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Title: geometry
Description: In order to solve geometric word problems, you will need to have memorized some geometric formulas for at least the basic shapes (circles, squares, right triangles, etc). You will usually need to figure out from the word problem which formula to use, and many times you will need more than one formula for one exercise. So make sure you have memorized any formulas that are used in the homework, because you may be expected to know them on the test. Some problems are just straightforward applications of basic geometric formulae. The radius of a circle is 3 centimeters. What is the circle's circumference? The formula for the circumference C of a circle with radius r is: C = 2(pi)r ...where "pi" (above) is of course the number approximately equal to 22 / 7 or 3.14159. They gave me the value of r and asked me for the value of C, so I'll just "plug-n-chug": C = 2(pi)(3) = 6pi Then, after re-checking the original exercise for the required units (so my answer will be complete): the circumference is 6pi cm. Note: Unless you are told to use one of the approximations for pi, or are told to round to some number of decimal places (from having used the "pi" button on your calculator), you are generally supposed to keep your answer in "exact" form, as shown above. If you're not sure if you should use the "pi" form or the decimal form, use both: "6pi cm, or about 18.85 cm". A square has an area of sixteen square centimeters. What is the length of each of its sides? The formula for the area A of a square with side- length s is: A = s2 They gave me the area, so I'll plug this value into the area formula, and see where this leads: 16 = s2 4 = s After re-reading the exercise to find the correct units, my answer is: The length of each side is 4 centimeters... Most geometry word problems are a bit more involved than the example above. For most exercises, you will be given at least two pieces of information, such as a statement about a square's perimeter and then a question about its area. To find the solution, you will need to know the equations related to the various pieces of information; you will then probably solve one of the equations for a useful bit of new information, and then plug the result into another of the equations. In other words, geometry word problems often aren't simple one-step exercises like the one shown above. But if you take all the information that you've been given, write down any applicable formulas, try to find ways to relate the various pieces, and see where this leads, then you'll almost always end up with a valid answer. A cube has a surface area of fifty-four square centimeters. What is the volume of the cube? The formula for the volume V of a cube with edge- length e is: V = e3 To find the volume, I need the edge-length. Can I use the surface-area information to get what I need? Let's see... A cube has six sides, each of which is a square; and the edges of the cube's faces are the sides of those squares. The formula for the area of a square with side-length e is A = e2. There are six faces so there are six squares, and the cube's total surface area SA must be: SA = 6e2 Plugging in the value they gave me, I get: 54 = 6e2 54 / 6 = (6e2) / 6 9 = e2 3 = e Since the volume is the cube of the edge-length, and since the units on this cube are centimeters, then: the volume is 27 cubic centimeters, or 27 cc, or 27 mL A circle has an area of 49pi square units. What is the length of the circle's diameter? The formula for the area A of a circle with radius r is: A = (pi)r2 I know that radius r is half of the length of the diameter d, so: 49pi = (pi)r2 (49pi) / pi = [(pi)r2] / pi 49 = r2 7 = r Then the radius r has a length of 7 units, and: the length of the diameter is 14 units Warning: You can not assume that you will always be given all the geometric formulae on your tests. At some point, you will need to know at least some of them "by heart". The basic formulae you should know include the formulae for the area and perimeter (or circumference) of squares, rectangles, triangles, and circles; and the surface areas and volumes of cubes, rectangular solids, spheres, and cylinders. Depending on the class, you may also need to know the formulae for cones and pyramids. If you're not sure what your instructor expects, ask now. Sometimes geometry word problems wrap the geometry in a thick layer of "real life". You will need to be able to "see" the geometry, and extract the relevant information. Suppose a water tank in the shape of a right circular cylinder is thirty feet long and eight feet in diameter. How much sheet metal was used in its construction? What they are asking for here is the surface area of the water tank. The total surface area of the tank will be the sum of the surface areas of the side (the cylindrical part) and of the ends. If the diameter is eight feet, then the radius is four feet. The surface area of each end is given by the area formula for a circle with radius r: A = (pi)r2. (There are two end pieces, so I will be multiplying this area by 2 when I find my total-surface-area formula.) The surface area of the cylinder is the circumference of the circle, multiplied by the height: A = 2(pi)rh. Side view of the cylindrical tank, showing the radius "r". An "exploded" view of the tank, showing the three separate surfaces whose areas I need to find. Then the total surface area of this tank is given by: 2 ×( (pi)r2 ) + 2(pi)rh (the two ends, plus the cylinder) = 2( (pi) (42) ) + 2(pi) (4)(30) = 2( (pi) × 16 ) + 240(pi) = 32(pi) + 240(pi) = 272(pi) Since the original dimensions were given in terms of feet, then my area must be in terms of square feet: the surface area is 272(pi) square feet. By the way, this is one of those exercises that doesn't translate well into "real life". In reality, the sheet metal has thickness, and adjustments would be required in order to account for this thickness. For example, if you needed to figure out the amount of sheet metal required to create a tank with a certain volume, you'd have to account for the fact that the volume is on the inside, while the surface area is on the outside. Since the walls of a real-world tank have thickness, the real-world answer would not match the "ideal" mathematical one. A piece of 16-gauge copper wire 42 cm long is bent into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle. ADVERTISEMENT Ads by Google Ad covers the page Stop seeing this ad Do I care that the wire is made of copper, or that the wire is a length of sixteen-gauge? No; all I care is that the length is forty-two units, that the units are centimeters, that the rectangle is twice as long in one direction as the other, and that I'm supposed to find the values of each of these directions. I can ignore the other information. Since the wire is 42 centimeters long, then the perimeter of the rectangle is 42 centimeters. That is: 2L + 2W = 42 I also know that the width is twice the length, so: W = 2L Then: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved 2L + 2(2L) = 42 (by substitution for W from the above equation) 2L + 4L = 42 6L = 42 L = 7 Since the width is related to the length by W = 2L, then W = 14, and the rectangle is 7 centimeters long and 14 centimeters wide. A circular swimming pool with a diameter of 28 feet has a deck of uniform width built around it. If the area of the deck is 60(pi) square feet, find its width. I have this situation: A pool is surrounded by a deck. The pool has radius 14, and the deck has width "d ". If the diameter of the pool is 28, then the radius is 14. The area of the pool is then: (pi)r2 = (pi)(14)2 = 196(pi) Then the total area of the pool plus the surrounding decking is: 196(pi) + 60(pi) = 256(pi) Working backwards from the area formula, I can find the radius of the whole pool-plus-deck area: 256(pi) = (pi)r2 256 = r2 16 = r Since I already know that the pool has a radius of 14 feet, and I now know that the whole area has a radius of 16, then clearly: the deck is two feet wide. If one side of a square is doubled in length and the adjacent side is decreased by two centimeters, the area of the resulting rectangle is 96 square centimeters larger than that of the original square. Find the dimensions of the rectangle. I'm starting from a square with sides of some unknown length. The sides of the rectangle are defined in terms of that unknown length. So I'll pick a variable for the unknown side-length, create expressions for the rectangle's sides, and then work from there. square's side length: x one side is doubled: 2x next side is decreased by two: x – 2 square's area: x2 rectangle's area: (2x)(x – 2) = 2x2 – 4x new area is 96 more than old area: 2x2 – 4x = x2 + 96 2x2 – 4x = x2 + 96 x2 – 4x – 96 = 0 (x – 12)(x + 8) = 0 x = 12 or x = –8 I'm supposed to find the dimensions of a rectangle, so can I just erase that one "minus" sign and say that the rectangle is 12 by 8? No! I defined "x" as standing for the side length of the square, not as one of the sides of the rectangle. Looking back at my definitions, I see what "x = 12" (the only reasonable solution for the square) means that the sides of the rectangle have lengths 2(12) and (12) – 2:
Description: In order to solve geometric word problems, you will need to have memorized some geometric formulas for at least the basic shapes (circles, squares, right triangles, etc). You will usually need to figure out from the word problem which formula to use, and many times you will need more than one formula for one exercise. So make sure you have memorized any formulas that are used in the homework, because you may be expected to know them on the test. Some problems are just straightforward applications of basic geometric formulae. The radius of a circle is 3 centimeters. What is the circle's circumference? The formula for the circumference C of a circle with radius r is: C = 2(pi)r ...where "pi" (above) is of course the number approximately equal to 22 / 7 or 3.14159. They gave me the value of r and asked me for the value of C, so I'll just "plug-n-chug": C = 2(pi)(3) = 6pi Then, after re-checking the original exercise for the required units (so my answer will be complete): the circumference is 6pi cm. Note: Unless you are told to use one of the approximations for pi, or are told to round to some number of decimal places (from having used the "pi" button on your calculator), you are generally supposed to keep your answer in "exact" form, as shown above. If you're not sure if you should use the "pi" form or the decimal form, use both: "6pi cm, or about 18.85 cm". A square has an area of sixteen square centimeters. What is the length of each of its sides? The formula for the area A of a square with side- length s is: A = s2 They gave me the area, so I'll plug this value into the area formula, and see where this leads: 16 = s2 4 = s After re-reading the exercise to find the correct units, my answer is: The length of each side is 4 centimeters... Most geometry word problems are a bit more involved than the example above. For most exercises, you will be given at least two pieces of information, such as a statement about a square's perimeter and then a question about its area. To find the solution, you will need to know the equations related to the various pieces of information; you will then probably solve one of the equations for a useful bit of new information, and then plug the result into another of the equations. In other words, geometry word problems often aren't simple one-step exercises like the one shown above. But if you take all the information that you've been given, write down any applicable formulas, try to find ways to relate the various pieces, and see where this leads, then you'll almost always end up with a valid answer. A cube has a surface area of fifty-four square centimeters. What is the volume of the cube? The formula for the volume V of a cube with edge- length e is: V = e3 To find the volume, I need the edge-length. Can I use the surface-area information to get what I need? Let's see... A cube has six sides, each of which is a square; and the edges of the cube's faces are the sides of those squares. The formula for the area of a square with side-length e is A = e2. There are six faces so there are six squares, and the cube's total surface area SA must be: SA = 6e2 Plugging in the value they gave me, I get: 54 = 6e2 54 / 6 = (6e2) / 6 9 = e2 3 = e Since the volume is the cube of the edge-length, and since the units on this cube are centimeters, then: the volume is 27 cubic centimeters, or 27 cc, or 27 mL A circle has an area of 49pi square units. What is the length of the circle's diameter? The formula for the area A of a circle with radius r is: A = (pi)r2 I know that radius r is half of the length of the diameter d, so: 49pi = (pi)r2 (49pi) / pi = [(pi)r2] / pi 49 = r2 7 = r Then the radius r has a length of 7 units, and: the length of the diameter is 14 units Warning: You can not assume that you will always be given all the geometric formulae on your tests. At some point, you will need to know at least some of them "by heart". The basic formulae you should know include the formulae for the area and perimeter (or circumference) of squares, rectangles, triangles, and circles; and the surface areas and volumes of cubes, rectangular solids, spheres, and cylinders. Depending on the class, you may also need to know the formulae for cones and pyramids. If you're not sure what your instructor expects, ask now. Sometimes geometry word problems wrap the geometry in a thick layer of "real life". You will need to be able to "see" the geometry, and extract the relevant information. Suppose a water tank in the shape of a right circular cylinder is thirty feet long and eight feet in diameter. How much sheet metal was used in its construction? What they are asking for here is the surface area of the water tank. The total surface area of the tank will be the sum of the surface areas of the side (the cylindrical part) and of the ends. If the diameter is eight feet, then the radius is four feet. The surface area of each end is given by the area formula for a circle with radius r: A = (pi)r2. (There are two end pieces, so I will be multiplying this area by 2 when I find my total-surface-area formula.) The surface area of the cylinder is the circumference of the circle, multiplied by the height: A = 2(pi)rh. Side view of the cylindrical tank, showing the radius "r". An "exploded" view of the tank, showing the three separate surfaces whose areas I need to find. Then the total surface area of this tank is given by: 2 ×( (pi)r2 ) + 2(pi)rh (the two ends, plus the cylinder) = 2( (pi) (42) ) + 2(pi) (4)(30) = 2( (pi) × 16 ) + 240(pi) = 32(pi) + 240(pi) = 272(pi) Since the original dimensions were given in terms of feet, then my area must be in terms of square feet: the surface area is 272(pi) square feet. By the way, this is one of those exercises that doesn't translate well into "real life". In reality, the sheet metal has thickness, and adjustments would be required in order to account for this thickness. For example, if you needed to figure out the amount of sheet metal required to create a tank with a certain volume, you'd have to account for the fact that the volume is on the inside, while the surface area is on the outside. Since the walls of a real-world tank have thickness, the real-world answer would not match the "ideal" mathematical one. A piece of 16-gauge copper wire 42 cm long is bent into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle. ADVERTISEMENT Ads by Google Ad covers the page Stop seeing this ad Do I care that the wire is made of copper, or that the wire is a length of sixteen-gauge? No; all I care is that the length is forty-two units, that the units are centimeters, that the rectangle is twice as long in one direction as the other, and that I'm supposed to find the values of each of these directions. I can ignore the other information. Since the wire is 42 centimeters long, then the perimeter of the rectangle is 42 centimeters. That is: 2L + 2W = 42 I also know that the width is twice the length, so: W = 2L Then: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved 2L + 2(2L) = 42 (by substitution for W from the above equation) 2L + 4L = 42 6L = 42 L = 7 Since the width is related to the length by W = 2L, then W = 14, and the rectangle is 7 centimeters long and 14 centimeters wide. A circular swimming pool with a diameter of 28 feet has a deck of uniform width built around it. If the area of the deck is 60(pi) square feet, find its width. I have this situation: A pool is surrounded by a deck. The pool has radius 14, and the deck has width "d ". If the diameter of the pool is 28, then the radius is 14. The area of the pool is then: (pi)r2 = (pi)(14)2 = 196(pi) Then the total area of the pool plus the surrounding decking is: 196(pi) + 60(pi) = 256(pi) Working backwards from the area formula, I can find the radius of the whole pool-plus-deck area: 256(pi) = (pi)r2 256 = r2 16 = r Since I already know that the pool has a radius of 14 feet, and I now know that the whole area has a radius of 16, then clearly: the deck is two feet wide. If one side of a square is doubled in length and the adjacent side is decreased by two centimeters, the area of the resulting rectangle is 96 square centimeters larger than that of the original square. Find the dimensions of the rectangle. I'm starting from a square with sides of some unknown length. The sides of the rectangle are defined in terms of that unknown length. So I'll pick a variable for the unknown side-length, create expressions for the rectangle's sides, and then work from there. square's side length: x one side is doubled: 2x next side is decreased by two: x – 2 square's area: x2 rectangle's area: (2x)(x – 2) = 2x2 – 4x new area is 96 more than old area: 2x2 – 4x = x2 + 96 2x2 – 4x = x2 + 96 x2 – 4x – 96 = 0 (x – 12)(x + 8) = 0 x = 12 or x = –8 I'm supposed to find the dimensions of a rectangle, so can I just erase that one "minus" sign and say that the rectangle is 12 by 8? No! I defined "x" as standing for the side length of the square, not as one of the sides of the rectangle. Looking back at my definitions, I see what "x = 12" (the only reasonable solution for the square) means that the sides of the rectangle have lengths 2(12) and (12) – 2:
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Title: geometry
Description: In order to solve geometric word problems, you will need to have memorized some geometric formulas for at least the basic shapes (circles, squares, right triangles, etc). You will usually need to figure out from the word problem which formula to use, and many times you will need more than one formula for one exercise. So make sure you have memorized any formulas that are used in the homework, because you may be expected to know them on the test. Some problems are just straightforward applications of basic geometric formulae. The radius of a circle is 3 centimeters. What is the circle's circumference? The formula for the circumference C of a circle with radius r is: C = 2(pi)r ...where "pi" (above) is of course the number approximately equal to 22 / 7 or 3.14159. They gave me the value of r and asked me for the value of C, so I'll just "plug-n-chug": C = 2(pi)(3) = 6pi Then, after re-checking the original exercise for the required units (so my answer will be complete): the circumference is 6pi cm. Note: Unless you are told to use one of the approximations for pi, or are told to round to some number of decimal places (from having used the "pi" button on your calculator), you are generally supposed to keep your answer in "exact" form, as shown above. If you're not sure if you should use the "pi" form or the decimal form, use both: "6pi cm, or about 18.85 cm". A square has an area of sixteen square centimeters. What is the length of each of its sides? The formula for the area A of a square with side- length s is: A = s2 They gave me the area, so I'll plug this value into the area formula, and see where this leads: 16 = s2 4 = s After re-reading the exercise to find the correct units, my answer is: The length of each side is 4 centimeters... Most geometry word problems are a bit more involved than the example above. For most exercises, you will be given at least two pieces of information, such as a statement about a square's perimeter and then a question about its area. To find the solution, you will need to know the equations related to the various pieces of information; you will then probably solve one of the equations for a useful bit of new information, and then plug the result into another of the equations. In other words, geometry word problems often aren't simple one-step exercises like the one shown above. But if you take all the information that you've been given, write down any applicable formulas, try to find ways to relate the various pieces, and see where this leads, then you'll almost always end up with a valid answer. A cube has a surface area of fifty-four square centimeters. What is the volume of the cube? The formula for the volume V of a cube with edge- length e is: V = e3 To find the volume, I need the edge-length. Can I use the surface-area information to get what I need? Let's see... A cube has six sides, each of which is a square; and the edges of the cube's faces are the sides of those squares. The formula for the area of a square with side-length e is A = e2. There are six faces so there are six squares, and the cube's total surface area SA must be: SA = 6e2 Plugging in the value they gave me, I get: 54 = 6e2 54 / 6 = (6e2) / 6 9 = e2 3 = e Since the volume is the cube of the edge-length, and since the units on this cube are centimeters, then: the volume is 27 cubic centimeters, or 27 cc, or 27 mL A circle has an area of 49pi square units. What is the length of the circle's diameter? The formula for the area A of a circle with radius r is: A = (pi)r2 I know that radius r is half of the length of the diameter d, so: 49pi = (pi)r2 (49pi) / pi = [(pi)r2] / pi 49 = r2 7 = r Then the radius r has a length of 7 units, and: the length of the diameter is 14 units Warning: You can not assume that you will always be given all the geometric formulae on your tests. At some point, you will need to know at least some of them "by heart". The basic formulae you should know include the formulae for the area and perimeter (or circumference) of squares, rectangles, triangles, and circles; and the surface areas and volumes of cubes, rectangular solids, spheres, and cylinders. Depending on the class, you may also need to know the formulae for cones and pyramids. If you're not sure what your instructor expects, ask now. Sometimes geometry word problems wrap the geometry in a thick layer of "real life". You will need to be able to "see" the geometry, and extract the relevant information. Suppose a water tank in the shape of a right circular cylinder is thirty feet long and eight feet in diameter. How much sheet metal was used in its construction? What they are asking for here is the surface area of the water tank. The total surface area of the tank will be the sum of the surface areas of the side (the cylindrical part) and of the ends. If the diameter is eight feet, then the radius is four feet. The surface area of each end is given by the area formula for a circle with radius r: A = (pi)r2. (There are two end pieces, so I will be multiplying this area by 2 when I find my total-surface-area formula.) The surface area of the cylinder is the circumference of the circle, multiplied by the height: A = 2(pi)rh. Side view of the cylindrical tank, showing the radius "r". An "exploded" view of the tank, showing the three separate surfaces whose areas I need to find. Then the total surface area of this tank is given by: 2 ×( (pi)r2 ) + 2(pi)rh (the two ends, plus the cylinder) = 2( (pi) (42) ) + 2(pi) (4)(30) = 2( (pi) × 16 ) + 240(pi) = 32(pi) + 240(pi) = 272(pi) Since the original dimensions were given in terms of feet, then my area must be in terms of square feet: the surface area is 272(pi) square feet. By the way, this is one of those exercises that doesn't translate well into "real life". In reality, the sheet metal has thickness, and adjustments would be required in order to account for this thickness. For example, if you needed to figure out the amount of sheet metal required to create a tank with a certain volume, you'd have to account for the fact that the volume is on the inside, while the surface area is on the outside. Since the walls of a real-world tank have thickness, the real-world answer would not match the "ideal" mathematical one. A piece of 16-gauge copper wire 42 cm long is bent into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle. ADVERTISEMENT Ads by Google Ad covers the page Stop seeing this ad Do I care that the wire is made of copper, or that the wire is a length of sixteen-gauge? No; all I care is that the length is forty-two units, that the units are centimeters, that the rectangle is twice as long in one direction as the other, and that I'm supposed to find the values of each of these directions. I can ignore the other information. Since the wire is 42 centimeters long, then the perimeter of the rectangle is 42 centimeters. That is: 2L + 2W = 42 I also know that the width is twice the length, so: W = 2L Then: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved 2L + 2(2L) = 42 (by substitution for W from the above equation) 2L + 4L = 42 6L = 42 L = 7 Since the width is related to the length by W = 2L, then W = 14, and the rectangle is 7 centimeters long and 14 centimeters wide. A circular swimming pool with a diameter of 28 feet has a deck of uniform width built around it. If the area of the deck is 60(pi) square feet, find its width. I have this situation: A pool is surrounded by a deck. The pool has radius 14, and the deck has width "d ". If the diameter of the pool is 28, then the radius is 14. The area of the pool is then: (pi)r2 = (pi)(14)2 = 196(pi) Then the total area of the pool plus the surrounding decking is: 196(pi) + 60(pi) = 256(pi) Working backwards from the area formula, I can find the radius of the whole pool-plus-deck area: 256(pi) = (pi)r2 256 = r2 16 = r Since I already know that the pool has a radius of 14 feet, and I now know that the whole area has a radius of 16, then clearly: the deck is two feet wide. If one side of a square is doubled in length and the adjacent side is decreased by two centimeters, the area of the resulting rectangle is 96 square centimeters larger than that of the original square. Find the dimensions of the rectangle. I'm starting from a square with sides of some unknown length. The sides of the rectangle are defined in terms of that unknown length. So I'll pick a variable for the unknown side-length, create expressions for the rectangle's sides, and then work from there. square's side length: x one side is doubled: 2x next side is decreased by two: x – 2 square's area: x2 rectangle's area: (2x)(x – 2) = 2x2 – 4x new area is 96 more than old area: 2x2 – 4x = x2 + 96 2x2 – 4x = x2 + 96 x2 – 4x – 96 = 0 (x – 12)(x + 8) = 0 x = 12 or x = –8 I'm supposed to find the dimensions of a rectangle, so can I just erase that one "minus" sign and say that the rectangle is 12 by 8? No! I defined "x" as standing for the side length of the square, not as one of the sides of the rectangle. Looking back at my definitions, I see what "x = 12" (the only reasonable solution for the square) means that the sides of the rectangle have lengths 2(12) and (12) – 2:
Description: In order to solve geometric word problems, you will need to have memorized some geometric formulas for at least the basic shapes (circles, squares, right triangles, etc). You will usually need to figure out from the word problem which formula to use, and many times you will need more than one formula for one exercise. So make sure you have memorized any formulas that are used in the homework, because you may be expected to know them on the test. Some problems are just straightforward applications of basic geometric formulae. The radius of a circle is 3 centimeters. What is the circle's circumference? The formula for the circumference C of a circle with radius r is: C = 2(pi)r ...where "pi" (above) is of course the number approximately equal to 22 / 7 or 3.14159. They gave me the value of r and asked me for the value of C, so I'll just "plug-n-chug": C = 2(pi)(3) = 6pi Then, after re-checking the original exercise for the required units (so my answer will be complete): the circumference is 6pi cm. Note: Unless you are told to use one of the approximations for pi, or are told to round to some number of decimal places (from having used the "pi" button on your calculator), you are generally supposed to keep your answer in "exact" form, as shown above. If you're not sure if you should use the "pi" form or the decimal form, use both: "6pi cm, or about 18.85 cm". A square has an area of sixteen square centimeters. What is the length of each of its sides? The formula for the area A of a square with side- length s is: A = s2 They gave me the area, so I'll plug this value into the area formula, and see where this leads: 16 = s2 4 = s After re-reading the exercise to find the correct units, my answer is: The length of each side is 4 centimeters... Most geometry word problems are a bit more involved than the example above. For most exercises, you will be given at least two pieces of information, such as a statement about a square's perimeter and then a question about its area. To find the solution, you will need to know the equations related to the various pieces of information; you will then probably solve one of the equations for a useful bit of new information, and then plug the result into another of the equations. In other words, geometry word problems often aren't simple one-step exercises like the one shown above. But if you take all the information that you've been given, write down any applicable formulas, try to find ways to relate the various pieces, and see where this leads, then you'll almost always end up with a valid answer. A cube has a surface area of fifty-four square centimeters. What is the volume of the cube? The formula for the volume V of a cube with edge- length e is: V = e3 To find the volume, I need the edge-length. Can I use the surface-area information to get what I need? Let's see... A cube has six sides, each of which is a square; and the edges of the cube's faces are the sides of those squares. The formula for the area of a square with side-length e is A = e2. There are six faces so there are six squares, and the cube's total surface area SA must be: SA = 6e2 Plugging in the value they gave me, I get: 54 = 6e2 54 / 6 = (6e2) / 6 9 = e2 3 = e Since the volume is the cube of the edge-length, and since the units on this cube are centimeters, then: the volume is 27 cubic centimeters, or 27 cc, or 27 mL A circle has an area of 49pi square units. What is the length of the circle's diameter? The formula for the area A of a circle with radius r is: A = (pi)r2 I know that radius r is half of the length of the diameter d, so: 49pi = (pi)r2 (49pi) / pi = [(pi)r2] / pi 49 = r2 7 = r Then the radius r has a length of 7 units, and: the length of the diameter is 14 units Warning: You can not assume that you will always be given all the geometric formulae on your tests. At some point, you will need to know at least some of them "by heart". The basic formulae you should know include the formulae for the area and perimeter (or circumference) of squares, rectangles, triangles, and circles; and the surface areas and volumes of cubes, rectangular solids, spheres, and cylinders. Depending on the class, you may also need to know the formulae for cones and pyramids. If you're not sure what your instructor expects, ask now. Sometimes geometry word problems wrap the geometry in a thick layer of "real life". You will need to be able to "see" the geometry, and extract the relevant information. Suppose a water tank in the shape of a right circular cylinder is thirty feet long and eight feet in diameter. How much sheet metal was used in its construction? What they are asking for here is the surface area of the water tank. The total surface area of the tank will be the sum of the surface areas of the side (the cylindrical part) and of the ends. If the diameter is eight feet, then the radius is four feet. The surface area of each end is given by the area formula for a circle with radius r: A = (pi)r2. (There are two end pieces, so I will be multiplying this area by 2 when I find my total-surface-area formula.) The surface area of the cylinder is the circumference of the circle, multiplied by the height: A = 2(pi)rh. Side view of the cylindrical tank, showing the radius "r". An "exploded" view of the tank, showing the three separate surfaces whose areas I need to find. Then the total surface area of this tank is given by: 2 ×( (pi)r2 ) + 2(pi)rh (the two ends, plus the cylinder) = 2( (pi) (42) ) + 2(pi) (4)(30) = 2( (pi) × 16 ) + 240(pi) = 32(pi) + 240(pi) = 272(pi) Since the original dimensions were given in terms of feet, then my area must be in terms of square feet: the surface area is 272(pi) square feet. By the way, this is one of those exercises that doesn't translate well into "real life". In reality, the sheet metal has thickness, and adjustments would be required in order to account for this thickness. For example, if you needed to figure out the amount of sheet metal required to create a tank with a certain volume, you'd have to account for the fact that the volume is on the inside, while the surface area is on the outside. Since the walls of a real-world tank have thickness, the real-world answer would not match the "ideal" mathematical one. A piece of 16-gauge copper wire 42 cm long is bent into the shape of a rectangle whose width is twice its length. Find the dimensions of the rectangle. ADVERTISEMENT Ads by Google Ad covers the page Stop seeing this ad Do I care that the wire is made of copper, or that the wire is a length of sixteen-gauge? No; all I care is that the length is forty-two units, that the units are centimeters, that the rectangle is twice as long in one direction as the other, and that I'm supposed to find the values of each of these directions. I can ignore the other information. Since the wire is 42 centimeters long, then the perimeter of the rectangle is 42 centimeters. That is: 2L + 2W = 42 I also know that the width is twice the length, so: W = 2L Then: Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved 2L + 2(2L) = 42 (by substitution for W from the above equation) 2L + 4L = 42 6L = 42 L = 7 Since the width is related to the length by W = 2L, then W = 14, and the rectangle is 7 centimeters long and 14 centimeters wide. A circular swimming pool with a diameter of 28 feet has a deck of uniform width built around it. If the area of the deck is 60(pi) square feet, find its width. I have this situation: A pool is surrounded by a deck. The pool has radius 14, and the deck has width "d ". If the diameter of the pool is 28, then the radius is 14. The area of the pool is then: (pi)r2 = (pi)(14)2 = 196(pi) Then the total area of the pool plus the surrounding decking is: 196(pi) + 60(pi) = 256(pi) Working backwards from the area formula, I can find the radius of the whole pool-plus-deck area: 256(pi) = (pi)r2 256 = r2 16 = r Since I already know that the pool has a radius of 14 feet, and I now know that the whole area has a radius of 16, then clearly: the deck is two feet wide. If one side of a square is doubled in length and the adjacent side is decreased by two centimeters, the area of the resulting rectangle is 96 square centimeters larger than that of the original square. Find the dimensions of the rectangle. I'm starting from a square with sides of some unknown length. The sides of the rectangle are defined in terms of that unknown length. So I'll pick a variable for the unknown side-length, create expressions for the rectangle's sides, and then work from there. square's side length: x one side is doubled: 2x next side is decreased by two: x – 2 square's area: x2 rectangle's area: (2x)(x – 2) = 2x2 – 4x new area is 96 more than old area: 2x2 – 4x = x2 + 96 2x2 – 4x = x2 + 96 x2 – 4x – 96 = 0 (x – 12)(x + 8) = 0 x = 12 or x = –8 I'm supposed to find the dimensions of a rectangle, so can I just erase that one "minus" sign and say that the rectangle is 12 by 8? No! I defined "x" as standing for the side length of the square, not as one of the sides of the rectangle. Looking back at my definitions, I see what "x = 12" (the only reasonable solution for the square) means that the sides of the rectangle have lengths 2(12) and (12) – 2: