Spheres and composite solids

Surface area 4πr² and volume (4/3)πr³ of spheres and hemispheres, and composite solids made of cylinders, cones and hemispheres.

Spheres

Sphere

$$SA = 4\pi r^2 \qquad V = \frac{4}{3}\pi r^3$$

Sphere surface area and volume

At r = 3 they are equal (both 36π). For larger spheres the volume grows faster.

Example 1●●

Find the volume of a sphere with radius $6$ cm.

$V = \tfrac{4}{3}\pi(216) = 288\pi \approx 904.78$ cm³

Example 2●●●

Find the total surface area of a solid hemisphere with radius $2$.

Curved part: $\tfrac{1}{2} \times 4\pi(4) = 8\pi$. Flat circle: $4\pi$.

Total $= 12\pi \approx 37.70$

Flashcards

Sphere surface area

$4\pi r^2$

Sphere volume

$\tfrac{4}{3}\pi r^3$

Solid hemisphere SA

$3\pi r^2$

Practice

Volume of a sphere, radius $6$ (2 d.p.)?
Total surface area of a solid hemisphere, radius $2$ (2 d.p.)?
Surface area of a sphere, radius $3$ (2 d.p.)?
A sphere has volume $36\pi$. Find its radius.
Volume of a ball with diameter $10$ cm (2 d.p.)?
A sphere has surface area $100\pi$. Radius?
Doubling a sphere's radius multiplies its volume by?
An ice-cream cone ($r = 3$, $h = 4$) is topped by a hemisphere of radius $3$. Exact total volume? Type like 30pi.