Geometry: Area, Volume & the Coordinate Plane

Step 1: One cut makes two rectangles: $6$ m by $4$ m and $3$ m by $2$ m.

Step 2: Big piece: $6 \times 4 = 24\text{ m}^2\text{.}$

Step 3: Small piece: $3 \times 2 = 6\text{ m}^2\text{.}$

Step 4: Add: $24 + 6 = \mathbf{30}\text{ m}^2\text{.}$

Step 1: Draw the rectangle around it: $8 \times 5 = 40\text{ cm}^2\text{.}$

Step 2: The triangle is half of that rectangle.

Step 3: $A = \tfrac{1}{2} \times 8 \times 5 = \mathbf{20}\text{ cm}^2\text{.}$

Step 1: Add the bases: $5 + 9 = 14\text{.}$

Step 2: Multiply by $4\text{:}$ $14 \times 4 = 56\text{.}$

Step 3: That's the doubled shape, so halve it: $\mathbf{28}\text{ m}^2\text{.}$

Step 1: $A$ to $B$ is flat, so compare $x$ values: $-3$ and $4\text{.}$

Step 2: They sit on opposite sides of $0\text{,}$ so add: $|{-3}| + |4| = 7\text{.}$

Step 3: $B$ to $C$ goes straight down: $|2| + |{-2}| = 4\text{.}$

Step 4: Area: $7 \times 4 = \mathbf{28}$ square units.

Step 1: Floor layer: $5 \times 3 = 15$ cubes.

Step 2: Stack $4$ layers: $15 \times 4 = 60$ cubes.

Step 3: $V = \ell w h = \mathbf{60}\text{ in}^3\text{.}$

Step 1: Top and bottom are $4$ by $3\text{:}$ $2 \times 12 = 24\text{.}$

Step 2: Front and back are $4$ by $2\text{:}$ $2 \times 8 = 16\text{.}$

Step 3: Left and right are $3$ by $2\text{:}$ $2 \times 6 = 12\text{.}$

Step 4: Add: $24 + 16 + 12 = \mathbf{52}$ square units.