diff --git a/src/bootstrap/src/core/build_steps/dist.rs b/src/bootstrap/src/core/build_steps/dist.rs
index 6d56492e41b56..3ee9ef6aba60b 100644
--- a/src/bootstrap/src/core/build_steps/dist.rs
+++ b/src/bootstrap/src/core/build_steps/dist.rs
@@ -398,10 +398,11 @@ impl Step for Rustc {
             let host = compiler.host;
             let src = builder.sysroot(compiler);
 
-            // Copy rustc/rustdoc binaries
+            // Copy rustc binary
+            builder.install(&src.join("bin").join(exe("rustc", host)), &image.join("bin"), 0o755);
             t!(fs::create_dir_all(image.join("bin")));
-            builder.cp_r(&src.join("bin"), &image.join("bin"));
 
+            // If enabled, copy rustdoc binary
             if builder
                 .config
                 .tools
diff --git a/src/bootstrap/src/core/build_steps/install.rs b/src/bootstrap/src/core/build_steps/install.rs
index 1c565e7f7cc06..0225f8f24a5ef 100644
--- a/src/bootstrap/src/core/build_steps/install.rs
+++ b/src/bootstrap/src/core/build_steps/install.rs
@@ -95,7 +95,7 @@ fn install_sh(
     }
 
     let datadir = prefix.join(default_path(&builder.config.datadir, "share"));
-    let docdir = prefix.join(default_path(&builder.config.docdir, "share/doc/rust"));
+    let docdir = prefix.join(default_path(&builder.config.docdir, &format!("share/doc/{package}")));
     let mandir = prefix.join(default_path(&builder.config.mandir, "share/man"));
     let libdir = prefix.join(default_path(&builder.config.libdir, "lib"));
     let bindir = prefix.join(&builder.config.bindir); // Default in config.rs