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Use Properties of Rectangles, Triangles, and Trapezoids

Understand linear, square, and cubic measure

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 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 ().

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 ().

Suppose the cube in 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

Try it.

For each item, state whether you would use linear, square, or cubic measure:

ⓐ amount of carpeting needed in a room
ⓑ extension cord length
ⓒ amount of sand in a sandbox
ⓓ length of a curtain rod
ⓔ amount of flour in a canister
ⓕ size of the roof of a doghouse.

Solution

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

Condensed — the full section is in OpenStax Prealgebra 2e.

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 .

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

\[\begin{array}{l}P=L+W+L+W \\ \text{or} \\ P=2L+2W\end{array}\]

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 . Since \(A=2⋅3,\) we see that the area, \(A,\) is the length, \(L,\) times the width, \(W,\) so the area of a rectangle is \(A=L⋅W.\)

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

Example

Try it.

Find the length of a rectangle with perimeter \(50\) inches and width \(10\) inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length of the rectangle
Step 3. Name. Choose a variable to represent it.	Let L = the length
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 15 inches.

Condensed — the full section is in OpenStax Prealgebra 2e.

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 , 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 (). 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 \(\text{90^{\circ}}\) angle with the base. shows three triangles with the base and height of each marked.

Example

Try it.

Find the area of a triangle whose base is \(11\) inches and whose height is \(8\) inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the triangle
Step 3. Name. Choose a variable to represent it.	let A = area of the triangle
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area is 44 square inches.

Example

Try it.

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?

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	length of the third side of a triangle
Step 3. Name. Choose a variable to represent it.	Let c = the third side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side is 11 feet long.

Condensed — the full section is in OpenStax Prealgebra 2e.

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 .

The formula for the area of a trapezoid is:

\[{\text{Area}}_{\text{trapezoid}}=\frac{1}{2}h(b+B)\]

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 .

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

The formula for the area of a trapezoid is

If we distribute, we get,

Example

Try it.

Find the area of a trapezoid whose height is 6 inches and whose bases are \(14\) and \(11\) inches.

Solution

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the trapezoid
Step 3. Name. Choose a variable to represent it.	Let \(A=\text{the area}\)
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check: Is this answer reasonable?

If we draw a rectangle around the trapezoid that has the same big base \(B\) and a height \(h,\) its area should be greater than that of the trapezoid.

If we draw a rectangle inside the trapezoid that has the same little base \(b\) and a height \(h,\) its area should be smaller than that of the trapezoid.

The area of the larger rectangle is \(84\) square inches and the area of the smaller rectangle is \(66\) square inches. So it makes sense that the area of the trapezoid is between \(84\) and \(66\) square inches

Step 7. Answer the question. The area of the trapezoid is \(75\) square inches.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Properties of Rectangles
- Rectangles have four sides and four right (90°) angles.
- The lengths of opposite sides are equal.
- The perimeter, \(P\), of a rectangle is the sum of twice the length and twice the width.
  - \(P=2L+2W\)
- The area, \(A\), of a rectangle is the length times the width.
  - \(A=L⋅W\)
Triangle Properties
- For any triangle \(\Delta ABC\), the sum of the measures of the angles is 180°.
  - \(m∠A+m∠B+m∠C=180^{\circ}\)
- The perimeter of a triangle is the sum of the lengths of the sides.
  - \(P=a+b+c\)
- The area of a triangle is one-half the base, b, times the height, h.
  - \(A=\frac{1}{2}bh\)

Use Properties of Rectangles, Triangles, and Trapezoids

Understand Linear, Square, and Cubic Measure

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

Try it.

amount of water in a fish tank

Solution

cubic

Try it.

length of dental floss

Try it.

living area of an apartment

Solution

square

Try it.

floor space of a bathroom tile

Try it.

height of a doorway

Solution

linear

Try it.

capacity of a truck trailer

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

Try it.

Solution

ⓐ 10 cm
ⓑ 4 sq. cm

Try it.

Solution

ⓐ 8 cm
ⓑ 3 sq. cm

Try it.

Solution

ⓐ 10 cm
ⓑ 5 sq. cm

Try it.

Use the Properties of Rectangles

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

Try it.

The length of a rectangle is \(85\) feet and the width is \(45\) feet.

Solution

ⓐ 260 ft
ⓑ 3825 sq. ft

Try it.

The length of a rectangle is \(26\) inches and the width is \(58\) inches.

Try it.

A rectangular room is \(15\) feet wide by \(14\) feet long.

Solution

ⓐ 58 ft
ⓑ 210 sq. ft

Try it.

A driveway is in the shape of a rectangle \(20\) feet wide by \(35\) feet long.

In the following exercises, solve.

Try it.

Find the length of a rectangle with perimeter \(124\) inches and width \(38\) inches.

Solution

24 inches

Try it.

Find the length of a rectangle with perimeter \(20.2\) yards and width of \(7.8\) yards.

Try it.

Find the width of a rectangle with perimeter \(92\) meters and length \(19\) meters.

Solution

27 meters

Try it.

Find the width of a rectangle with perimeter \(16.2\) meters and length \(3.2\) meters.

Try it.

The area of a rectangle is \(414\) square meters. The length is \(18\) meters. What is the width?

Solution

23 m

Try it.

The area of a rectangle is \(782\) square centimeters. The width is \(17\) centimeters. What is the length?

Try it.

The length of a rectangle is \(9\) inches more than the width. The perimeter is \(46\) inches. Find the length and the width.

Solution

7 in., 16 in.

Try it.

The width of a rectangle is \(8\) inches more than the length. The perimeter is \(52\) inches. Find the length and the width.

Try it.

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.

Solution

17 m, 12 m

Try it.

The perimeter of a rectangle is \(62\) feet. The width is \(7\) feet less than the length. Find the length and the width.

Try it.

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.

Solution

13.5 m, 12.8 m

Try it.

The length of the rectangle is \(1.1\) meters less than the width. The perimeter of a rectangle is \(49.4\) meters. Find the dimensions of the rectangle.

Try it.

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.

Solution

25 ft, 50 ft

Try it.

The length of a rectangle is three times the width. The perimeter is \(72\) feet. Find the length and width of the rectangle.

Try it.

The length of a rectangle is \(3\) meters less than twice the width. The perimeter is \(36\) meters. Find the length and width.

Solution

l = 11 m, w = 7 m

Try it.

The length of a rectangle is \(5\) inches more than twice the width. The perimeter is \(34\) inches. Find the length and width.

Try it.

The width of a rectangular window is \(24\) inches. The area is \(624\) square inches. What is the length?

Solution

26 in.

Try it.

The length of a rectangular poster is \(28\) inches. The area is \(1316\) square inches. What is the width?

Try it.

The area of a rectangular roof is \(2310\) square meters. The length is \(42\) meters. What is the width?

Solution

55 m

Try it.

The area of a rectangular tarp is \(132\) square feet. The width is \(12\) feet. What is the length?

Try it.

The perimeter of a rectangular courtyard is \(160\) feet. The length is \(10\) feet more than the width. Find the length and the width.

Solution

35 ft, 45 ft

Try it.

The perimeter of a rectangular painting is \(306\) centimeters. The length is \(17\) centimeters more than the width. Find the length and the width.

Try it.

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.

Solution

76 in., 36 in.

Try it.

The width of a rectangular playground is \(7\) meters less than the length. The perimeter of the playground is \(46\) meters. Find the length and the width.

Use the Properties of Triangles

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

Try it.

Find the area of a triangle with base \(12\) inches and height \(5\) inches.

Solution

30 sq. in.

Try it.

Find the area of a triangle with base \(45\) centimeters and height \(30\) centimeters.

Try it.

Find the area of a triangle with base \(8.3\) meters and height \(6.1\) meters.

Solution

25.315 sq. m

Try it.

Find the area of a triangle with base \(24.2\) feet and height \(20.5\) feet.

Try it.

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

Solution

0.75 sq. ft

Try it.

A triangular window has base of \(8\) feet and height of \(6\) feet. What is its area?

Try it.

If a triangle has sides of \(6\) feet and \(9\) feet and the perimeter is \(23\) feet, how long is the third side?

Solution

8 ft

Try it.

If a triangle has sides of \(14\) centimeters and \(18\) centimeters and the perimeter is \(49\) centimeters, how long is the third side?

Try it.

What is the base of a triangle with an area of \(207\) square inches and height of \(18\) inches?

Solution

23 in.

Try it.

What is the height of a triangle with an area of \(893\) square inches and base of \(38\) inches?

Try it.

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?

Solution

11 yd

Try it.

A triangular courtyard has perimeter of \(120\) meters. The lengths of two sides are \(30\) meters and \(50\) meters. How long is the third side?

Try it.

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

Solution

28 cm

Try it.

An isosceles triangle has a base of \(25\) inches. If the perimeter is \(95\) inches, find the length of each of the other sides.

Try it.

Find the length of each side of an equilateral triangle with a perimeter of \(51\) yards.

Solution

17 yd

Try it.

Find the length of each side of an equilateral triangle with a perimeter of \(54\) meters.

Try it.

The perimeter of an equilateral triangle is \(18\) meters. Find the length of each side.

Solution

6 m

Try it.

The perimeter of an equilateral triangle is \(42\) miles. Find the length of each side.

Try it.

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.

Solution

15 ft

Try it.

The perimeter of an isosceles triangle is \(83\) inches. The length of the shortest side is \(24\) inches. Find the length of the other two sides.

Try it.

A dish is in the shape of an equilateral triangle. Each side is \(8\) inches long. Find the perimeter.

Solution

24 in.

Try it.

A floor tile is in the shape of an equilateral triangle. Each side is \(1.5\) feet long. Find the perimeter.

Try it.

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.

Solution

27.5 in.

Try it.

A scarf in the shape of an isosceles triangle has a base of \(0.75\) meters. If the perimeter is \(2\) meters, find the length of each of the other sides.

Try it.

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.

Solution

12 ft, 13 ft, 14 ft

Try it.

The perimeter of a triangle is \(35\) feet. One side of the triangle is \(5\) feet longer than the second side. The third side is \(3\) feet longer than the second side. Find the length of each side.

Try it.

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.

Solution

3 ft, 6 ft, 8 ft

Try it.

One side of a triangle is three times the smallest side. The third side is \(3\) feet more than the shortest side. The perimeter is \(13\) feet. Find the lengths of all three sides.

Use the Properties of Trapezoids

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

Try it.

The height of a trapezoid is \(12\) feet and the bases are \(9\) and \(15\) feet. What is the area?

Solution

144 sq. ft

Try it.

The height of a trapezoid is \(24\) yards and the bases are \(18\) and \(30\) yards. What is the area?

Try it.

Find the area of a trapezoid with a height of \(51\) meters and bases of \(43\) and \(67\) meters.

Solution

2805 sq. m

Try it.

Find the area of a trapezoid with a height of \(62\) inches and bases of \(58\) and \(75\) inches.

Try it.

The height of a trapezoid is \(15\) centimeters and the bases are \(12.5\) and \(18.3\) centimeters. What is the area?

Solution

231 sq. cm

Try it.

The height of a trapezoid is \(48\) feet and the bases are \(38.6\) and \(60.2\) feet. What is the area?

Try it.

Find the area of a trapezoid with a height of \(4.2\) meters and bases of \(8.1\) and \(5.5\) meters.

Solution

28.56 sq. m

Try it.

Find the area of a trapezoid with a height of \(32.5\) centimeters and bases of \(54.6\) and \(41.4\) centimeters.

Try it.

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?

Solution

13.5 sq. ft

Try it.

Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with width \(5\) feet and lengths \(5\) feet and \(8\) feet. What is the area of the floor?

Try it.

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?

Solution

1036 sq. in.

Try it.

Elena is knitting a scarf. The scarf will be shaped like a trapezoid with width \(8\) inches and lengths \(48.2\) inches and \(56.2\) inches. What is the area of the scarf?

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

The length of a rectangle is \(3\) less than the width. Let \(w\) represent the width. Write an expression for the length of the rectangle.
If you missed this problem, review .

Odhaliť odpoveď

\(w-3\)
Simplify: \(\frac{1}{2}(6h).\)
If you missed this problem, review .

Odhaliť odpoveď

\(3h\)
Simplify: \(\frac{5}{2}(10.3-7.9).\)
If you missed this problem, review .

Odhaliť odpoveď

\(6\)

For each item, state whether you would use linear, square, or cubic measure:

ⓐ amount of carpeting needed in a room
ⓑ extension cord length
ⓓ length of a curtain rod
ⓔ amount of flour in a canister
ⓕ size of the roof of a doghouse.

Odhaliť odpoveď

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

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
Odhaliť odpoveď
1. ⓐ cubic
2. ⓑ linear
3. ⓒ square
4. ⓓ linear
5. ⓔ square
6. ⓕ cubic
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
Odhaliť odpoveď
1. ⓐ cubic
2. ⓑ square
3. ⓒ cubic
4. ⓓ linear
5. ⓔ square
6. ⓕ linear
Each of two square tiles is \(1\) square inch. Two tiles are shown together.
1. ⓐ What is the perimeter of the figure?
2. ⓑ What is the area?
Odhaliť odpoveď

ⓐ The perimeter is the distance around the figure. The perimeter is \(6\) inches.
ⓑ The area is the surface covered by the figure. There are \(2\) square inch tiles so the area is \(2\) square inches.
Each box in the figure below is 1 square inch. Find the ⓐ perimeter and ⓑ area of the figure:
Odhaliť odpoveď
1. ⓐ 8 inches
2. ⓑ 3 sq. inches
Each box in the figure below is 1 square inch. Find the ⓐ perimeter and ⓑ area of the figure:
Odhaliť odpoveď
1. ⓐ 8 centimeters
2. ⓑ 4 sq. centimeters

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

Odhaliť odpoveď

ⓐ
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the perimeter of a rectangle
Step 3. Name. Choose a variable to represent it.	Let P = the perimeter
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The perimeter of the rectangle is 104 meters.

ⓑ
Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of a rectangle
Step 3. Name. Choose a variable to represent it.	Let A = the area
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area of the rectangle is 640 square meters.

The length of a rectangle is \(120\) yards and the width is \(50\) yards. Find ⓐ the perimeter and ⓑ the area.
Odhaliť odpoveď
1. ⓐ 340 yd
2. ⓑ 6000 sq. yd
The length of a rectangle is \(62\) feet and the width is \(48\) feet. Find ⓐ the perimeter and ⓑ the area.
Odhaliť odpoveď
1. ⓐ 220 ft
2. ⓑ 2976 sq. ft

Find the length of a rectangle with perimeter \(50\) inches and width \(10\) inches.

Odhaliť odpoveď

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length of the rectangle
Step 3. Name. Choose a variable to represent it.	Let L = the length
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 15 inches.

Find the length of a rectangle with a perimeter of \(80\) inches and width of \(25\) inches.

Odhaliť odpoveď

15 in.
Find the length of a rectangle with a perimeter of \(30\) yards and width of \(6\) yards.

Odhaliť odpoveď

9 yd

The width of a rectangle is two inches less than the length. The perimeter is \(52\) inches. Find the length and width.

Odhaliť odpoveď

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width of the rectangle
Step 3. Name. Choose a variable to represent it. Now we can draw a figure using these expressions for the length and width.	Since the width is defined in terms of the length, we let L = length. The width is two feet less that the length, so we let L − 2 = width
Step 4.Translate. Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information. Substitute in the given information.
Step 5. Solve the equation.	\(52=2L+2L-4\)
Combine like terms.	\(52=4L-4\)
Add 4 to each side.	\(56=4L\)
Divide by 4.	\(\frac{56}{4}=\frac{4L}{4}\)
	\(14=L\)
	The length is 14 inches.
Now we need to find the width.
The width is L − 2.	The width is 12 inches.
Step 6. Check: Since \(14+12+14+12=52\), this works!
Step 7. Answer the question.	The length is 14 feet and the width is 12 feet.

The width of a rectangle is seven meters less than the length. The perimeter is \(58\) meters. Find the length and width.

Odhaliť odpoveď

18 m, 11 m
The length of a rectangle is eight feet more than the width. The perimeter is \(60\) feet. Find the length and width.

Odhaliť odpoveď

11 ft , 19 ft

The length of a rectangle is four centimeters more than twice the width. The perimeter is \(32\) centimeters. Find the length and width.

Odhaliť odpoveď

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the length and width
Step 3. Name. Choose a variable to represent it.	let W = width The length is four more than twice the width. 2w + 4 = length
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length is 12 cm and the width is 4 cm.

The length of a rectangle is eight more than twice the width. The perimeter is \(64\) feet. Find the length and width.

Odhaliť odpoveď

8 ft, 24 ft
The width of a rectangle is six less than twice the length. The perimeter is \(18\) centimeters. Find the length and width.

Odhaliť odpoveď

5 cm, 4 cm

The area of a rectangular room is \(168\) square feet. The length is \(14\) feet. What is the width?

Odhaliť odpoveď

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the width of a rectangular room
Step 3. Name. Choose a variable to represent it.	Let W = width
Step 4.Translate. Write the appropriate formula and substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The width of the room is 12 feet.

The area of a rectangle is \(598\) square feet. The length is \(23\) feet. What is the width?

Odhaliť odpoveď

26 ft
The width of a rectangle is \(21\) meters. The area is \(609\) square meters. What is the length?

Odhaliť odpoveď

29 m

The perimeter of a rectangular swimming pool is \(150\) feet. The length is \(15\) feet more than the width. Find the length and width.

Odhaliť odpoveď

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the length and width of the pool
Step 3. Name. Choose a variable to represent it. The length is 15 feet more than the width.	Let \(W=\text{width}\) \(W+15=\text{length}\)
Step 4.Translate. Write the appropriate formula and substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The length of the pool is 45 feet and the width is 30 feet.

The perimeter of a rectangular swimming pool is \(200\) feet. The length is \(40\) feet more than the width. Find the length and width.

Odhaliť odpoveď

30 ft, 70 ft
The length of a rectangular garden is \(30\) yards more than the width. The perimeter is \(300\) yards. Find the length and width.

Odhaliť odpoveď

60 yd, 90 yd

Find the area of a triangle whose base is \(11\) inches and whose height is \(8\) inches.

Odhaliť odpoveď

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	the area of the triangle
Step 3. Name. Choose a variable to represent it.	let A = area of the triangle
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The area is 44 square inches.

Find the area of a triangle with base \(13\) inches and height \(2\) inches.

Odhaliť odpoveď

13 sq. in.
Find the area of a triangle with base \(14\) inches and height \(7\) inches.

Odhaliť odpoveď

49 sq. in.

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?

Odhaliť odpoveď

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	length of the third side of a triangle
Step 3. Name. Choose a variable to represent it.	Let c = the third side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side is 11 feet long.

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?

Odhaliť odpoveď

8 ft
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?

Odhaliť odpoveď

6 ft

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?

Odhaliť odpoveď

Step 1. Read the problem. Draw the figure and label it with the given information.
Step 2. Identify what you are looking for.	height of a triangle
Step 3. Name. Choose a variable to represent it.	Let h = the height
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The height of the triangle is 12 meters.

The area of a triangular painting is \(126\) square inches. The base is \(18\) inches. What is the height?

Odhaliť odpoveď

14 in.
A triangular tent door has an area of \(15\) square feet. The height is \(5\) feet. What is the base?

Odhaliť odpoveď

6 ft

The perimeter of an equilateral triangle is \(93\) inches. Find the length of each side.

Odhaliť odpoveď

Step 1. Read the problem. Draw the figure and label it with the given information.	Perimeter = 93 in.
Step 2. Identify what you are looking for.	length of the sides of an equilateral triangle
Step 3. Name. Choose a variable to represent it.	Let s = length of each side
Step 4.Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	Each side is 31 inches.

Find the length of each side of an equilateral triangle with perimeter \(39\) inches.

Odhaliť odpoveď

13 in.
Find the length of each side of an equilateral triangle with perimeter \(51\) centimeters.

Odhaliť odpoveď

17 cm

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?

Odhaliť odpoveď

Step 1. Read the problem. Draw the figure and label it with the given information.	P = 156 in.
Step 2. Identify what you are looking for.	the lengths of the two equal sides
Step 3. Name. Choose a variable to represent it.	Let s = the length of each side
Step 4.Translate. Write the appropriate formula. Substitute in the given information.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	Arianna can make each of the two equal sides 48 inches long.

Symbols used here

^\circ

degrees
1/360 of a full turn. 180° = π radians.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Use Properties of Rectangles, Triangles, and Trapezoids

Understand linear, square, and cubic measure
Use properties of rectangles
Use properties of triangles
Use properties of trapezoids

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Vyskúšajte si vlastné

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Viac v kategórii Arithmetic

Order of operations Fractions Percentages Powers and roots