Perimeter and Area: Additional Practice (Set 2)

Split the L with a vertical cut into a tall left rectangle and a short right rectangle.

$A_{\text{left}} = 4 \times 10 = 40 \text{ cm}^2, \qquad A_{\text{right}} = 4 \times 4 = 16 \text{ cm}^2$

$A = 40 + 16 = 56 \text{ cm}^2$

As a check, the full $8 \times 10$ rectangle is $80$, minus the missing $4 \times 6 = 24$ corner, leaves $56$.

First find the width from the area.

$w = \frac{72}{12} = 6 \text{ m}$

$P = 2(12 + 6) = 2 \times 18 = 36 \text{ m}$

The height is drawn perpendicular to the base (the right-angle mark), so use it directly.

$A = \frac{1}{2} \times 12 \times 7 = \frac{1}{2} \times 84 = 42 \text{ cm}^2$

The square's area is $6^2 = 36 \text{ cm}^2$, and the rectangle must match it.

$\text{length} = \frac{36}{4} = 9 \text{ cm}$

Both areas are base times perpendicular height, and the base and height match.

$A = b \times h = 10 \times 6 = 60 \text{ cm}^2 \text{ for each}$

A parallelogram is a sheared rectangle of the same base and height, so the areas are equal.

Let the width be $w$; then the length is $2w$, and the perimeter relation is

$2(2w + w) = 48 \;\Rightarrow\; 6w = 48 \;\Rightarrow\; w = 8 \text{ cm}$

So the length is $16$ cm, and

$A = 16 \times 8 = 128 \text{ cm}^2$

Find the side from the area, then use $P = 4s$.

$s = \sqrt{144} = 12 \text{ m}$

$P = 4 \times 12 = 48 \text{ m}$

Find the side from the area, then use $P = 4s$.

$s = \sqrt{169} = 13 \text{ cm}$

$P = 4 \times 13 = 52 \text{ cm}$

Add the rectangle and the triangle.

$A_{\text{rect}} = 6 \times 4 = 24 \text{ cm}^2$

$A_{\text{tri}} = \frac{1}{2} \times 8 \times 3 = 12 \text{ cm}^2$

$A = 24 + 12 = 36 \text{ cm}^2$

With base $b$ and height $b$, the area is $\tfrac{1}{2} b \times b = \tfrac{1}{2} b^2$.

$\frac{1}{2} b^2 = 32 \;\Rightarrow\; b^2 = 64 \;\Rightarrow\; b = 8 \text{ cm}$

Find the area, then the number of $100 \text{ cm}^2$ blocks, then the cost.

$A = 60 \times 40 = 2{,}400 \text{ cm}^2$

$\frac{2{,}400}{100} = 24 \text{ blocks}, \qquad 24 \times 0.50 = 12$

So the glass costs $12$ dollars.

Average the parallel sides, then multiply by the height.

$\frac{6 + 12}{2} = \frac{18}{2} = 9 \text{ cm}$

$A = 9 \times 8 = 72 \text{ cm}^2$