Area

The area of a rectangle is its length times its width. Reading the two sides off the diagram above, $$A = l \times w = 8 \times 5 = \boxed{40}.$$ The area is 40 square units.

A square's area is the side squared, \(A = s^2\). Here the side is 7, so $$A = 7^2 = \boxed{49}.$$ The area is 49 square units.

The area of a rectangle is \(l \times w\), so \(9 \times w = 72\). To find the missing width, divide the area by the length. $$w = \frac{72}{9} = 8$$ The width is \(\boxed{8}\).

A square's area is the side times itself, so to undo that we take the square root of the area. $$s = \sqrt{81} = 9$$ The side is \(\boxed{9}\).

The two legs are the base and the height, so $$A = \frac{1}{2}\times 8 \times 6 = \boxed{24}.$$ In a right triangle the legs already meet at a right angle, so you never have to hunt for the height.

The area of a triangle is \(\frac{1}{2}\times \text{base}\times \text{height}\). The base is \(10\) and the dashed height is \(6\), so $$A = \frac{1}{2}\times 10 \times 6 = \boxed{30}.$$

Half the base times the height. $$A = \frac{1}{2}\times 14 \times 9 = \boxed{63}$$ The half is there because the triangle fills exactly half of the \(14\) by \(9\) rectangle drawn around it.

Split the L into two rectangles and add their areas. The bottom rectangle is \(9\) wide and \(3\) tall, so it holds \(9 \times 3 = 27\). The top rectangle is \(4\) wide and \(3\) tall, so it holds \(4 \times 3 = 12\). Adding them gives $$27 + 12 = \boxed{39}.$$ You could also take the full \(9 \times 6\) box and subtract the \(5 \times 3\) notch, \(54 - 15 = 39\), which lands on the same answer.

The whole rectangle is \(10\times 8 = 80\). The corner sliced off is a right triangle with legs \(6\) and \(4\), so it covers \(\frac{1}{2}\times 6\times 4 = 12\). $$80 - 12 = \boxed{68}$$ When a piece has been cut away, subtracting is usually faster than chopping the leftover shape into parts.

Take the big rectangle and subtract the cut-out piece. The outer rectangle has area \(10\times 8 = 80\), and the hole has area \(6\times 4 = 24\). Subtract to get what remains, $$10\times 8 - 6\times 4 = 80 - 24 = \boxed{56}.$$ The shaded area is \(56\) square units.

The region covers \(12\times 6 = 72\) square units and each tile covers \(2\times 3 = 6\). $$\frac{72}{6} = \boxed{12}$$ Tile counts are area divided by area, and that works here because the \(2\) by \(3\) tiles fit the region with no gaps or overlaps.

Fill in the notch to make a \(12\) by \(9\) rectangle, area \(12\times 9 = 108\). The corner that was missing is \(4\) by \(3\), area \(12\). $$108 - 12 = \boxed{96}$$ Splitting the floor into two rectangles gets there too, \(12\times 6 = 72\) plus \(8\times 3 = 24\).

Split the house at the dashed line and add the two pieces. The rectangle below has area \(10\times 6 = 60\). The roof is a triangle with base \(10\) and height \(4\), so its area is \(\frac{1}{2}\times 10 \times 4 = 20\). Add them for the total, $$60 + 20 = \boxed{80}.$$

The area of a rectangle is its width times its height. Multiply the two lengths marked on the diagram, \(11\times 6 = 66\). So the area is \(\boxed{66}\) square units.

A square's area is \(s^2\), the side multiplied by itself. The side here is 8, so $$8^2 = 8 \times 8 = \boxed{64}.$$ The area is 64 square units.

Divide the area by the length. $$\frac{90}{15} = \boxed{6}$$ The width is whatever the \(15\) gets multiplied by to reach \(90\), so dividing undoes that multiplication.

A square with side \(s\) has area \(s^2\). Here \(s^2 = 100\), so \(s = \sqrt{100}\). Since \(10 \times 10 = 100\), the side is $$s = \sqrt{100} = \boxed{10}.$$

Take half the product of the two legs. $$\frac{1}{2}\times 12 \times 5 = \frac{60}{2} = \boxed{30}$$ The legs are already perpendicular, so one is the base and the other is the height with no extra work.

A triangle's area is half its base times its height. The diagram gives a base of 16 and a height of 5, so $$\frac{1}{2}\times 16 \times 5 = 40.$$ The area is \(\boxed{40}\) square units.

Half the base times the height. $$\frac{1}{2}\times 9 \times 8 = \frac{72}{2} = \boxed{36}$$ Multiplying the two numbers first and halving at the end keeps the arithmetic in whole numbers.

Split the L into two rectangles. The bottom one is \(8\times 4 = 32\), and the narrow part sitting on top of it is \(3\times 3 = 9\). $$32 + 9 = \boxed{41}$$ Subtracting works too. The full \(8\times 7\) box is \(56\) and the notch is \(5\times 3 = 15\), and \(56 - 15 = 41\).

Take the full rectangle and subtract the triangle that was cut away. The rectangle is \(9\times 8 = 72\). The removed corner is a right triangle with legs 6 and 4, so its area is \(\frac{1}{2}\times 6\times 4 = 12\). What is left is $$72 - 12 = \boxed{60}$$

The region is \(10\times 8 = 80\) square units and each tile is \(2\times 2 = 4\). $$\frac{80}{4} = \boxed{20}$$ Dividing area by area gives the tile count whenever the tiles fit with no gaps or overlaps.

The shaded frame is the big rectangle with the hole taken out, so take the big area and subtract the hole. The outer rectangle is \(12\times 9 = 108\), and the hole is \(8\times 5 = 40\). Subtracting gives $$108 - 40 = \boxed{68}.$$

Fill in the notch to make a full rectangle, then subtract the piece you added. The whole rectangle is \(11\times 8 = 88\), and the missing corner is \(3\times 2 = 6\). So the patio is $$88 - 6 = \boxed{82}.$$ The area is 82 square units.

Split the house into the rectangle and the triangular roof, then add the two areas. The rectangle is \(8\times 5 = 40\). The roof has base \(8\) and height \(3\), so its area is \(\frac{1}{2}\times 8\times 3 = 12\). Adding them, $$40 + 12 = \boxed{52}$$ The total area is \(52\) square units.