Geometry Calculator – Area & Volume

Calculate the area and volume of 11 geometric shapes instantly. Supports circle, square, rectangle, triangle, trapezoid, parallelogram, cube, rectangular prism, cylinder, cone, and sphere with step-by-step solutions.

What is Geometry Calculator – Area & Volume?

Geometry is one of the oldest and most practical branches of mathematics, dealing with shapes, sizes, positions, and properties of figures. Whether you are a student solving homework, an architect planning a building, or a DIY enthusiast measuring materials, calculating the area and volume of geometric shapes is an essential skill.

This geometry calculator handles 11 of the most common two-dimensional and three-dimensional shapes. In Area mode, you can compute the area (and perimeter or circumference) of circles, squares, rectangles, triangles, trapezoids, and parallelograms. In Volume mode, you can find the volume and surface area of cubes, rectangular prisms, cylinders, cones, and spheres. Every result includes a full step-by-step derivation so you can learn the formulas and understand how the answer was reached.

Formula

Area formulas:

Circle: A = πr², C = 2πr

Square: A = s², P = 4s

Rectangle: A = l × w, P = 2(l + w)

Triangle: A = ½bh

Trapezoid: A = ½(b₁ + b₂)h

Parallelogram: A = bh

Volume formulas:

Cube: V = s³, SA = 6s²

Rectangular Prism: V = lwh, SA = 2(lw + lh + wh)

Cylinder: V = πr²h, SA = 2πr(r + h)

Cone: V = ⅓πr²h, SA = πr(r + √(r² + h²))

Sphere: V = ⁴⁄₃πr³, SA = 4πr²

How to Calculate

Choose Area or Volume mode depending on what you need to calculate.
Select the geometric shape from the available options.
Enter the required dimensions (radius, side length, base, height, etc.).
The calculator instantly computes the result and shows a step-by-step derivation.

Example

A cylinder with radius r = 5 and height h = 10: Volume = π × 5² × 10 = 250π ≈ 785.40 cubic units. Surface area = 2π × 5 × (5 + 10) = 150π ≈ 471.24 square units.

Key Benefits

Covers 11 shapes — both 2D areas and 3D volumes in one tool
Animated SVG shape previews that scale with your input values
Full step-by-step solutions using KaTeX-rendered math formulas
Instant real-time results as you type
Copy, share, and export results with one click

Common Mistakes to Avoid

Using the diameter instead of the radius — always divide the diameter by 2 before using it in formulas
Confusing surface area (square units) with volume (cubic units) — they measure different things
Forgetting to square the radius in circle and cylinder formulas — πr² is not π × 2r
Using the wrong height in a cone — the perpendicular height, not the slant height, goes into the volume formula

Pro Tips

Always use consistent units throughout your calculation — mixing centimeters and meters will give wrong answers
For a cone, the slant height l = √(r² + h²) is needed for surface area but not for volume
A sphere always has the minimum surface area for a given volume among all shapes — this is why bubbles are spherical
When solving real-world problems, round your final answer but keep full precision during intermediate steps

Key Terms Explained

Radius: The distance from the center of a circle or sphere to any point on its surface
Diameter: A straight line passing through the center, equal to twice the radius (d = 2r)
Circumference: The perimeter or boundary length of a circle (C = 2πr)
Surface Area: The total area of all faces or surfaces of a three-dimensional object
Volume: The amount of three-dimensional space occupied by an object, measured in cubic units
Slant Height: The distance from the apex of a cone or pyramid to any point on the base edge

Common Use Cases

Solving geometry homework and exam problems
Architecture and construction — calculating material quantities
Packaging design — determining box dimensions and volumes
Landscaping — computing garden bed areas and mulch volumes
Cooking — converting between volume measurements for recipes

Frequently Asked Questions

How do I calculate the area of a circle?

Use the formula A = πr², where r is the radius. If you are given the diameter, divide it by 2 to get the radius first. For example, if the diameter is 10, the radius is 5, and the area is π × 5² ≈ 78.54 square units.

What is the difference between surface area and volume?

Surface area measures the total area of all outer faces of a 3D shape (expressed in square units like cm²), while volume measures how much space is inside the shape (expressed in cubic units like cm³). Think of surface area as the amount of paint needed to cover the outside, and volume as the amount of water the shape can hold.

How do I find the volume of a cone?

The volume of a cone is V = ⅓πr²h, where r is the radius of the circular base and h is the perpendicular height. A cone holds exactly one-third the volume of a cylinder with the same radius and height.

Why is the area of a triangle ½bh?

Any triangle can be thought of as half of a rectangle. A rectangle with base b and height h has area bh. Cutting it diagonally produces two identical triangles, each with area ½bh. This formula works regardless of the triangle's shape.

Can I use this calculator for irregular shapes?

This calculator covers standard geometric shapes with well-known formulas. For irregular shapes, you can often decompose them into simpler shapes (triangles, rectangles, circles), calculate each area or volume separately, and then add or subtract the results.