × h (or ½absin C) | | Parallelogram | b × h | | Trapezoid | ½(b₁ + b₂) × h | | Circle | πr² | | Ellipse | π × a × b (semi-major, semi-minor) | | Regular polygon | ½ × perimeter × apothem | | Sector of circle | ½ × r² × θ (θ in radians) | **Worked example: rectangle 10 × 5** A = 10 × 5 = 50 square units **Worked example: circle radius 7** A = π × 7² = π × 49 ≈ 153.94 square units **Worked example: triangle base 8, height 6** A = ½ × 8 × 6 = 24 square units Triangle area is always half the area of the rectangle that bounds it. **Worked example: trapezoid bases 10 and 6, height 4** A = ½ × (10 + 6) × 4 = ½ × 16 × 4 = 32 square units **Heron's formula (triangle with sides a, b, c):** s = (a + b + c) / 2 (semi-perimeter) A = √(s(s-a)(s-b)(s-c)) For triangle sides 3, 4, 5: s = 6 A = √(6 × 3 × 2 × 1) = √36 = 6. **Common unit conversions:** | From | To | Factor | |---|---|---| | m² | ft² | 10.764 | | ft² | m² | 0.0929 | | ft² | in² | 144 | | in² | cm² | 6.452 | | m² | km² | 10⁻⁶ | | acre | m² | 4,047 | | acre | ft² | 43,560 | | hectare | m² | 10,000 | | hectare | acre | 2.471 | | km² | hectare | 100 | | km² | mi² | 0.386 | | mi² | km² | 2.59 | **Common areas (perspective):** | Object | Area | |---|---| | Letter paper (8.5×11") | 93.5 in² ≈ 0.060 m² | | Standard parking space | ~14 m² (~150 ft²) | | Bedroom (typical) | 10-20 m² (110-220 ft²) | | Small house | 100-150 m² | | Football field | ~5,350 m² (~57,600 ft²) | | Basketball court | ~420 m² | | Tennis court | ~261 m² | | Acre | 4,047 m² (~43,560 ft²) | | Hectare | 10,000 m² (~107,640 ft²) | | Central Park NYC | 3.41 km² (843 acres) | | Manhattan | 59.1 km² | | State of Rhode Island | 4,001 km² | | Continental US | 8 million km² | | Earth's surface | 510 million km² | **Circle calculations:** Area: A = π × r² Circumference: C = 2 × π × r = π × d Diameter: d = 2r For a circle with d = 10 ft (5 ft radius): A = π × 25 ≈ 78.54 ft² **Sector area:** For circle sector with angle θ: - Radians: A = ½ × r² × θ - Degrees: A = (θ/360) × πr² Half circle: A = πr²/2. Quarter circle: A = πr²/4. **Annulus (ring) area:** A = π(R² - r²) Where R = outer radius, r = inner radius. Used for washers, pipes, donut shapes. **Polygon area (Shoelace formula):** For polygon with vertices (x₁, y₁), (x₂, y₂), ..., (xₙ, yₙ): A = ½ × |Σ(xᵢ × yᵢ₊₁ - xᵢ₊₁ × yᵢ)| Works for any simple polygon (no self-intersections). **Surface area of 3D objects:** For exterior surface of solids: - Cube: 6s² - Sphere: 4πr² - Cylinder: 2πr² + 2πrh = 2πr(r+h) - Cone (with apex): πr² + πrl (l = slant height) **Irregular shape methods:** - **Grid method**: count squares. - **Trapezoidal rule**: numerical integration. - **Simpson's rule**: better numerical integration. - **CAD software**: automatic from drawing. - **GIS tools**: from polygon coordinates. - **Mobile apps**: phone-based area measurement. **Practical application: paint coverage:** Most paints cover ~10 m² per liter (~400 ft² per gallon). For a wall 3 m × 4 m = 12 m², need ~1.2 L of paint (one coat). **Practical application: lawn calculations:** For lawn 20 m × 15 m = 300 m². Fertilizer rate 25 g/m² → 7.5 kg fertilizer needed. **Composite shapes:** For irregular shapes: 1. Divide into basic shapes (rectangles + triangles). 2. Calculate each area. 3. Sum (or subtract for cutouts). Example: L-shaped room = rectangle - rectangle. **Common applications:** - **Construction**: drywall, flooring, paint, roofing, insulation. - **Real estate**: room/house/lot square footage. - **Landscaping**: lawn, mulch, sod, fertilizer, irrigation. - **Agriculture**: field acreage, fertilizer, seed quantities. - **Sewing**: fabric requirements. - **Packaging**: surface area for labels. - **HVAC**: heating/cooling load (BTU/ft²). - **Solar**: panel area for energy generation.

How to use this calculator

Choose the shape: rectangle, circle, triangle, or trapezoid.
Enter required dimensions.
Rectangle: length × width. Circle: radius. Triangle: base + height. Trapezoid: both bases + height.
Calculator returns area in your chosen units (squared).
For irregular shapes, divide into basic shapes and sum.
For composite shapes with cutouts, subtract.

Worked examples

Living room flooring

**Scenario:** Living room 5 m × 4 m. Hardwood flooring at $80/m². Cost? **Calculation:** A = 5 × 4 = 20 m². Cost = 20 × 80 = $1,600. Add 10% for waste: $1,760. **Result:** ~$1,760 for hardwood materials. Add labor (~$30-50/m² for installation): ~$600-1,000. Total: $2,400-3,000. Engineered floors cheaper; solid oak more expensive.

Circular patio

**Scenario:** Round patio 8 m diameter. Pavers cost $30/m². How much? **Calculation:** r = 4 m. A = π × 16 ≈ 50.27 m². Cost = 50.27 × 30 = $1,508. Add 5% waste: $1,584. **Result:** ~$1,584 for pavers (~$3,000 with labor). Circles waste more material in cutting than rectangular shapes due to curved edges. Hexagonal pavers minimize waste for circular layouts.

Triangular garden bed

**Scenario:** Triangular flower bed, base 4 m, height 3 m. Mulch volume at 5 cm thickness? **Calculation:** Area = ½ × 4 × 3 = 6 m². Volume = 6 × 0.05 = 0.3 m³. **Result:** Need 0.3 m³ (300 L) of mulch — about 8 standard 40 L bags. Garden centers sell bulk by m³ for larger projects. For 5 cm coverage, 0.3 m³ is correct; doubling depth doubles the requirement.

When to use this calculator

**Use area calculations for:**

- **Construction**: paint, flooring, insulation, drywall, roofing. - **Landscaping**: lawn maintenance products, mulch, gravel. - **Real estate**: property valuations, room sizes. - **Agriculture**: field area, planting density. - **Manufacturing**: material requirements, packaging. - **HVAC**: heating/cooling capacity sizing. - **Solar**: panel area for energy generation. - **Sewing/fashion**: fabric estimation.

**Area unit choices:**

- **Small**: cm², in² (cabinetry, small objects). - **Room/house**: m², ft² (residential). - **Land (small)**: acres, hectares. - **Large land**: km², square miles.

US uses mostly imperial; rest of world uses metric.

**Conversions to remember:**

- 1 acre = 43,560 ft² = 4,047 m². - 1 hectare = 10,000 m² = 2.47 acres. - 1 km² = 100 hectares. - 1 mi² ≈ 640 acres ≈ 2.59 km².

**Composite shapes:**

Real-world shapes are often composites: - L-shaped rooms: divide into two rectangles. - T-shaped lots: three rectangles. - Curved property: rectangle + sector. - Buildings with cutouts: subtract cutout area.

Break complex shapes into rectangles, triangles, and circular sectors. Sum (or subtract) to get total.

**Common applications:**

- **Paint estimation**: 1 gal ≈ 400 ft² (1 L ≈ 10 m²) for one coat. - **Carpet/flooring**: include 5-10% waste for cuts. - **Mulch**: 1 yd³ covers 100 ft² at 3" deep. - **Concrete**: depth × area = volume. - **Roofing**: include 10% for waste and overlap. - **Lawn seed**: 1 lb covers ~250 ft² for new lawn. - **Land taxes**: often based on lot area. - **Insurance**: building values per ft².

**Surface area vs area:**

- **Area**: 2D measure, e.g., wall area. - **Surface area**: total exterior area of 3D object, e.g., box has 6 faces.

For painting a room, you sum wall + ceiling areas. For wrapping a gift, you compute box surface area.

**Practical tips:**

- Always include waste factor (5-15% depending on material). - Round up for material orders. - Verify with multiple measurements. - Use proper units consistently. - Sketch the shape with dimensions to avoid errors.

**Software:**

- **CAD packages**: AutoCAD, SolidWorks, SketchUp. - **GIS**: ArcGIS, QGIS for land calculations. - **Mobile apps**: AR-based area measurement. - **Spreadsheets**: simple formula calculators. - **Google Earth**: estimate property areas.

**Land measurement tools:**

- **Tape measure**: small areas. - **Laser distance meter**: medium rooms/buildings. - **GPS receiver**: outdoor land surveying. - **Drones with photogrammetry**: large/complex sites. - **Total station**: professional surveying.

**Pitfalls:**

- **Mixing units**: m × ft = wrong. - **Forgetting waste**: order materials short. - **Confusing base and height in triangles**: height must be perpendicular to base. - **Using diameter for circle area**: must convert to radius first. - **Forgetting to subtract cutouts**: doors, windows, etc. - **Rounding too aggressively**: 0.5 m² error in 100 m² is significant. - **Area vs perimeter**: completely different concepts. - **Curve treated as straight**: circular surfaces need π.

Common mistakes to avoid

Confusing diameter and radius (πr² uses radius, not diameter).
Using non-perpendicular height in triangle area (must be perpendicular to base).
Mixing units (meters with feet, etc.).
Forgetting to subtract cutouts (doors, windows, etc.).
Forgetting waste factor when ordering materials.
Confusing area with perimeter or volume.
Using circumference instead of area for circles.
Computing area of irregular shapes without breaking into pieces.

Frequently Asked Questions

Sources & further reading

SponsoredShop Top Deals on AmazonSupport CalcMountain — browse top-rated products at no extra cost to you.