합성 도형과 도형의 일부

Jigsaw shapes

• A patio is a rectangle with a semicircular flower bed on one end. How much paving do you need?
• Compound shapes 复合图形 are built from simple ones. Split, calculate, then add or subtract.

Strategy: split into known parts

• Break the shape into rectangles, triangles, circles — anything you have a formula for.
• Calculate each area separately, then add them together (or subtract if a piece is cut out).

Split a compound shape into parts you know, here a rectangle plus a semicircle, then add the areas

Parts of solids

• A hemisphere 半球 is half a sphere: volume $= \dfrac{1}{2} \times \dfrac{4}{3}\pi r^3 = \dfrac{2}{3}\pi r^3$.
• A frustum 平截头体 is a cone or pyramid with the top sliced off parallel to the base.
• Find it by subtracting: whole cone minus small top cone.

Worked example — hemisphere

• Hemisphere of radius $6\text{ cm}$:
• Volume $= \dfrac{2}{3}\pi(6)^3 = \dfrac{2}{3}\pi(216) = 144\pi\text{ cm}^3$.

Compound shapes are built from simple ones — here two squares of different sizes, like a rectangle made of two parts.

Why the frustum matters

• The frustum is the shape of most lampshades, buckets, and drinking glasses. Engineers and architects use the formula constantly.

Exposed area at a join

• Two cubes of edge 3 cm join face to face. Their volume adds: $V=2(3^3)=54\text{ cm}^3$. Their area does not simply add: the two touching faces disappear from the outside, so $S=12(3^2)-2(3^2)=90\text{ cm}^2$.
• Half a sphere of radius 6 cm has curved area $2\pi r^2=72\pi\text{ cm}^2$; including the flat circular base gives total $3\pi r^2=108\pi\text{ cm}^2$. State whether the flat base is exposed.

You've got it