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Deciphering Perimeter, Area, and Volume
When a problem involves perimeter, area, or volume, take the formula and fill in what you know. Where do you find the formulas? One place is in this book, of course (they’re sprinkled throughout the chapters). Geometry books and almanacs also have formulas. Or you can do like my neighbors and call me. (Just not during dinnertime, please.)
In general, perimeter is a linear measure of the distance around a figure, area is a square measure of how many squares tile the inside of a figure, and volume is a cubic measure of how many cubes it takes to fill a three-dimensional figure.
Using perimeter formulas to get around
The perimeter of a figure is the distance around its outside. So how do you apply perimeter? You can always just add up the measures of the sides, or you can use perimeter formulas when the amount of fencing you need is for a rectangular yard, or the railing is around a circular track, or the amount of molding is around an octagonal room.
You don’t need to memorize all the perimeter formulas; just realize that there are formulas for the perimeters of standard-shaped objects. Perimeter formulas are helpful for doing the needed computing, and you can alter them to solve for the desired value. Other formulas are available in geometry books, almanacs, and books of math tables.
Here are the perimeter (P) formulas for rectangles, squares, and triangles:
Rectangle: P = 2(l + w), where l is the length and w is the width.
Square: P = 4s, where s represents the length of a side.
Triangle: P = a + b + c, where a, b, and c are the sides.
Q. If you know that the perimeter of a particular rectangle is 20 yards and that the length is 8 yards, then what is the width?
A. 2 yards. You can find a rectangle’s perimeter by using the formula P = 2(l + w) where l and w are the length and width of the rectangle. Substitute what you know into the formula and solve for the unknown. In this case, you know P and l. The formula now reads 20 = 2(8 + w). Divide each side of the equation by 2 to get 10 = 8 + w. Subtract 8 from each side, and you get the width, w, of 2 yards.
Q. An isosceles triangle has a perimeter of 40 yards and two equal sides, each 5 yards longer than the base. How long is the base?
A. The base is 10 yards long. First, you can write the triangle’s perimeter as P = 2s + b. The two equal sides, s, are 5 yards longer than the base, b, which means you can write the lengths of the sides as b + 5. Putting b + 5 in for the s in the formula and putting the 40 in for P, the problem now involves solving the equation 40 = 2(b + 5) + b. Distribute the 2 to get 40 = 2b + 10 + b. Simplify on the right to get 40 = 3b + 10. Subtracting 10 from each side gives you 30 = 3b. Dividing by 3, you get 10 = b. So the base is 10 yards. The two equal sides are then 15 yards each. If you add the two 15-yard sides to the 10-yard base, you get (drum roll, please) 15 + 15 + 10 = 40, the perimeter.
5. If a rectangle has a length that’s 3 inches greater than twice the width, and if the perimeter of the rectangle is 36 inches, then what is its length?
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6. You have 400 feet of fencing to fence in a rectangular yard. If the yard is 30 feet wide and you’re going to use all 400 feet to fence in the yard, then how long is the yard?
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7. A square and an equilateral triangle (all three sides equal in length) have sides that are the same length. If the sum of their perimeters is 84 feet, then what is the perimeter of the square?
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8. A triangle has one side that’s twice as long as the shortest side, and a third side that’s 8 inches longer than the shortest side. Its perimeter is 60 inches. What are the lengths of the three sides?
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Squaring off with area formulas
You measure the area of a figure in square inches, square feet, square yards, and so on. Some of the more commonly found figures, such as rectangles, circles, and triangles, have standard area formulas. Obscure figures even have formulas, but they aren’t used very often, especially in an algebra class. Here are the area formulas for rectangles, squares, circles, and triangles:
Rectangle: A = lw, where l and w represent the length and width
Square: A = s2, where s represents the length of a side
Circle: A = πr2, where r is the radius
Triangle: , where b is the base and h is the height
Q. Find the area of a circle with a circumference of 1,256 feet.
A. 125,600 square feet. You’re told that the distance around the outside (circum-ference) of a circular field is 1,256 feet. The formula for the circumference of a circle is C = πd = 2πr, which says that the circumference is π (about 3.14) times the diameter or two times π times the radius. To find the area of a circle, you need the formula A = πr2. So to find the area of this circular field, you first find the radius by putting the 1,256 feet into the circumference formula: 1,256 = 2πr. Replace the π with 3.14 and solve for r:
The radius is 200 feet. Putting that into the area formula, you get that the area is 125,600 square feet.
Q. A builder is designing a house with a square room. If she increases the sides of the room by 8 feet, the area increases by 224 square feet. What are the dimensions of the expanded room?
A. 18 by 18 feet. You can find the area of a square with A = s2, where s is the length of the sides. Start by letting the original room have sides measuring s feet. Its area is A = s2. The larger room has sides that measure s + 8 feet. Its area is A = (s + 8)2. The difference between these two areas is 224 feet, so subtract the smaller area from the larger and write the equation showing the 224 as a difference: (s + 8)2 – s2 = 224. Simplify the left side of the equation: . Subtract 64 from each side and then divide by 16: . The original room has walls measuring 10 feet. Eight feet more than that is 18 feet.
9. If a rectangle is 4 inches longer than it is wide and the area is 60 square inches, then what are the dimensions of the rectangle?
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10. You can find the area of a trapezoid with . Determine the length of the base b1 if the trapezoid has an area of 30 square yards, a height of 5 yards, and the base b2 of 3 yards.
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11. The perimeter of a square is 40 feet. What is its area? (Remember: P = 4s and A = s2.)
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12. You can find the area of a triangle with , where the base and the height are perpendicular to one another. If a right triangle has legs measuring 10 inches and 24 inches and a hypotenuse of 26 inches, what is its area?
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Working with volume formulas
The volume of an object is a three-dimensional measurement. In a way, you’re asking, “How many little cubes can I fit into this object?” Cubes won’t fit into spheres, pyramids, or other structures with slants and curves, so you have to accept that some of these little cubes are getting shaved off or cut into pieces to fit. Having a formula is much easier than actually sitting and trying to fit all those little cubes into an often large or unwieldy object. Here are the important volume formulas:
Box (rectangular prism): V = lwh
Sphere:
Cylinder: V = πr2h
Q. What are the possible dimensions of a refrigerator that has a capacity of 8 cubic feet?
A. It could be 1 foot deep, 1 foot wide, and 8 feet high, or it could be 2 feet deep, 2 feet wide, and 2 feet high (there are lots of answers). A refrigerator with these suggested dimensions isn’t very efficient — or easy to find. Maybe 1 foot deep by 2 feet wide by 4 feet high would be better. More likely than not, it’s something more like feet.
Q. Find the volume of an orange traffic cone that’s 30 inches tall and has a diameter of 18 inches.
A. A little more than 2,543 cubic inches. A right circular cone (that’s what those traffic cones outlining a construction area look like) has a volume you can find if you know its radius and its height. The formula is . As you can see, the multiplier π is in this formula because the base is a circle. Use 3.14 as an
estimate of π; because the diameter is 18 inches, use 9 inches for the radius. To find this cone’s volume, put those dimensions into the formula to get . The cone’s volume is over 2,500 cubic inches.
13. You can find the volume of a box (right rectangular prism) with V = lwh. Find the height of the box if the volume is 200 cubic feet and the square base is 5 feet on each side (length and width are each 5).
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14. The volume of a sphere (ball) is , where r is the radius of the sphere — the measure from the center to the outside. What is the volume of a sphere with a radius of 6 inches?
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15. You can find the volume of a right circular cylinder (soda pop can) with V = pr2h, where r is the radius, and h is the height of the cylinder — the distance between the two circular bases (the top and bottom of the can). Which has the greater volume: a cylinder with a radius of 6 cm and a height of 9 cm or a cylinder with a radius of 9 cm and a height of 4 cm?
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16. The volume of a cube is 216 cubic centimeters. What is the new volume if you double the length of each side?
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Getting Interested in Using Percent
Percentages are a form of leveling the playing field. They’re great for comparing ratios of numbers that have different bases. For example, if you want to compare the fact that 45 men out of 80 bought a Kindle with the fact that 33 women out of 60 bought a Kindle, you can change both of these to percentages to determine who is more likely to buy a Kindle. (In this case, it’s % men and 55% women.)
To change a ratio or fraction to a percent, divide the part by the whole (numerator by denominator) and multiply by 100. For instance, in the case of the Kindles, you divide 45 by 80 and get .5625. Multiplying that by 100, you get 56.25 which you can write as %.
Percents also show up in interest formulas because you earn interest on an investment or pay interest on a loan based on a percentage of the initial amount. The formula for simple interest is I = Prt, which is translated as “Interest earned is equal to the principal invested times the interest rate (written as a decimal) times time (the number of years).”
Compare the total amount of money earning simple interest with the total you’d have if you invested in an account that compounded interest. Compounding means that the interest is added to the initial amount at certain intervals, and the interest is then figured on the new sum. The formula for compound interest is . The A is the total amount — the principal plus the interest earned. The P is the principal, the r is the interest rate written as a decimal, the n is the number of times each year that compounding occurs, and the t is the number of years.
Q. How much money do you have after 5 years if you invest $1,000 at 4% simple interest?
A. $1,200. Multiplying 1000(0.04)(5), you get that you’ll earn 200 dollars in simple interest. Add that to the amount you started with for a total of $1,200.
Q. How much money will you have if you invest that same $1,000 at 4% for 5 years compounded quarterly (4 times each year)?
A. $1,220.19. Putting the numbers in the formula, you get . Using a calculator, the result comes out to be $1,220.19. True, that’s not all that much more than using simple interest, but the more money you invest, the bigger difference it makes.
17. If 60% of the class has the flu and that 60% is 21 people, then how many are in the class?
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18. How much simple interest will you earn on $4,000 invested at 3% for 10 years? What is the total amount of money at the end of the 10 years?
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19. How much money will be in an account that started with $4,000 and earned 3% compounded quarterly for 10 years?
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20. If you earned $500 in simple interest on an investment that was deposited at 2% interest for 5 years, how much had you invested?
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Answers to Problems on Using Formulas
The following are the answers (in bold) to the practice problems in this chapter.
1. The simple interest formula is I = Prt. Find I if the principal, P, is $10,000, the rate, r, is 2%, and the time, t, is 4 years. The answer is $800.
Inserting the numbers into the formula (and changing 2% to the decimal equivalent 0.02), you get .
2. The simple interest formula is I = Prt. Solve for t so you can find out how long it takes for $10,000 to earn $1,000 interest when the rate is 2%. The answer is 5 years.
Solving for t, you divide each side of the formula by Pr:
Plug the values into the new equation and solve:
3. The formula for the perimeter of a rectangle is P = 2(l + w). Find the perimeter, P, when the length is 7 feet and the width is one yard. The answer is 20 feet.
The length and width are in different units. Changing one yard to 3 feet and substituting into the formula, you get P = 2(7 + 3) = 2(10) = 20.
4. The formula for the area of a trapezoid is . Solve for h so you can determine how high a trapezoid is if the area is 56 square centimeters and the two bases are 6 cm. and 8 cm. The answer is 8 cm.
To solve for h, multiply each side of the formula by 2 and the divide by the sum in the parentheses:
Now use the new formula to solve for h:
5. If a rectangle has a length that’s 3 inches greater than twice the width, and if the perimeter of the rectangle is 36 inches, then what is its length? The length is 13 inches. Use this figure to help you solve this problem.
Let w = the width of the rectangle, which makes the length, . A rectangle’s perimeter is . Substituting in 3 + 2w for the l in this formula and replacing P with 36, you get . Simplifying, you get . Now divide each side of the equation by 2 to get . Subtract 3 from each side: 15 = 3w.Divide each side by 3: 5 = w. The length is . The rectangle is 5 inches by 13 inches.
6. You have 400 feet of fencing to fence in a rectangular yard. If the yard is 30 feet wide and you’re going to use all 400 feet, then how long is the yard? 170 feet.
You know that the total distance around the yard (its perimeter) is 400 and the width of the yard is 30, so plug those values into the formula P = 2 (l + w) to get 400 = 2(l + 30). Divide by 2 to get and then subtract 30 from each side to get . So l = 170 feet.
7. A square and an equilateral triangle (all three sides equal in length) have sides that are the same length. If the sum of their perimeters is 84 feet, then what is the perimeter of the square? 48 feet.
The perimeter of the square is 4l, and the perimeter of the equilateral triangle is 3l. Adding these together, you get . Dividing each side of 7l = 84 by 7 gives you l = 12 feet. So the perimeter of the square is .
8. A triangle has one side that’s twice as long as the shortest side and a third side that’s 8 inches longer than the shortest side. Its perimeter is 60 inches. What are the lengths of the three sides? 13 inches, 26 inches, 21 inches.
This problem doesn’t need a special perimeter formula. The perimeter of a triangle is just the sum of the measures of the three sides. Letting the shortest side be x, twice that is 2x, and 8 more than that is x + 8. Add the three measures together to get x + 2x + x + 8 = 60, which works out to be 4x + 8 = 60, or 4x = 52. Dividing by 4 to x = 13. Twice that is 26, and eight more than that is 21.
9. If a rectangle is 4 inches longer than it is wide and the area is 60 square inches, what are the dimensions of the rectangle? 6 inches by 10 inches.
Using the area formula for a rectangle, A = lw, and letting l = w + 4, you get the equation . Simplifying, setting the equation equal to 0, and then factoring gives you . The two solutions of the equation are –10 and 6. Only the +6 makes sense, so the width is 6, and the length is 4 more than that or 10.
10. You can find the area of a trapezoid with . Determine the length of the base b1 if the trapezoid has an area of 30 square yards, a height of 5 yards, and the base b2 of 3 yards. 9 yards.
Plug in the values and then multiply each side by two: . Then divide each side by 5 to get . Do the math to find that 12 – 3 = b1, so b1 = 9 yards
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Algebra I Workbook For Dummies Page 16