Why is the geometry package messing my margins up?

Question

I want to write a document making use of as much of the page as I can. The code I am using is this (disclaimer: I took the sample text from here):

\documentclass[12pt, oneside, a4paper]{memoir}

\usepackage{layout}
\usepackage{showframe}

%\usepackage[a4paper]{geometry}

\usepackage{amssymb, amsfonts, amsthm}
\newtheorem{problem}{Problem}

\begin{document}
\title{Sample Document}
\author{John Doe}
\maketitle

\layout{}

Here is some sample text to show you what LaTeX does. 

To start a new paragraph, you need to leave a line of white space in your tex file.

To include math, you have two options. The first is called ``in line,'' and you do this by putting your math between two dollar signs (\$). For example, Fermat's Little Theorem tells us that if $p$ is a prime and $a$ is an integer such that $p \nmid a$, then $a^{p-1} \equiv 1 \pmod{p}$. Note that the exponent on $a$ has to go in braces (in the tex file). 

But if you really want an equation (or congruence) to stand out, then you ``display'' the math. It looks like this:
\[
    a^{p-1} \equiv 1 \pmod{p}.
\]
Nice, right?

Finally, we can make things look really nice for homeworks as follows:

\begin{problem}
    Let $X$ and $Y$ be blah blah blah \ldots
\end{problem}

\begin{problem}
    Let $X$ and $Y$ be as in the previous problem. What is $Z$?
\end{problem}

Notice that LaTeX automatically numbers the problems for us. 

\end{document}

It looks nice, but with a lots of unused paper on the sides:

However as soon as I uncomment the line invoking the geometry package the margins get messed up, even though I haven't even issued any command to alter them.

I tried looking at some sources but they do not mention anything like this happening, so I would like to understand why geometry is behaving that way.

Also, if possible, some help getting the code right to leave as little unused paper space as possible.

Answer 1

memoir has its own means of defining the page layout and its best not to use geometry with it. The MWE below increases the size of the textblock by decreasing the margins.

% memgeomprob.tex  SE 562072

\documentclass[12pt, oneside, a4paper]{memoir}

\setlrmarginsandblock{0.8in}{*}{1.0} % set left/right margins to 0.8in
\setulmarginsandblock{1in}{*}{1.0}   % set upper/lower margins to 1in
\checkandfixthelayout

\usepackage{layout}
\usepackage{showframe}

%\usepackage[a4paper]{geometry}
%\usepackage{geometry}

\usepackage{amssymb, amsfonts, amsthm}
\newtheorem{problem}{Problem}

\begin{document}
\title{Sample Document}
\author{John Doe}
\maketitle

\layout{}

Here is some sample text to show you what LaTeX does. 

To start a new paragraph, you need to leave a line of white space in your tex file.

To include math, you have two options. The first is called ``in line,'' and you do this by putting your math between two dollar signs (\$). For example, Fermat's Little Theorem tells us that if $p$ is a prime and $a$ is an integer such that $p \nmid a$, then $a^{p-1} \equiv 1 \pmod{p}$. Note that the exponent on $a$ has to go in braces (in the tex file). 

But if you really want an equation (or congruence) to stand out, then you ``display'' the math. It looks like this:
\[
    a^{p-1} \equiv 1 \pmod{p}.
\]
Nice, right?

Finally, we can make things look really nice for homeworks as follows:

\begin{problem}
    Let $X$ and $Y$ be blah blah blah \ldots
\end{problem}

\begin{problem}
    Let $X$ and $Y$ be as in the previous problem. What is $Z$?
\end{problem}

Notice that LaTeX automatically numbers the problems for us. 

\end{document}

Read the manual (> texdoc memoir) for more information.