Notes
Area
| Figure | Area |
|---|---|
| Parallelogram | $bh$ |
| Triangle | $\displaystyle \frac{1}{2}bh$ |
| Trapezoid | $\displaystyle \frac{1}{2}(b_1 + b_2)h$ |
| Rhombus or kite | $\displaystyle \frac{1}{2}d_1 d_2$ |
| Regular polygon | $\displaystyle \frac{1}{2}aP$ |
| Circle | $\pi r^2$ |
For composite figures, add pieces or subtract cut-out parts.
Surface area
- Prism: $Ph + 2B$. Cylinder: $2\pi r h + 2\pi r^2$.
- Pyramid: $\displaystyle \frac{1}{2}P\ell + B$. Cone: $\pi r \ell + \pi r^2$. Use the slant height $\ell$.
- Sphere: $4\pi r^2$.
Volume
- Prism and cylinder: $V = Bh$. Pyramid and cone: $\displaystyle V = \frac{1}{3}Bh$. Sphere: $\displaystyle V = \frac{4}{3}\pi r^3$.
- Cavalieri's Principle: same height and same cross-section areas give the same volume, so oblique solids use the same formulas.
- Similar solids with scale factor $k$: surface areas scale by $k^2$, volumes by $k^3$.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle \frac{1}{2}(10)(16) = 80$.
A cone holds one third of the matching cylinder.
$4\pi(5)^2 = 100\pi$.
$\pi(3)^2(5) = 45\pi \approx 141.4$.
$\displaystyle k^2 = \frac{9}{25}$ gives $\displaystyle k = \frac{3}{5}$, so $\displaystyle k^3 = \frac{27}{125}$.
Box: $4 \cdot 4 \cdot 6 = 96$. Pyramid: $\displaystyle \frac{1}{3}(16)(3) = 16$. Total 112.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A grain silo is a cylinder with radius 10 feet and height 30 feet, topped by a hemisphere with radius 10 feet. Round answers to the nearest tenth.
(a) Find the total volume of the silo.
Show the answer and scoring guide
Cylinder: $\pi(10)^2(30) = 3000\pi$. Hemisphere: $\displaystyle \frac{1}{2} \cdot \frac{4}{3}\pi(10)^3 = \frac{2000}{3}\pi$. Total $\displaystyle = \frac{11000}{3}\pi \approx 11519.2$ cubic feet.
Check off the points you earned:
(b) The outside of the silo (side and dome, not the floor) will be painted. Find the area to paint.
Show the answer and scoring guide
Cylinder side: $2\pi(10)(30) = 600\pi$. Dome: $\displaystyle \frac{1}{2} \cdot 4\pi(10)^2 = 200\pi$. Total $= 800\pi \approx 2513.3$ square feet.
Check off the points you earned:
(c) Paint costs $0.35 per square foot. Find the cost to paint the silo, and explain why the floor was left out.
Show the answer and scoring guide
$2513.3 \times 0.35 \approx 879.65$, so about $879.65. The floor sits on the ground and is not part of the outside surface that gets painted.
Check off the points you earned: