Surface Area Volume Calculator

Shape Properties

Understanding Surface Area and Volume

Surface area and volume are fundamental geometric concepts used to describe three-dimensional objects. Surface area refers to the total area of all the surfaces of an object, essentially the amount of "skin" it has. Volume, on the other hand, measures the amount of space an object occupies.

These calculations are crucial in various fields, including engineering, architecture, physics, chemistry, and even everyday tasks like packaging, painting, or calculating the capacity of containers.

Key Formulas and Concepts

Cube

A cube is a three-dimensional shape with six equal square faces.

Let 'a' be the length of one side.
Surface Area (SA) = 6 * a²
Volume (V) = a³

Cuboid

A cuboid (or rectangular prism) is a six-faced three-dimensional shape with rectangular faces.

Let 'l' be the length, 'w' be the width, and 'h' be the height.
Surface Area (SA) = 2 * (lw + lh + wh)
Volume (V) = l * w * h

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space.

Let 'r' be the radius.
Surface Area (SA) = 4 * π * r²
Volume (V) = (4/3) * π * r³

Cylinder

A cylinder is a basic three-dimensional shape with two parallel circular bases connected by a curved surface.

Let 'r' be the radius of the base and 'h' be the height.
Surface Area (SA) = 2 * π * r * (r + h) (includes top and bottom bases)
Volume (V) = π * r² * h

Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

Let 'r' be the radius of the base, 'h' be the height, and 'l' be the slant height (l = sqrt(r^2 + h^2)).
Surface Area (SA) = π * r * (r + l) (includes base)
Volume (V) = (1/3) * π * r² * h

Rectangular Pyramid

A pyramid with a rectangular base.

Let 'l' be the length of the base, 'w' be the width of the base, 'h' be the height.
Let 's_l' be the slant height for the faces with width 'w' (s_l = sqrt(h^2 + (l/2)^2))
Let 's_w' be the slant height for the faces with length 'l' (s_w = sqrt(h^2 + (w/2)^2))
Surface Area (SA) = lw + l*s_l + w*s_w (area of base + area of 4 triangular faces)
Volume (V) = (1/3) * l * w * h

Use Cases

• Construction & Architecture: Calculating paint needed for walls (surface area) or concrete for foundations (volume).
• Manufacturing & Packaging: Determining material needed for products and packaging, or the capacity of containers.
• Science & Engineering: Analyzing fluid dynamics, heat transfer, material strength, and chemical reactions.
• Everyday Life: Estimating garden soil needs (volume), or the amount of fabric for a project (surface area).