# What is the volume of the extension?

the answer is 63.7. My calculations are much bigger than the correct answer, I may have not understood the question correctly quiet possibly.

finding height of the triangle: 1/2*2*3*(sin65)=3 volume of prism: 1/2*5.5*3^2=25 volume: lwh = 3.5*5.5*5.5=106

• You'd need to show your calculation in order for us to spot any possible errors. – Angina Seng Jun 14 '19 at 5:29
• where has the $\sin(65^\circ)$ factor gone in "1/2*2*3*(sin65)=3"? (By the way, don't forget the degree sign) – user10354138 Jun 14 '19 at 5:44

The base of the triangle is given by $$\sqrt{2^2 + 3^2 - (2)(2)(3)\cos(65)}=2.81$$

The area of the triangle is given by $$\frac{1}{2}(2)(3)\sin65=2.718$$.

So the volume of the triangular prism is $$(2.718)(3.5)=9.51$$.

Also, the volume of the rectangular prism is given by $$(2.81)(5.5)(3.5)=54.1$$.

Adding together these volumes gives us $$63.6$$.

Hint: We get $$V_1=\frac{1}{2}3m\cdot 2m\sin(65^{\circ})\cdot 3.5m$$ (the volume of the triangular Prisma) and $$V_2=3.5m\cdot 5.5m\cdot x$$ (the volume of the rectangular Prisma) where $$x^2=(2m)^2+(5m)^2-2\cdot 2m\cdot 5m\cos(65^{\circ})$$ Can you proceed?

Assuming the obvious missing edge is horizontal. Dividing the pentagonal face into a rectangle and the top triangle, the area of the triangle is $$\frac{1}{2}\cdot 2\cdot 3\cdot\sin(65^\circ)=3\sin(65^\circ)\approx 2.72\,\mathrm{m}^2$$ and the length of the missing side (the base) is $$\sqrt{2^2+3^2-2\cdot 2\cdot 3\cdot\cos(65^\circ)}\approx 2.82\,\mathrm{m}$$ so the area of the pentagonal face is $$3\sin(65^\circ)+5.5\times\sqrt{2^2+3^2-2\cdot 2\cdot 3\cdot\cos(65^\circ)}\approx 18.2\,\mathrm{m}^2$$ Hence the volume is $$3.5\times \left(3\sin(65^\circ)+5.5\times\sqrt{2^2+3^2-2\cdot 2\cdot 3\cdot\cos(65^\circ)}\right)\approx 63.7\,\mathrm{m}^3$$