Calculate surface area and volume of cylinders, cones, and spheres, and solve problems involving combinations of solids.
Cone volume=⅓πr²h, curved surface area=πrl (l=slant height). Sphere volume=4/3πr³, surface area=4πr².
Find the volume of a cone with radius 6 cm and height 14 cm (π=22/7).
⅓×22/7×36×14=528 cm³
Find the surface area of a sphere with radius 7 cm (π=22/7).
4×22/7×49=616 cm²
Many problems combine solids — like a cone on top of a cylinder — requiring the volumes or surface areas of each part to be added appropriately.
A solid is a cylinder (radius 5cm, height 10cm) topped with a cone (same radius, height 6cm). Find the total volume (π=3.14).
Cylinder=3.14×25×10=785; cone=⅓×3.14×25×6=157; total=942 cm³
Why is the curved surface area (not total surface area) of the cone used when it's attached to a cylinder?
Because the base of the cone is internal (touching the cylinder), so only the exposed curved surface contributes to the combined solid's total surface
1. Find the volume of a cone with radius 7 cm and height 24 cm (π=22/7).
2. Find the volume of a sphere with radius 3 cm (π=3.14).
3. Find the curved surface area of a cone with radius 3cm and slant height 5cm (π=3.14).
4. A solid hemisphere has radius 6 cm. Find its volume (π=22/7).
5. A cylinder (radius 4cm, height 10cm) has a cone (same radius, height 3cm) removed from the top. Find the remaining volume (π=3.14).