SOLIDS & VOLUME Common 3D Shapes sphere cuboid cylinder cube cone

SPHERES diameter radius A sphere is like a circle in 3D so can have a radius and diameter. So calculations involving spheres use . It can be shown that the volume occupied/filled by a sphere is given by the formula V = 4/3r3

Example 12cm V = 4/3r3 = 4/3 123 = 4  3 x  x 12 ^ 3 r = 12cm = 7238.2cm3

Example 20m V = 4/3r3 = 4/3103 = 4  3 x  x 10 ^ 3 = 4188.8m3 d = 20m so r = 10m

HEMISPHERES A hemisphere is a half sphere. The equator divides the earth into the northern & southern hemispheres. For a sphere we have V = 4/3r3 So for a hemisphere V = 2/3r3

Example An igloo has a diameter of 3m. Find its volume. d = 3m so r = 1.5m V = 2/3r3 = 2  3 x  x 1.5 ^ 3 = 7.069m3

CONES radius It can be shown that the volume of a cone is given by height V =1/3r2h This is the same as Vol = 1/3 X area base X height

Example 2cm A cone for ice-cream is 10cm deep and has a radius of 2cm. If ice-cream is packed inside this how much will it hold? 10cm V =1/3r2h = 3.14 X 2 X 2 X 10  3 = 41.8666… = 41.9cm3 ( to 3 sfs )

Example A traffic cone is 40cm high with a base of diameter 25cm. 40cm Find its volume! NB: d= 25 so r =12.5 25cm V =1/3r2h = 3.14 X 12.5 X 12.5 X 40  3 = 6541.66666… = 6540cm3 ( to 3 sfs )

Example A pencil sharpener consists of a metal cuboid with a conical hole drilled out. The sharpener is 2cm long, 1.5cm wide and 1cm thick The hole is 2cm long with a diameter of 0.8cm. Find the volume of metal in the sharpener. ************* Cuboid Cone Vol metal V = l X b X h d = 0.8 so r = 0.4 = 3 – 0.334… = 2 X 1.5 X 1 V =1/3r2h = 2.665… = 3.14 X 0.4 X 0.4 X 2  3 = 3cm3 = 2.67cm3 = 0.334933….cm3