A free area and volume calculator computes the surface measure or capacity of common geometric shapes from the dimensions you enter. On Yaya Tools you pick area or volume mode, choose a shape, type positive lengths in millimeters through yards, and read the result instantly—along with the formula, a plain-language summary, and a conversion table across metric, imperial, and specialty units. Everything runs locally in your browser; nothing is uploaded to a server.
People search for an area calculator, a volume calculator, or a geometry calculator online when homework, construction estimates, packaging plans, or lab work require a quick check without opening a spreadsheet. This page covers every shape and formula built into the tool, worked examples you can verify on the page, and practical tips for units and common mistakes. For general length conversions between systems, the US unit converter complements this calculator when you need to normalize measurements before you paste them here.
What is an area and volume calculator?
An area and volume calculator applies standard geometry formulas to find two-dimensional area (square units) or three-dimensional volume (cubic units) from linear measurements such as radius, base, height, or side length. A useful online version shows the formula used, updates results as you type, and converts the answer into units you actually report—square meters for flooring, cubic feet for shipping, liters for tank capacity, or hectares and mu (亩) for land.
Yaya Tools’ calculator is built for that everyday workflow:
- Two modes — switch between Area (7 shapes) and Volume (6 shapes)
- Real-time results — no calculate button; output refreshes on every input change
- Formula display — the exact expression used for the selected shape
- Summary sentence — a readable recap of shape, parameters, and result in your chosen length unit
- Unit conversion table — the same result expressed in all supported area or volume units, with your natural unit highlighted
- Length input units — mm, cm, m, km, in, ft, yd (default cm)
- Area output units — mm², cm², m², km², in², ft², yd², hectare (ha), mu (亩)
- Volume output units — mm³, cm³, ml, L, m³, in³, ft³, yd³, US gallon
- Validation — all dimensions must be positive numbers; sector central angle must be greater than 0 and at most 360 degrees
- Local processing — math stays in your current browser session
It does not compute surface area of 3D solids, composite shapes, or irregular polygons. For fractional dimensions or scaling a result by a ratio, pair this page with the fraction calculator or percentage calculator after you confirm the base area or volume here.
How to use this area and volume calculator
The workflow is short so you can answer a geometry question in under a minute.
- Open the area and volume calculator on Yaya Tools.
- Click Area or Volume at the top to set the mode.
- Choose a shape from the dropdown (circle and cube are the defaults for area and volume).
- Select a length unit for your inputs—cm is preselected; switch to inches, feet, meters, or another option if your blueprint uses different labels.
- Enter every dimension field shown for that shape. Results update automatically as you type.
- Read the main result card, the formula line beneath it, and the summary sentence describing shape, inputs, and answer.
- Scroll to the unit conversion table to copy the value in m², ft³, liters, hectares, gallons, or other supported units.
- Click Clear to reset to area mode, circle shape, cm units, and default sample dimensions.
If any field is empty, non-numeric, zero, or negative, the tool clears results and shows an error asking for valid positive dimensions. For a sector, the central angle must be strictly greater than 0° and no greater than 360°. If the computed value overflows or becomes invalid, an error prompts you to reduce input magnitudes.
Privacy note: dimensions are processed only in the browser. Yaya Tools does not receive or store what you enter. On a shared device, click Clear when you finish if measurements are sensitive.
Area formulas this calculator uses
Area mode covers seven plane figures. Notation below matches the on-page formulas. All linear inputs use the unit you select; the raw result appears in the matching square unit (for example, cm inputs → cm² in the main result, with conversions in the table).
Circle
A = π × r²
Inputs: radius (r). Example: radius 5 cm → A = π × 25 ≈ 78.54 cm².
Triangle
A = b × h ÷ 2
Inputs: base (b), height (h)—perpendicular height, not slant side length.
Rectangle
A = w × h
Inputs: length (w), width (h).
Parallelogram
A = b × h
Inputs: base (b), perpendicular height (h).
Trapezoid
A = (a + b) × h ÷ 2
Inputs: top base (a), bottom base (b), perpendicular height (h).
Ellipse
A = π × a × b
Inputs: semi-major axis (a), semi-minor axis (b).
Sector (circular sector)
A = π × r² × θ ÷ 360
Inputs: radius (r), central angle θ in degrees (must be > 0 and ≤ 360). Example: r = 5 cm, θ = 90° gives one quarter of a full disk: A = π × 25 × 90 ÷ 360 ≈ 19.63 cm².
Volume formulas this calculator uses
Volume mode covers six three-dimensional solids. Results display in the cubic unit that matches your length unit (cm → cm³ by default), with ml and L treated as volume equivalents in the conversion table.
Cube
V = a³
Inputs: edge length (a).
Box (rectangular prism)
V = l × w × h
Inputs: length (l), width (w), height (h).
Cylinder
V = π × r² × h
Inputs: radius (r) of the circular base, height (h) along the axis. Right circular cylinder only.
Sphere
V = 4/3 × π × r³
Inputs: radius (r).
Cone
V = 1/3 × π × r² × h
Inputs: radius (r) of the base, height (h) from base to apex along the axis. Right circular cone.
Pyramid (square base)
V = 1/3 × a² × h
Inputs: base edge length (a)—the tool assumes a square base—and height (h) from base to apex. This is one third of base area times height, consistent with the cone formula structure.
Unit conversion: how the table works
You choose one length unit for every dimension field. The calculator converts internally to meters for area (m²) or volume (m³), then divides by each output unit’s factor to fill the table.
Length units (input): mm, cm, m, km, in, ft, yd.
Area units (output table): mm², cm², m², km², in², ft², yd², hectare (ha)—10,000 m²—and mu (亩)—2000/3 m², a Chinese land measure.
Volume units (output table): mm³, cm³, ml, L, m³, in³, ft³, yd³, US gallon.
Because 1 cm³ = 1 ml, those rows match numerically. The highlighted natural row matches your input unit squared or cubed; other rows quote the same geometry in units your audience expects. For length pairs outside this table, use the US unit converter.
Worked examples
These examples use default sample values on the page. Open the area and volume calculator and compare.
Example 1: Circle area in cm² and m²
Shape: Circle, unit: cm, radius = 5.
A = π × 5² = 25π ≈ 78.5398 cm²
In m²: 78.5398 × 10⁻⁴ ≈ 0.007854 m²
Example 2: Trapezoid area (land plot sketch)
Shape: Trapezoid, unit: m, top base = 8, bottom base = 5, height = 4.
A = (8 + 5) × 4 ÷ 2 = 13 × 2 = 26 m²
In hectares: 26 ÷ 10,000 = 0.0026 ha
In mu: 26 ÷ (2000/3) ≈ 0.039 mu
Example 3: Sector area for a pie slice
Shape: Sector, unit: cm, radius = 5, angle = 90°.
A = π × 25 × 90 ÷ 360 = 25π/4 ≈ 19.635 cm²
Example 4: Box volume for shipping
Shape: Box, unit: in, length = 5, width = 4, height = 3.
V = 5 × 4 × 3 = 60 in³
Example 5: Cylinder volume in liters
Shape: Cylinder, unit: cm, radius = 3, height = 8.
V = π × 9 × 8 = 72π ≈ 226.195 cm³
Since 1 cm³ = 1 ml, volume ≈ 226.2 ml ≈ 0.226 L
Example 6: Square-base pyramid
Shape: Pyramid, unit: m, base edge = 4, height = 6.
V = 1/3 × 4² × 6 = 1/3 × 16 × 6 = 32 m³
Common mistakes when calculating area and volume
Mixing units across fields. The tool applies one length unit to every input. Do not enter feet in one box and inches in another without converting first; use the US unit converter to normalize.
Using diameter instead of radius for circle, cylinder, sphere, or cone. Halve the diameter before entry, or use the fraction calculator to divide by two when your source lists diameter only.
Entering angle in radians for a sector. The sector formula expects degrees. Convert radians × (180 ÷ π) first.
Zero or negative dimensions. The page requires strictly positive values. Open boundaries (angle 0°) are rejected for sectors.
Wrong height on triangles and trapezoids. Height must be perpendicular to the chosen base, not a side length along the slope.
Assuming any pyramid formula. Only a square base with edge a is supported (V = 1/3 × a² × h). Rectangular-base pyramids need a different base area.
Ignoring the one-third factor on cone and pyramid volumes. Cones and pyramids are exactly one third of the corresponding cylinder or prism with the same base and height—a frequent exam trap.
References
- Area formulas (Math is Fun — Area) — Plane shape definitions aligned with standard textbook notation.
- Volume formulas (Math is Fun — Volume) — Prisms, cylinders, spheres, cones, and pyramids.
- NIST Guide for the Use of the International System of Units (SI) — Official guidance on metric units and unit conversion discipline.