diff --git a/doc/Makefile b/doc/Makefile
index ef2d60e5f..ddeadbd7e 100644
--- a/doc/Makefile
+++ b/doc/Makefile
@@ -2,6 +2,7 @@
 #
 
 # You can set these variables from the command line.
+BUILDDIR      = build
 SPHINXOPTS    = -W
 SPHINXBUILD   = sphinx-build
 PAPER         =
@@ -9,7 +10,7 @@ PAPER         =
 # Internal variables.
 PAPEROPT_a4     = -D latex_paper_size=a4
 PAPEROPT_letter = -D latex_paper_size=letter
-ALLSPHINXOPTS   = -d build/doctrees $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) source
+ALLSPHINXOPTS   = -d $(BUILDDIR)/doctrees $(PAPEROPT_$(PAPER)) $(SPHINXOPTS) source
 
 .PHONY: help clean html web pickle htmlhelp latex changes linkcheck
 
@@ -24,52 +25,52 @@ help:
 	@echo "  linkcheck to check all external links for integrity"
 
 clean:
-	-rm -rf build/*
+	-rm -rf $(BUILDDIR)/*
 
 html:
-	mkdir -p build/html build/doctrees
-	$(SPHINXBUILD) -b html $(ALLSPHINXOPTS) build/html
+	mkdir -p $(BUILDDIR)/html $(BUILDDIR)/doctrees
+	$(SPHINXBUILD) -b html $(ALLSPHINXOPTS) $(BUILDDIR)/html
 	@echo
-	@echo "Build finished. The HTML pages are in build/html."
+	@echo "Build finished. The HTML pages are in $(BUILDDIR)/html."
 
 pickle:
-	mkdir -p build/pickle build/doctrees
-	$(SPHINXBUILD) -b pickle $(ALLSPHINXOPTS) build/pickle
+	mkdir -p $(BUILDDIR)/pickle $(BUILDDIR)/doctrees
+	$(SPHINXBUILD) -b pickle $(ALLSPHINXOPTS) $(BUILDDIR)/pickle
 	@echo
 	@echo "Build finished; now you can process the pickle files."
 
 web: pickle
 
 json:
-	mkdir -p build/json build/doctrees
-	$(SPHINXBUILD) -b json $(ALLSPHINXOPTS) build/json
+	mkdir -p $(BUILDDIR)/json $(BUILDDIR)/doctrees
+	$(SPHINXBUILD) -b json $(ALLSPHINXOPTS) $(BUILDDIR)/json
 	@echo
 	@echo "Build finished; now you can process the JSON files."
 
 htmlhelp:
-	mkdir -p build/htmlhelp build/doctrees
-	$(SPHINXBUILD) -b htmlhelp $(ALLSPHINXOPTS) build/htmlhelp
+	mkdir -p $(BUILDDIR)/htmlhelp $(BUILDDIR)/doctrees
+	$(SPHINXBUILD) -b htmlhelp $(ALLSPHINXOPTS) $(BUILDDIR)/htmlhelp
 	@echo
 	@echo "Build finished; now you can run HTML Help Workshop with the" \
-	      ".hhp project file in build/htmlhelp."
+	      ".hhp project file in $(BUILDDIR)/htmlhelp."
 
 latex:
-	mkdir -p build/latex build/doctrees
-	$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) build/latex
+	mkdir -p $(BUILDDIR)/latex $(BUILDDIR)/doctrees
+	$(SPHINXBUILD) -b latex $(ALLSPHINXOPTS) $(BUILDDIR)/latex
 	@echo
-	@echo "Build finished; the LaTeX files are in build/latex."
+	@echo "Build finished; the LaTeX files are in $(BUILDDIR)/latex."
 	@echo "Run \`make all-pdf' or \`make all-ps' in that directory to" \
 	      "run these through (pdf)latex."
 
 changes:
-	mkdir -p build/changes build/doctrees
-	$(SPHINXBUILD) -b changes $(ALLSPHINXOPTS) build/changes
+	mkdir -p $(BUILDDIR)/changes $(BUILDDIR)/doctrees
+	$(SPHINXBUILD) -b changes $(ALLSPHINXOPTS) $(BUILDDIR)/changes
 	@echo
-	@echo "The overview file is in build/changes."
+	@echo "The overview file is in $(BUILDDIR)/changes."
 
 linkcheck:
-	mkdir -p build/linkcheck build/doctrees
-	$(SPHINXBUILD) -b linkcheck $(ALLSPHINXOPTS) build/linkcheck
+	mkdir -p $(BUILDDIR)/linkcheck $(BUILDDIR)/doctrees
+	$(SPHINXBUILD) -b linkcheck $(ALLSPHINXOPTS) $(BUILDDIR)/linkcheck
 	@echo
 	@echo "Link check complete; look for any errors in the above output " \
-	      "or in build/linkcheck/output.txt."
+	      "or in $(BUILDDIR)/linkcheck/output.txt."
diff --git a/tox.ini b/tox.ini
index 518577183..47a7a6209 100644
--- a/tox.ini
+++ b/tox.ini
@@ -1,6 +1,6 @@
 [tox]
 requires = tox>=4
-env_list = py{37,38,39,310,311,312}, lint, mypy
+env_list = py{37,38,39,310,311,312}, lint, mypy, html
 
 [testenv]
 description = Run unit tests
@@ -22,11 +22,11 @@ base_python = py39
 commands = mypy -p git
 ignore_outcome = true
 
-# Run "tox -e html" for this. It is deliberately excluded from env_list, as
-# unlike the other environments, this one writes outside the .tox/ directory.
 [testenv:html]
 description = Build HTML documentation
 base_python = py39
 deps = -r doc/requirements.txt
 allowlist_externals = make
-commands = make -C doc html
+commands =
+    make BUILDDIR={env_tmp_dir}/doc/build -C doc clean
+    make BUILDDIR={env_tmp_dir}/doc/build -C doc html