Use Properties of Rectangles, Triangles, and 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. 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 in square units. We often use square inches, square feet, square centimeters, or square miles. A square centimeter is a square that is one centimeter on each side, and a square inch is a square that is one inch on each side.

Picture a rectangular rug that is $2$ feet long by $3$ feet wide, made of $1$-foot squares. The rug is made of $6$ squares, so its area 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. A cubic centimeter is a cube that measures one centimeter on each side, and a cubic inch is a cube that measures one inch on each side.

Suppose a cube measures $3$ inches on each side and is cut into unit cubes. If we took the big cube apart, we would find $27$ little cubes, each 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. For each item, state whether you would use linear, square, or cubic measure: (a) amount of carpeting needed in a room, (b) extension cord length, (c) amount of sand in a sandbox, (d) length of a curtain rod, (e) amount of flour in a canister, (f) size of the roof of a doghouse.

(a) You are measuring how much surface the carpet covers, which is the area — square measure. (b) You are measuring how long the extension cord is, which is the length — linear measure. (c) You are measuring the volume of the sand — cubic measure. (d) You are measuring the length of the curtain rod — linear measure. (e) You are measuring the volume of the flour — cubic measure. (f) You are measuring the area of the roof — square measure.

Many geometry applications involve finding the perimeter or the area of a figure, and it’s important to understand what each means. 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 are laid. To figure out how many strips are needed, you must know the distance around the room — this distance is the perimeter.

A square tile that is $1$ inch on each side has a perimeter of $4$ inches (if an ant walked around its edge, it would walk $4$ inches) and an area of $1$ square inch.

Example. Each of two square tiles is $1$ square inch, shown together side by side as a $2 \times 1$ rectangle of tiles. (a) What is the perimeter of the figure? (b) What is the area?

(a) The perimeter is the distance around the figure. Walking around the outside edge of the two joined tiles covers $6$ inches, so the perimeter is $6$ inches. (b) The area is the surface covered by the figure. There are $2$ square-inch tiles, so the area is $2$ 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, $W$.

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 $P = L + W + L + W$, or $P = 2L + 2W$.

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. Since $6 = 2 \cdot 3$, 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, here is the Problem Solving Strategy for Geometry Applications restated:

Example. The length of a rectangle is $32$ meters and the width is $20$ meters. Find (a) the perimeter, and (b) the area.

(a) Let $P =$ the perimeter. Substituting into $P = 2L + 2W$:

Checking: $20 + 32 + 20 + 32 = 104$. The perimeter of the rectangle is $104$ meters.

(b) Let $A =$ the area. Substituting into $A = L \cdot W$:

Checking: $32 \cdot 20 = 640$. The area of the rectangle is $640$ square meters.

In the next example, the width is defined in terms of the length, so we wait to draw the figure until we have expressions for both.

Example. The width of a rectangle is two inches less than the length. The perimeter is $52$ inches. Find the length and width.

Let $L =$ length, so the width is $L - 2$. Substituting into $P = 2L + 2W$ with $P = 52$:

The length is $14$ inches. The width is $L - 2 = 12$ inches. Checking: $14 + 12 + 14 + 12 = 52$. The length is $14$ inches and the width is $12$ inches.

Example. The length of a rectangle is four centimeters more than twice the width. The perimeter is $32$ centimeters. Find the length and width.

Let $w =$ width, so $2w + 4 =$ length. Substituting into $P = 2L + 2W$ with $P = 32$:

The width is $4$ cm. The length is $2w + 4 = 2(4) + 4 = 12$ cm. Checking: $P = 2(12) + 2(4) = 32$. The length is $12$ cm and the width is $4$ cm.

Example. The area of a rectangular room is $168$ square feet. The length is $14$ feet. What is the width?

Let $W =$ width. Substituting into $A = LW$: $168 = 14W$. Dividing both sides by $14$ gives $W = 12$. Checking: $14 \cdot 12 = 168$. The width of the room is $12$ feet.

Example. The perimeter of a rectangular swimming pool is $150$ feet. The length is $15$ feet more than the width. Find the length and width.

Let $w =$ width, so $w + 15 =$ length. Substituting into $P = 2L + 2W$ with $P = 150$:

The width of the pool is $30$ feet, and the length is $w + 15 = 45$ feet. Checking: $150 = 2(45) + 2(30)$. The length of the pool is $45$ feet and the width is $30$ feet.

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: label the rectangle’s length $b$ and width $h$, so its area is $bh$. We can divide this rectangle into two congruent triangles (triangles with identical side lengths and angles, so their areas are equal). The area of each triangle is one-half the area of the rectangle, or $\tfrac{1}{2}bh$. This is why the formula for the area of a triangle is $A = \tfrac{1}{2}bh$.

To find the area of a 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 $90°$ angle with the base.

Example. Find the area of a triangle whose base is $11$ inches and whose height is $8$ inches.

Let $A =$ the area. Substituting into $A = \tfrac{1}{2}bh$:

Checking: $44 \overset{?}{=} \tfrac{1}{2}(11)(8)$, and $44 = 44$. The area is $44$ square inches.

Example. The perimeter of a triangular garden is $24$ feet. The lengths of two sides are $4$ feet and $9$ feet. How long is the third side?

Let $c =$ the third side. Substituting into $P = a + b + c$ with $P = 24$, $a = 4$, $b = 9$:

Checking: $4 + 9 + 11 = 24$. The third side is $11$ feet long.

Example. The area of a triangular church window is $90$ square meters. The base of the window is $15$ meters. What is the window’s height?

Let $h =$ the height. Substituting into $A = \tfrac{1}{2}bh$ with $A = 90$, $b = 15$:

Checking: $90 \overset{?}{=} \tfrac{1}{2} \cdot 15 \cdot 12$, and $90 = 90$. The height of the triangle is $12$ meters.

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 — the third side is the base. A triangle that has three sides of equal length is called an equilateral triangle.

Example. The perimeter of an equilateral triangle is $93$ inches. Find the length of each side.

Let $s =$ length of each side. Substituting into $P = a + b + c$, with all three sides equal to $s$:

Checking: $31 + 31 + 31 = 93$. Each side is $31$ inches.

Example. Arianna has $156$ inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of $60$ inches. How long can she make the two equal sides?

Let $s =$ the length of each equal side. Substituting into $P = a + b + c$ with $P = 156$ and base $60$:

Checking: $48 + 60 + 48 = 156$. Arianna can make each of the two equal sides $48$ inches long.

Use the properties of trapezoids

A trapezoid is a 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.

The formula for the area of a trapezoid is:

Splitting the trapezoid into two triangles (each with height $h$, one with base $b$ and one with base $B$) may help you understand the formula: the area of the trapezoid is the sum of the areas of the two triangles, $\tfrac{1}{2}bh + \tfrac{1}{2}Bh$, which factors to $\tfrac{1}{2}h(b + B)$.

Example. Find the area of a trapezoid whose height is $6$ inches and whose bases are $14$ and $11$ inches.

Let $A =$ the area. Substituting into $A = \tfrac{1}{2}h(b + B)$ with $h = 6$, $b = 11$, $B = 14$:

Checking: this should be reasonable — a rectangle with the same big base $B = 14$ and height $6$ has area $84$ square inches, and a rectangle with the same little base $b = 11$ and height $6$ has area $66$ square inches, so the trapezoid’s area should be between $66$ and $84$. Indeed $66 < 75 < 84$. The area of the trapezoid is $75$ square inches.

Example. Find the area of a trapezoid whose height is $5$ feet and whose bases are $10.3$ and $13.7$ feet.

Checking: this is reasonable since it’s less than a rectangle with base $13.7$ and height $5$ ($68.5$ sq ft), and more than a rectangle with base $10.3$ and height $5$ ($51.5$ sq ft). The area of the trapezoid is $60$ square feet.

Example. Vinny has a garden shaped like a trapezoid, with a height of $3.4$ yards and bases of $8.2$ and $5.6$ yards. How many square yards will be available to plant?

Checking: this is reasonable — less than a rectangle with base $8.2$ and height $3.4$ ($27.88$ sq yd), and more than a rectangle with base $5.6$ and height $3.4$ ($19.04$ sq yd). Vinny has $23.46$ square yards in which he can plant.

Key terms

linear measure — a measure of length, in units such as inches, feet, or centimeters. area (square measure) — a measure of the surface covered by a figure, in square units. volume (cubic measure) — a measure of the space filled by a solid, in cubic units. perimeter — the distance around a figure. rectangle — a four-sided figure with four right angles and equal opposite sides. congruent — having identical side lengths and angles. base and height (of a triangle) — the side a triangle sits on, and the perpendicular distance from that side to the opposite vertex. isosceles triangle — a triangle with two sides of equal length. equilateral triangle — a triangle with three sides of equal length. trapezoid — a four-sided figure with exactly two parallel sides (the bases).

This section is adapted from Prealgebra 2e, Section 9.4: Use Properties of Rectangles, Triangles, and Trapezoids by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the trapezoid figure as an accessible inline graphic and described other figures (tape measures, cubes, tile grids) in prose instead of hotlinking images; omitted the Be Prepared quiz, Manipulative Mathematics and Links to Literacy callouts, Media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.