Calculating volumes

## General information

"Calculate volumes, including prisms, pyramids, cones, and spheres, using formulae." Is part of the Geometry and Measurement section of Level 6 Mathematics and Statistics. One is often given the dimensions of a shape (which can be composite) and asked to calculate its volume, or asked to design their own shape and calculate its volume. These questions are often linked to cost of materials, and real-life situations. Formulas used to calculate these volumes are given in exams and therefore are not required to be memorised.

Dimensions will often be given to you in a different unit in which the answer is required. Remember that since you are calculating a volume, conversions will be very different from those on one or two dimensions. See unit conversion for more information.

## What You Need to Know: Rectangular Prism

The formula for the volume of a rectangular prism is length\*width\*height. All three dimensions must be known in order to calculate the volume of a normal rectangular prism, but only one is required for a cube, as the three dimensions are identical. Note that length\*width=base, so the formula for the volume of a general prism (found below) also applies.

## General Prism

The formula is base\*height, base here being the area of the cross-section of the prism. Keep in mind that a cross-section must be a surface that remains the same at any point of the prism. This means that for many prisms, only the surface cut along one side will be a cross-section. Three dimensions of the shape are generally given.

## Pyramids

The formula for the volume of a pyramid is very similar to that for a general prism, although it is different in that it lacks a cross-section. Therefore, the formula for the volume of a pyramid with any-shaped base (triangle, square, rectangle) is base\*height/3. Simply calculate the base area of the pyramid, multiply that by its height, and divide by 3. You may be required to calculate the height of a base triangle using Pythagoras' Theorem (A2+B2=C2)

## Cylinders and Cones

The formula for a general prism applies for the volume of cylinders, and a cone is essentially a circle-based pyramid. As such, the formula for the volume of a cone is base (πr2)\*height/3, while the formula for a cylinder is without the '/3' part. The radius and height will generally be given in a question.

## Spheres

This shape's formula is arguably the most complicated of these, but luckily, you are not required to memorise it. The formula is: 4/3πr3. You will inevitably have to round in these calculations. When doing so, remember to *only round your final answer*, as using rounded numbers in calculations will result in an inaccurate answer. A useful function is the 'save' on your calculator (which is permitted in this exam), which usually allows one to assign a number to a letter, and type the letter when the number is required. Show an appropriately rounded answer in your calculations (but *do not* leave π in), but use the saved number on your calculator. Consult the internet for how to do this on your specific calculator type.

## Links

The following are some useful links that may help you.

https://www.calculator.net/volume-calculator.html (a volume calculator. Simply insert the dimensions and select the unit, and it will calculate the volume of many common shapes!)

https://www.skillsyouneed.com/num/volume.html (useful guide for the calculation of many shapes)

https://www.khanacademy.org/math/basic-geo/basic-geo-volume-sa (Often linked to in this Wiki, Khan Academy has a series of practice questions on different shapes (scroll down for more shapes))