Volume of a hip roof.

Level 2

A hip roof has a rectangular base and four sides that have the same pitch.

Which is the correct volume formula in terms of l , w , l, w, and h ? h?

2 w l h 3 \frac{2wlh}3 h ( 2 l + w ) 3 \frac{h(2l+w)}3 w h ( 3 l − w ) 6 \frac{wh(3l-w)}6 ( h + l ) ( h + w ) ( h + l − w ) 6 \frac{(h+l)(h+w)(h+l-w)}6

Jeremy Galvagni by Jeremy Galvagni , 48, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Jeremy Galvagni
Mar 15, 2018

The shape can be thought of as two half-pyramids stuck onto either end of a triangular prism. The whole pyramid has a square base of side w and height h so its volume is w 2 h 3 \frac{w^{2}h}{3} . The prism has rectangular sides w and (l-w) with height h so its volume is w ( l − w ) h 2 \frac{w(l-w)h}{2} . Add these together and simplify:

w 2 h 3 + w ( l − w ) h 2 \frac{w^{2}h}{3}+\frac{w(l-w)h}{2}

= w 2 h 3 + w l h 2 − w 2 h 2 =\frac{w^{2}h}{3}+\frac{wlh}{2}-\frac{w^{2}h}{2}

= 3 l w h 6 − w 2 h 6 =\frac{3lwh}{6}-\frac{w^{2}h}{6}

V = w h ( 3 l − w ) 6 \boxed{V=\frac{wh(3l-w)}{6 }}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...