Surface Area Calculator

Surface area is the total area of the outside of a three-dimensional object. It is the area you would cover if you covered the object with something (e.g. wrapping paper or paint). This calculation tool allows you to calculate the surface area for eleven different types of objects. Select an object and enter in the measurements to instantly see the surface area, as well as the volume that the object can hold.

Since area covers a two-dimensional surface, it is always expressed in square units (e.g. square centimeters or square feet). Volume, on the other hand, fills up three-dimensional space and therefore is expressed in cubic units. You can change the units for each element and the calculator will do the conversion so that you can measure in inches and check your results in square feet or liters.

How to use this calculator tool:

Select a body from the form menu. The input fields change according to the form so that only those measurements actually required for this body are shown.

Enter each measurement and set the unit. As soon as you enter something, the surface will be updated and shown in a well visible number field and also in a copyable field with volume next to it. To try another shape just select a new shape from the menu.

Here are the formulas for calculating surface area and volume of each shape.

Every body has its own formula. This calculator uses the following formulas and always shows total surface area. All areas are added up, both flat and curved, including the base.

The difference between surface area and total area:

For bodies such as cylinders, cones, prisms and pyramids where the top and bottom are clearly defined it is customary to divide the surface into two parts. The lateral surface refers only to the side walls, that is, the part of the body surrounding it. The total area includes in addition the flat top and bottom sides.

This calculator always shows the total area. This is the area that must be covered or painted if you were to wrap the entire body. If only lateral surface area is needed, subtract the base area. For a cylinder, both circles are each pi*r^2. For a cone or pyramid, one base area is equal to the entered circle or square.

For cones and pyramids, use slant height.

For cones and pyramids, the height along the slant is longer than the height that goes straight up to the center. This slanted distance is called the slant height, and it's what actually determines the area of the curved surface or triangular faces. You can use the Pythagorean theorem to find the slant height from the vertical height and the base length.

For a cone with radius 3 and height 4 the slant height is the square root of 9 plus 16 which is 5. The total surface area is pi times the radius times the slant height plus the area of the base circle.

Bodies associated with spheres.

Of all bodies of equal volume, a sphere has the least surface area. This is why bubbles or water droplets naturally form into spherical shapes. The surface area and volume of a sphere are determined solely by its radius.

A hemisphere is half of a sphere and is placed with its flat, circular side facing down. Therefore the total surface area is made up of half the surface area (so pi r squared) plus the area of the circle in the base (pi r squared), which makes three pi r squared in total. A capsule is a cylinder whose ends are each closed by a hemisphere. When you put those two hemispheres together, the resulting surface area is both the volume and the surface area of a full sphere.

Examples of calculations:

Since these examples use default values they can be reproduced individually. The surface of each body is calculated in different ways.

Unit selection and conversion

The unit chosen for a measurement determines the units of the result. If a storage container is measured in meters, then results for surface area will be given in square meters and volume in cubic meters. These units cannot simply be converted directly. This is because area increases proportionally to the square of the length unit while volume increases proportionally to the third power of the length unit.

One meter is approximately three feet and a quarter, but one square meter is nearly eleven square feet, and one cubic meter exceeds thirty-five cubic feet. When the units are changed, the calculator maintains accuracy in conversion. Also volumes can be converted into liters or gallons. As one liter is exactly one thousand cubic centimeters, a cube with sides of ten centimeters will hold exactly one liter.

Surface area to volume ratio

When the surface area of an object is divided by its volume, a number results that has significance not only in geometry but also in a much broader context. For objects of similar shape, a smaller object will have a larger surface-area-to-volume ratio than a larger object. This is because as an object gets bigger, its volume increases faster than its surface area.

This explains why crushed ice melts faster than a large block of ice, why small animals lose heat more rapidly than larger ones, and why cells must remain small in order to obtain nutrients from their surfaces. For the same shape this effect can be directly observed by comparing the values for surface area and volume shown by the calculator for two different sizes.

Areas of application for surface calculation

Surface area determines the amount of material needed to cover or paint an object. It informs how much sheet metal is required for a storage container, how much paint is needed for a dome, how much wrapping paper is required for a box and how much cardboard is required for a moving carton. All these materials are sold or weighed in square units.

The surface area also affects various processes that take place at interfaces. Coolers and heaters are designed to have as large a surface area as possible in order to dissipate heat quickly. Insulation and cold rooms, on the other hand, are designed to have as small a surface area as possible in order to maintain a constant temperature. The larger the surface area, the faster chemical reactions, drying, and dissolution take place.

Tips for accurate results:

Before you read off the result, make sure all measurements of the same body are in the same units. When measuring the height of a pyramid or cone, measure along the center line perpendicular to the base, not along the slant. This is because the formula uses vertical height and the base area to calculate the slant height.

Since surface area increases as the square of length and volume as the cube of length, a small error in measurement leads to a large error in the result. When accuracy is required, measure twice.

This tool is for general education and planning purposes only. For technical applications, manufacturing or other high risk activities these values should be checked against the requirements of your project and tolerances.