Surface area and volume of prisms and cylinders

A right prism has two identical, parallel end faces joined by rectangles.

$V=\text{area of base}\times \text{height}$

$\text{Surface area}=2\times \text{area of base}+\text{perimeter of base}\times \text{height}$

• Rectangular prism: $V=lbh$, $S=2(lb+lh+bh)$
• Triangular prism: $V=\left(\frac{1}{2}bh\right)\times l$
• Cylinder: $V=\pi r^{2}h$, $S=2\pi r^2+2\pi rh$ (closed)

Example. A box 5 cm by 4 cm by 3 cm: $V=60$ cm³ and $S=2(20+15+12)=94$ cm².

Example. A cylinder with $r=3,5$ cm and $h=10$ cm: $V=\pi (3,5{)}^2(10)\approx 384,85$ cm³.

Units: 1 m³ = 1 000 litres, and 1 cm³ = 1 mℓ.

A composite object is made of two or more simple shapes.

• Volume: add the volumes of the parts (or subtract a hole).
• Surface area: add only the faces on the outside. Faces where the parts join are hidden.

Example. A closed cylindrical water tank with diameter 1,2 m and height 1,5 m stands on the ground:

• Volume: $\pi (0,6{)}^2(1,5)\approx 1,696$ m³, about 1 696 litres
• Outside area to paint (curved surface and lid, but not the base on the ground): $2\pi (0,6)(1,5)+\pi (0,6{)}^2\approx 6,79$ m²
• If the tank were open at the top, you would paint only the curved surface: $2\pi (0,6)(1,5)\approx 5,65$ m²

Read the wording carefully: "open", "no lid" and "standing on the ground" all change which faces count.