9.4 Use Properties of Rectangles, Triangles, and Trapezoids

Learning objectives

Learning objectives

• Understand linear, square, and cubic measure

• Use properties of rectangles

• Use properties of triangles

• Use properties of trapezoids

In this section, we’ll continue working with geometry applications. We will add some more properties of triangles, and we’ll learn about the properties of rectangles and trapezoids.

Understand Linear, Square, and Cubic Measure

When you measure your height or the length of a garden hose, you use a ruler or tape measure (Figure 9.13). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimeter and meter are units of linear measure.

When you want to know how much tile is needed to cover a floor, or the size of a wall to be painted, you need to know the area, a measure of the region needed to cover a surface. Area is measured is square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm) on each side. A square inch is a square that is one inch on each side (Figure 9.14).

Figure 9.15 shows a rectangular rug that is $2$ feet long by $3$ feet wide. Each square is $1$ foot wide by $1$ foot long, or $1$ square foot. The rug is made of $6$ squares. The area of the rug is $6$ square feet.

When you measure how much it takes to fill a container, such as the amount of gasoline that can fit in a tank, or the amount of medicine in a syringe, you are measuring volume. Volume is measured in cubic units such as cubic inches or cubic centimeters. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimeters, cubic inches, and cubic feet. A cubic centimeter is a cube that measures one centimeter on each side, while a cubic inch is a cube that measures one inch on each side (Figure 9.16).

Suppose the cube in Figure 9.17 measures $3$ inches on each side and is cut on the lines shown. How many little cubes does it contain? If we were to take the big cube apart, we would find $27$ little cubes, with each one measuring one inch on all sides. So each little cube has a volume of $1$ cubic inch, and the volume of the big cube is $27$ cubic inches.

Example 9.25

Solution

1. Classify the carpeting measurement. Carpeting covers a floor surface, so the amount needed is measured by area and uses square measure.
2. Classify the extension cord. The cord is measured along its length, so its length uses linear measure.
3. Classify the sand. Sand occupies space in the sandbox, so the amount of sand uses volume and therefore cubic measure.
4. Classify the curtain rod. The rod is measured from one end to the other, so its length uses linear measure.
5. Classify the flour. Flour occupies the space inside the canister, so its amount uses volume and therefore cubic measure.
6. Classify the roof. A roof is a surface that must be covered, so its size uses area and therefore square measure.

Takeaway: Length uses linear measure, surface uses square measure, and space occupied uses cubic measure, so carpet and roof area are square while sand and flour volume are cubic.

Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean.

Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor.

Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter.

Figure 9.18 shows a square tile that is $1$ inch on each side. If an ant walked around the edge of the tile, it would walk $4$ inches. This distance is the perimeter of the tile.

Since the tile is a square that is $1$ inch on each side, its area is one square inch. The area of a shape is measured by determining how many square units cover the shape.

Example 9.26

Solution

1. Trace the exposed edges. The two square tiles form a rectangle that is $2$ inches wide and $1$ inch tall, with perimeter $2(2)+2(1)=6$ inches.
2. State the perimeter. The distance around the combined figure is $6$ inches, since all four outer edges must be included.
3. Count the square units. Each tile covers $1$ square inch, and the figure contains two such tiles, so its area is $1+1=2$ square inches.
4. State the area. The area of the two-tile figure is $2$ square inches, distinct from its $6$-inch perimeter.

Takeaway: Perimeter measures the distance around a figure, while area counts the square inches it covers; this figure has perimeter $6$ inches and area $2$ square inches.

Use the Properties of Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, $L,$ and the adjacent side as the width, $W.$ See Figure 9.19.

The perimeter, $P,$ of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk $L+W+L+W$ units, or two lengths and two widths. The perimeter then is

What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was $2$ feet long by $3$ feet wide, and its area was $6$ square feet. See Figure 9.20. Since $A=2\cdot{}3,$ we see that the area, $A,$ is the length, $L,$ times the width, $W,$ so the area of a rectangle is $A=L\cdot{}W.$

For easy reference as we work the examples in this section, we will restate the Problem Solving Strategy for Geometry Applications here.

Example 9.27

Solution

1. Calculate the perimeter. The formula is $P=2L+2W$. Substituting $L=32$ meters and $W=20$ meters gives $P=2(32\\text{ m})+2(20\\text{ m})=64\\text{ m}+40\\text{ m}=104\\text{ m}$. The first figures show the rectangle and this calculation.
2. Calculate the area. The formula is $A=LW$. Substituting the same dimensions gives $A=(32\\text{ m})(20\\text{ m})=640\\text{ m}^2$.
3. Check both results. The figures verify the perimeter $20+32+20+32=104$ meters and the area $32\\times20=640$ square meters.
4. State the answers. ⓐ The perimeter is $104$ meters, and ⓑ the area is $640$ square meters.

Takeaway: Perimeter is measured in units such as meters, while area is measured in square units such as square meters.

Example 9.28

Solution

1. Use the perimeter formula. The rectangle has perimeter $50$ inches and width $10$ inches, so $P=2L+2W$ becomes $50\\text{ in}=2L+2(10\\text{ in})$.
2. Simplify the equation. Subtracting $20$ inches from both sides gives $30\\text{ in}=2L$. The figures display this algebraic step.
3. Solve for the length. Dividing both sides by $2$ gives $L=15$ inches.
4. Check the perimeter. Adding the four sides gives $15+10+15+10=50$ inches, which the final figure confirms. The length is $15$ inches.

Takeaway: When solving for a rectangle's length, substitute the known width into $P=2L+2W$ before isolating $L$.

In the next example, the width is defined in terms of the length. We’ll wait to draw the figure until we write an expression for the width so that we can label one side with that expression.

Example 9.29

Solution

1. Express the dimensions. Let $L$ be the length in inches. The width is two inches less, so $W=L-2$ inches, as the first figure labels.
2. Apply the perimeter formula. Using $P=2L+2W$, substitute the perimeter and width expression to get $52\\text{ in}=2L+2(L-2\\text{ in})$.
3. Solve for the length. Combining terms gives $52=2L+2L-4$, so $56=4L$ and $L=14$ inches. The figure shows the width becoming $14-2=12$ inches.
4. State and check the dimensions. The length is $14$ inches and the width is $12$ inches; they give perimeter $2(14)+2(12)=52$ inches.

Takeaway: Translate a dimension that is a fixed amount less than another before substituting it into a geometry formula.

Example 9.30

Solution

1. Represent the dimensions. Let $w$ be the width in centimeters. The length is $2w+4$ centimeters, which is shown in the rectangle diagram.
2. Apply the perimeter formula. Using $P=2L+2W$, substitute to obtain $32\\text{ cm}=2(2w+4\\text{ cm})+2w$.
3. Solve for the width. Distributing and combining terms gives $32=4w+8+2w$, then $24=6w$, so $w=4$ centimeters. The length is $2(4)+4=12$ centimeters, as the figure shows.
4. Check the perimeter. Substituting gives $2(12)+2(4)=32$ centimeters. The width is $4$ centimeters and the length is $12$ centimeters.

Takeaway: Distribute the factor $2$ across an algebraic side length before solving the perimeter equation.

Example 9.31

Solution

1. Use the area formula. The formula is $A=LW$, so the known dimensions give $168\\text{ ft}^2=(14\\text{ ft})W$.
2. Divide to solve for the width. Dividing both sides by $14$ feet gives $W=\\frac{168\\text{ ft}^2}{14\\text{ ft}}=12\\text{ ft}$, as the figures show.
3. Check the area. Multiplying the length and width gives $14\\text{ ft}\\times12\\text{ ft}=168\\text{ ft}^2$, which the final figure confirms.
4. State the width. The room is $12$ feet wide.

Takeaway: Dividing area by length leaves width, and the square units cancel to leave a length unit.

Example 9.32

Solution

1. Express the dimensions. Let $w$ be the width in feet. The length is $w+15$ feet, as the figure labels.
2. Apply the perimeter formula. Using $P=2L+2W$, substitute to get $150\\text{ ft}=2(w+15\\text{ ft})+2w$.
3. Solve for the width. Combining terms gives $150=4w+30$, so $120=4w$ and $w=30$ feet. The length is $w+15=30+15=45$ feet.
4. Check the perimeter. The figures verify $2(45)+2(30)=150$ feet. The pool is $30$ feet wide and $45$ feet long.

Takeaway: After solving a related-dimension equation, substitute the basic variable back to find the dimension defined in terms of it.

Use the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in Figure 9.20, we’ve labeled the length $b$ and the width $h,$ so it’s area is $bh.$

We can divide this rectangle into two congruent triangles (Figure 9.22). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or $\frac{1}{2}bh.$ This example helps us see why the formula for the area of a triangle is $A=\frac{1}{2}bh.$

The formula for the area of a triangle is $A=\frac{1}{2}bh,$ where $b$ is the base and $h$ is the height.

To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a $\mathrm{90^{\circ}}$ angle with the base. Figure 9.23 shows three triangles with the base and height of each marked.

Example 9.33

Solution

1. State the area formula. For a triangle, $A=\\frac{1}{2}bh$, where $b$ is the base and $h$ is the perpendicular height.
2. Substitute the measurements. With $b=11$ inches and $h=8$ inches, the formula becomes $A=\\frac{1}{2}(11\\text{ in})(8\\text{ in})$, as the figures show.
3. Calculate the area. Multiplying gives $A=44\\text{ in}^2$.
4. Check the result. The final figure verifies $\\frac{1}{2}(11)(8)=44$ square inches. The area is $44$ square inches.

Takeaway: Triangle area uses one-half of base times perpendicular height and therefore has square units.

Example 9.34

Solution

1. Use the perimeter formula. The formula is $P=a+b+c$. Let $c$ be the third side in feet, so $24\\text{ ft}=4\\text{ ft}+9\\text{ ft}+c$.
2. Solve for the missing side. Combining the known sides gives $24\\text{ ft}=13\\text{ ft}+c$, hence $c=11\\text{ ft}$. The figures display this calculation.
3. Check the perimeter. Adding all three sides gives $4+9+11=24$ feet, which the final figure confirms.
4. State the side length. The third side of the garden is $11$ feet long.

Takeaway: To find a missing side of a polygon, subtract the sum of the known sides from the perimeter.

Example 9.35

Solution

1. Use the area formula. For a triangle, $A=\\frac{1}{2}bh$. Substituting $A=90\\text{ m}^2$ and $b=15\\text{ m}$ gives $90\\text{ m}^2=\\frac{1}{2}(15\\text{ m})h$.
2. Solve for the height. Dividing both sides by $\\frac{15}{2}$ meters gives $h=12$ meters, as the calculation figures show.
3. Check the area. The final figure verifies $\\frac{1}{2}(15\\text{ m})(12\\text{ m})=90\\text{ m}^2$.
4. State the height. The window's height is $12$ meters.

Takeaway: When solving for height, divide the triangle's area by one-half of its base and retain the correct square and length units.

Isosceles and Equilateral Triangles

Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an isosceles triangle. A triangle that has three sides of equal length is called an equilateral triangle. Figure 9.24 shows both types of triangles.

Example 9.36

Solution

1. Use equal side lengths. Let $s$ be the length of each side in inches. Since all three sides are equal, $P=a+b+c$ becomes $93\\text{ in}=s+s+s$.
2. Simplify and solve. Combining the terms gives $93=3s$, so $s=31$ inches. The solution figure shows this division.
3. Check the perimeter. Adding the three equal sides gives $31+31+31=93$ inches, which the final figure confirms.
4. State the side length. Each side is $31$ inches long.

Takeaway: Divide the perimeter of an equilateral triangle by three to find its side length.

Example 9.37

Solution

1. Represent the equal sides. Let each of the two congruent sides have length $s$ inches. The base is $60$ inches, as the triangle diagram shows.
2. Use the perimeter formula. Since $P=a+b+c$, substitute the measurements to get $156\\text{ in}=s+60\\text{ in}+s$.
3. Solve for each equal side. Combining terms gives $156=2s+60$, so $96=2s$ and $s=48$ inches. The figures display these steps.
4. Check the perimeter. The final figure verifies $48+60+48=156$ inches. Each equal side can be $48$ inches long.

Takeaway: For an isosceles triangle, subtract the base from the perimeter and divide the remainder by two.

Use the Properties of Trapezoids

A trapezoid is four-sided figure, a quadrilateral, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base $b,$ and the length of the bigger base $B.$ The height, $h,$ of a trapezoid is the distance between the two bases as shown in Figure 9.25.

The formula for the area of a trapezoid is:

Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. See Figure 9.26.

The height of the trapezoid is also the height of each of the two triangles. See Figure 9.27.

The formula for the area of a trapezoid is

If we distribute, we get,

Example 9.38

Solution

1. State the trapezoid area formula. Use $A=\frac{1}{2}h(b+B)$, where $h$ is the perpendicular height and $b$ and $B$ are the two bases.
2. Substitute the measurements. With $h=6$, $b=11$, and $B=14$, the formula becomes $A=\frac{1}{2}(6)(11+14)$.
3. Calculate the area. The bases sum to $25$, so $A=\frac{1}{2}(6)(25)=3(25)=75$ square inches, as the calculation figure shows.
4. Check against bounding rectangles. The same-height rectangles with bases $11$ and $14$ have areas $66$ and $84$ square inches. Since $75$ lies between them, the trapezoid area is reasonable.
5. State the area. The area of the trapezoid is $75$ square inches.

Takeaway: A trapezoid's area is the average of its two bases times its height, so bases $14$ and $11$ with height $6$ give $75$ square inches.

Example 9.39

Solution

1. State the area formula. For a trapezoid, $A=\\frac{1}{2}h(b+B)$, where $b$ and $B$ are the two bases.
2. Substitute the measurements. With $h=5$ feet, $b=10.3$ feet, and $B=13.7$ feet, the formula becomes $A=\\frac{1}{2}(5\\text{ ft})(10.3\\text{ ft}+13.7\\text{ ft})$.
3. Calculate the area. The bases sum to $24$ feet, so $A=\\frac{1}{2}(5)(24)=60\\text{ ft}^2$. The calculation figures show this simplification.
4. Check with bounding rectangles. The final figure shows areas $68.5$, $60$, and $51.5$ square feet, confirming that the trapezoid's area lies between the smaller and larger rectangles. The area is $60$ square feet.

Takeaway: The trapezoid area is the average of its bases multiplied by its perpendicular height.

Example 9.40

Solution

1. State the area formula. For a trapezoid, $A=\\frac{1}{2}h(b+B)$, where $h$ is the height and $b$ and $B$ are the bases.
2. Substitute the measurements. With $h=3.4$ yards, $b=5.6$ yards, and $B=8.2$ yards, the formula becomes $A=\\frac{1}{2}(3.4\\text{ yd})(5.6\\text{ yd}+8.2\\text{ yd})$.
3. Calculate the area. The bases sum to $13.8$ yards, so $A=\\frac{1}{2}(3.4)(13.8)=23.46\\text{ yd}^2$. The calculation figure shows this result.
4. Check with bounding rectangles. The comparison figure gives rectangle areas $27.88$ and $19.04$ square yards, and $23.46$ lies between them. Vinny has $23.46$ square yards available to plant.

Takeaway: A trapezoid's area is always between the areas of the same-height rectangles built on its shorter and longer bases.

Section 9.4 Exercises

Section 9.4 Exercises

Understand Linear, Square, and Cubic Measure

In the following exercises, determine whether you would measure each item using linear, square, or cubic units.

In the following exercises, find the ⓐ perimeter and ⓑ area of each figure. Assume each side of the square is $1$ cm.

Use the Properties of Rectangles

In the following exercises, find the ⓐ perimeter and ⓑ area of each rectangle.

In the following exercises, solve.

Use the Properties of Triangles

In the following exercises, solve using the properties of triangles.

Use the Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

1. Determine whether you would use linear, square, or cubic measure for each item.

ⓐ amount of paint in a can ⓑ height of a tree ⓒ floor of your bedroom ⓓ diameter of bike wheel ⓔ size of a piece of sod ⓕ amount of water in a swimming pool

2. Determine whether you would use linear, square, or cubic measure for each item.

ⓐ volume of a packing box ⓑ size of patio ⓒ amount of medicine in a syringe ⓓ length of a piece of yarn ⓔ size of housing lot ⓕ height of a flagpole

3. Each box in the figure below is 1 square inch. Find the ⓐ perimeter and ⓑ area of the figure:

4. Each box in the figure below is 1 square inch. Find the ⓐ perimeter and ⓑ area of the figure:

5. The length of a rectangle is $120$ yards and the width is $50$ yards. Find ⓐ the perimeter and ⓑ the area.

1. The length of a rectangle is $62$ feet and the width is $48$ feet. Find ⓐ the perimeter and ⓑ the area.

2. Find the length of a rectangle with a perimeter of $80$ inches and width of $25$ inches.

3. Find the length of a rectangle with a perimeter of $30$ yards and width of $6$ yards.

4. The width of a rectangle is seven meters less than the length. The perimeter is $58$ meters. Find the length and width.

5. The length of a rectangle is eight feet more than the width. The perimeter is $60$ feet. Find the length and width.

1. The length of a rectangle is eight more than twice the width. The perimeter is $64$ feet. Find the length and width.

2. The width of a rectangle is six less than twice the length. The perimeter is $18$ centimeters. Find the length and width.

3. The area of a rectangle is $598$ square feet. The length is $23$ feet. What is the width?

4. The perimeter of a rectangular swimming pool is $200$ feet. The length is $40$ feet more than the width. Find the length and width.

5. The length of a rectangular garden is $30$ yards more than the width. The perimeter is $300$ yards. Find the length and width.

1. Find the area of a triangle with base $13$ inches and height $2$ inches.

2. Find the area of a triangle with base $14$ inches and height $7$ inches.

3. The perimeter of a triangular garden is $48$ feet. The lengths of two sides are $18$ feet and $22$ feet. How long is the third side?

4. The lengths of two sides of a triangular window are $7$ feet and $5$ feet. The perimeter is $18$ feet. How long is the third side?

5. The area of a triangular painting is $126$ square inches. The base is $18$ inches. What is the height?

1. A triangular tent door has an area of $15$ square feet. The height is $5$ feet. What is the base?

2. Find the length of each side of an equilateral triangle with perimeter $39$ inches.

3. Find the length of each side of an equilateral triangle with perimeter $51$ centimeters.

4. A backyard deck is in the shape of an isosceles triangle with a base of $20$ feet. The perimeter of the deck is $48$ feet. How long is each of the equal sides of the deck?

5. A boat’s sail is an isosceles triangle with base of $8$ meters. The perimeter is $22$ meters. How long is each of the equal sides of the sail?

1. The height of a trapezoid is $14$ yards and the bases are $7$ and $16$ yards. What is the area?

2. The height of a trapezoid is $18$ centimeters and the bases are $17$ and $8$ centimeters. What is the area?

3. The height of a trapezoid is $7$ centimeters and the bases are $4.6$ and $7.4$ centimeters. What is the area?

4. The height of a trapezoid is $9$ meters and the bases are $6.2$ and $7.8$ meters. What is the area?

5. Lin wants to sod his lawn, which is shaped like a trapezoid. The bases are $10.8$ yards and $6.7$ yards, and the height is $4.6$ yards. How many square yards of sod does he need?

1. Kira wants cover his patio with concrete pavers. If the patio is shaped like a trapezoid whose bases are $18$ feet and $14$ feet and whose height is $15$ feet, how many square feet of pavers will he need?

2. amount of water in a fish tank

3. living area of an apartment

4. height of a doorway

5. The length of a rectangle is $85$ feet and the width is $45$ feet.

1. A rectangular room is $15$ feet wide by $14$ feet long.

2. Find the length of a rectangle with perimeter $124$ inches and width $38$ inches.

3. Find the width of a rectangle with perimeter $92$ meters and length $19$ meters.

4. The area of a rectangle is $414$ square meters. The length is $18$ meters. What is the width?

5. The length of a rectangle is $9$ inches more than the width. The perimeter is $46$ inches. Find the length and the width.

1. The perimeter of a rectangle is $58$ meters. The width of the rectangle is $5$ meters less than the length. Find the length and the width of the rectangle.

2. The width of the rectangle is $0.7$ meters less than the length. The perimeter of a rectangle is $52.6$ meters. Find the dimensions of the rectangle.

3. The perimeter of a rectangle of $150$ feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.

4. The length of a rectangle is $3$ meters less than twice the width. The perimeter is $36$ meters. Find the length and width.

5. The width of a rectangular window is $24$ inches. The area is $624$ square inches. What is the length?

1. The area of a rectangular roof is $2310$ square meters. The length is $42$ meters. What is the width?

2. The perimeter of a rectangular courtyard is $160$ feet. The length is $10$ feet more than the width. Find the length and the width.

3. The width of a rectangular window is $40$ inches less than the height. The perimeter of the doorway is $224$ inches. Find the length and the width.

4. Find the area of a triangle with base $12$ inches and height $5$ inches.

5. Find the area of a triangle with base $8.3$ meters and height $6.1$ meters.

1. A triangular flag has base of $1$ foot and height of $1.5$ feet. What is its area?

2. If a triangle has sides of $6$ feet and $9$ feet and the perimeter is $23$ feet, how long is the third side?

3. What is the base of a triangle with an area of $207$ square inches and height of $18$ inches?

4. The perimeter of a triangular reflecting pool is $36$ yards. The lengths of two sides are $10$ yards and $15$ yards. How long is the third side?

5. An isosceles triangle has a base of $20$ centimeters. If the perimeter is $76$ centimeters, find the length of each of the other sides.

1. Find the length of each side of an equilateral triangle with a perimeter of $51$ yards.

2. The perimeter of an equilateral triangle is $18$ meters. Find the length of each side.

3. The perimeter of an isosceles triangle is $42$ feet. The length of the shortest side is $12$ feet. Find the length of the other two sides.

4. A dish is in the shape of an equilateral triangle. Each side is $8$ inches long. Find the perimeter.

5. A road sign in the shape of an isosceles triangle has a base of $36$ inches. If the perimeter is $91$ inches, find the length of each of the other sides.

1. The perimeter of a triangle is $39$ feet. One side of the triangle is $1$ foot longer than the second side. The third side is $2$ feet longer than the second side. Find the length of each side.

2. One side of a triangle is twice the smallest side. The third side is $5$ feet more than the shortest side. The perimeter is $17$ feet. Find the lengths of all three sides.

3. The height of a trapezoid is $12$ feet and the bases are $9$ and $15$ feet. What is the area?

4. Find the area of a trapezoid with a height of $51$ meters and bases of $43$ and $67$ meters.

5. The height of a trapezoid is $15$ centimeters and the bases are $12.5$ and $18.3$ centimeters. What is the area?

1. Find the area of a trapezoid with a height of $4.2$ meters and bases of $8.1$ and $5.5$ meters.

2. Laurel is making a banner shaped like a trapezoid. The height of the banner is $3$ feet and the bases are $4$ and $5$ feet. What is the area of the banner?

3. Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width $18.5$ inches and lengths $62$ and $50$ inches. What is the area of the counter?

4. Fence Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a $50$ foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be $10$ feet. How long can he make the other side if he wants to use the entire roll of fence?

5. Fence Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are $6$ feet, $8$ feet, and $10$ feet. The fence costs $\$10$ per foot. How much will it cost for Christa to fence in her flowerbed?

1. If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning.

2. Look at the two figures.

ⓐ Which figure looks like it has the larger area? Which looks like it has the larger perimeter?

ⓑ Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter?