Taking the following square with side length 6 inches, calculate the perimeter.

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1 Area and perimeter are two calculations performed on many geometric shapes. Perimeter is a measure of distance around a shape; for example, someone might want to figure out the perimeter around their garden before buying material to make a fence so that they know how much material to buy. Area is a measure of the amount of surface something covers; for example, someone might want to know how much space their garden takes up. Area and perimeter are often grouped together because one can be used to help you figure out the other. For example, if you know the perimeter of a square, you can easily figure out the area, and vice versa. Perimeter Perimeter simply measures the distance around an area. It can be measured in inches, feet, yards, miles, centimeters, meters, kilometers, and so on (any standard distance measurement). You can measure the perimeter of nearly any shape, you just add together the measure of each of its sides. Much of the ability to figure out perimeter lies in remembering the properties of certain shapes. We ll go through several examples. Perimeter of a Square Taking the following square with side length 6 inches, calculate the perimeter. In order to calculate perimeter, you need to add together the lengths of all four sides of the square. You are given the length of one side. Remember, all sides of a square are equal, so really you already have the measures of each side. Then, you add them together, so = 24 inches. Thus, 24 inches is your final answer. Perimeter of a Rectangle Taking the following rectangle with length 8 inches and width 4 inches, calculate the perimeter.

2 In order to calculate perimeter, you need to add together the lengths of all four sides of the rectangle. You are given the length of one side and the width of one side. Remember, opposite sides of a rectangle are equal, so really you already have the measures of each side. Then, you add them together, so = 24 inches. Thus, 24 inches is your final answer. Perimeter of a Polygon The perimeter of a polygon is calculated using the same method of adding together each side. Remember that if all the sides are equal, you only need to know one side of the polygon. If the sides are unequal, however, you do need to know the length of each different side. Taking the following pentagon with side length 7, calculate the perimeter. A pentagon has five sides, and all of these sides are equal, therefore you can perform the following calculation: = 35 Example 1 Michelle was planting a garden. She wanted her garden to be fenced in, so she went to the hardware store to buy fencing material. The salesperson asked Michelle how big her garden would be. She thought about it, and then replied that her garden would be 4 feet wide, and that it would be 2 feet longer (in length) than it is wide. Answer the following questions: 1. What shape is Michelle s garden? 2. How long is Michelle s garden? 3. What is the perimeter of Michelle s garden?

3 4. Draw and label Michelle s garden. Solution Once you've worked out the answers, click "Next Step" to show the answer to each answer to each question! 1. Michelle s garden is a rectangle. We know this because it talks about length and width (4-sided shapes have these) and we can conclude it is not a square, because the length and width are different, therefore all four sides are not equal. 2. The problem stated that Michelle s garden is two feet longer than it is wide. We know Michelle s garden is 4 feet wide, so we know that we have to add 2 to that number, resulting in 6. Thus, Michelle s garden is 6 feet long. 3. In order to find the perimeter of Michelle s garden, we have to add together all four sides. We know that two sides are 4 feet long, and the other two sides are 6 feet long. Therefore, we can solve the addition problem: = 20 feet. Our answer is that the perimeter of her garden is 20 feet. This means that, when Michelle buys the material to build her fence, she ll need 20 feet of material in order for the fence to be complete. 4. Example 2 Andrew is going to build a box to hold his hats. He decides that each side should be 5 inches long. He also decides to make this box in the shape of a regular hexagon. Answer the following questions: 1. How many sides does Andrew s box have? 2. Are all sides the same length? How do you know? 3. What is the perimeter of to Andrew s box?

4 Solution Once you've worked out the answers, click "Next Step" to show the answer to each answer to each question! 1. We know that Andrew s box is in the shape of a hexagon, and a hexagon has 6 sides. Therefore, Andrew s box also has 6 sides. 2. All sides of Andrew s box are the same length. We know this because the problem stated that we have a regular hexagon, and we know that regular means all sides are the same. 3. We can easily calculate the perimeter of Andrew s lid to the box by using the following addition problem: = 30 inches. Area Area is the measure of the amount of surface covered by something. Area formulas for different shapes are sometimes different, but for the most part, area is calculated by multiplying length times width. This is used when calculating area of squares and rectangles. Once you have the number answer to the problem, you need to figure out the units. When calculating area, you will take the units given in the problem (feet, yards, etc) and square them, so your unit measure would be in square feet (ft. 2 ) (or whatever measure they gave you). Area Example 1 Let s try an example. Nancy has a vegetable garden that is 6 feet long and 4 feet wide. It looks like this:

5 Nancy wants to cover the ground with fresh dirt. How many square feet of dirt would she need? We know that an answer in square feet would require us to calculate the area. In order to calculate the area of a rectangle, we multiply the length times the width. So, we have 6 x 4, which is 24. Therefore, the area (and amount of dirt Nancy would need) is 24 square feet. Area Example 2 Let s try that one more time. Zachary has a wall that he would like to paint. The wall is 10 feet wide and 16 feet long. It looks like this: Using Area and Perimeter Together Sometimes, you will be given either the area or the perimeter in a problem and you will be asked to calculate the value you are not given. For example, you may be given the perimeter and be asked to calculate area; or, you may be given the area and be asked to calculate the perimeter. Let s go through a few examples of what this would look like: Area and Perimeter Example 1 Valery has a large, square room that she wants to have carpeted. She knows that the perimeter of the room is 100 feet, but the carpet company wants to know the area. She knows that she can use the perimeter to calculate the area. What is the area of her room? We know that all four sides of a square are equal. Therefore, in order to find the length of each side, we would divide the perimeter by 4. We would do this because we know a square has four sides, and they are each the same length and we want the division to be equal. So, we do our division 100 divided by 4 and get 25 as our answer. 25 is the length of each side of the room. Now, we just have to figure out the area. We know that the area of a square is length times width, and since all sides of a square are the same, we would multiply 25 x 25, which is 625. Thus, she would be carpeting 625 square feet.

6 Area and Perimeter Example 2 Now let s see how we would work with area to figure out perimeter. Let s say that John has a square sandbox with an area of 100 square feet. He wants to put a short fence around his sandbox, but in order to figure out how much fence material he should buy, he needs to know the perimeter. He knows that he can figure out the perimeter by using the area. What is the perimeter of his sandbox? We know that the area of a square is length times width. In the case of squares, these two numbers are the same. Therefore, we need to think, what number times itself gives us 100? We know that 10 x 10 = 100, so we know that 10 is the length of one side of the sandbox. Now, we just need to find the perimeter. We know that perimeter is calculated by adding together the lengths of all the sides. Therefore, we have = 40 (or, 10 x 4 = 40), so we know that our perimeter is 40 ft. John would need to buy 40 feet of fencing material to make it all the way around his garden. Calculating Area and Perimeter Using Algebraic Equations So far, we have been calculating area and perimeter after having been given the length and the width of a square or rectangle. Sometimes, however, you will be given the total perimeter, and a ratio of one side to the other, and be expected to set up an algebraic equation (using variables) in order to solve the problem. We ll show you how to set this up so that you can be successful in solving these types of problems. Eleanor has a room that is not square. The length of the room is five feet more than the width of the room. The total perimeter of the room is 50 ft. Eleanor wants to tile the floor of the room. How many square feet (ft 2 ) will she be tiling? In this problem, we will be calculating area, but first we re going to use the perimeter to figure out the length and width of the room. First, we have to assign variables to each side of the rectangle. X is the most often used variable, but you can pick any letter of the alphabet that you d like to use. For now, we ll just keep things simple and use x. To assign a variable to a side, you first need to figure out which side they give you the least information about. In this problem, it says the length is five feet longer than the width. That means that you have no information about the width, but you do have information about the length based on the width. Therefore, you re going to call the width (the side with the least information) x. Now, the width = x, and x simply stands for a number you don t know yet. Now, you can assign a variable to the length. We can t call the length x, because we already named the width x, and we know that these two measurements are not equal. However, the problem said that the length is five feet longer than the width. Therefore, whatever the width (x) is, we need to add 5 to get the length. So, we re going to call the length x + 5.

7 Now that we ve named each side, we can say that width = x, and length = x + 5. Here s a picture of what this would look like: Next, we need to set up an equation using these variables and the perimeter in order to figure out the length of each side. Remember, when calculating perimeter you add all four sides together. Our equation is going to look the same way, just with x s instead of numbers. So, our equation looks like this: x + x + x x + 5 = 50 Now, we need to make this look more like an equation we can solve. Our first step is to combine like terms, which simply means to add all the x s together, and then add the whole numbers together (for more help on this, see Combining Like Terms). Once we combine like terms, our equation looks like this: 4x + 10 = 50 Next, we follow the steps for solving equations. (For additional help with this, see Solving Equations). We subtract 10 from each side of the equation, which leaves us with the following: 4x = 40 Now, we have to get x by itself, which means getting rid of the 4. In order to do this, we need to perform the opposite operation of what s in the equation. So, since 4x means multiplication, we need to divide by 4 to get x alone. But remember, what we do to one side, we have to do to the other side. After dividing each side by 4, we get: x = 10 Next, we have to interpret what this means. We look back and recall that we named the width x, so the width is 10. Now, we need to figure out the length. We named the length x + 5, so that means we have

8 to substitute 10 in for x, and complete the addition. Therefore, we have , which gives us 15. So, our length is 15. Now, we need to look back and remember that the problem asked us to calculate the area of the floor that Eleanor will be tiling. We know that in order to calculate area, we need to multiply the length times the width. We now have both the length and the width, so we simply set up a multiplication problem, like this: 10 x 15 =? We multiply the two numbers together, and get 150. Thus, your final answer is 150 ft 2. Area and Perimeter Practice Problems Now, we ll give you several practice problems so that you can try calculating area and perimeter on your own. 1. Leah has a flower garden that is 4 meters long and 2 meters wide. Leah would like to put bricks around the garden, but she needs to know the perimeter of the garden before she buys the bricks. What is the perimeter of Leah s garden (in meters)? {12 12 m 12 meters 12 meter} 2. David has a rug that is square, and the length of one side is 5 feet. He has an open floor space in his living room that is 36 square feet. Would the rug fit perfectly, be too big, or be too small for the space he has? (Answer Choices: fit perfectly, too big, too small) too small 3. Debbie has pool in her back yard that has a perimeter of 64 feet. The length of the pool is 2 feet longer than the width. Debbie wants to buy a cover for the pool, and needs to know how many square feet she needs to cover. How many square feet (ft 2 ) is Debbie s pool? (hint: if you can, set up an algebraic equation to solve this problem!) { sq ft 255 square feet 255 ft2 255 ft² 255 ft 2 } 4. Hector is planting a square garden in front of his house. He wants to plant carrots in the garden. He knows he can plant the carrots one foot apart. He has six feet across his yard (length) and he can plant carrots four feet deep (width). How many square feet (ft 2 ) does he have to plant carrots?

9 {24 24 ft sq 24 sq ft 24 ft²} 5. Amanda is building a house, and she s trying to calculate the area of her bedroom. She knows that the living room is 22 feet long and 20 feet wide. She was told that her bedroom should be half of the area of the living room. What will the area of her bedroom be?

10 Perimeter, Area and Volume This topic makes use of formulae for perimeters, areas and volumes of shapes and solids. If you need help with working with formulae see the topic Formulae and Algebra in the menu to the left of the screen before reviewing this topic. Note - this topic contains specialised formatting and symbols. If you copy parts of it to the Scratch Pad this formatting may be lost. You can print the topic without losing the formatting and symbols. There are 10 pages. You can scroll down to read all the help in this topic or click on one of the links below to go straight to a specific area. Click on one of these links to go to help on working with perimeter, area and volume, alternatively you may want to check a particular formula, click on any of the following to go straight to it: Calculating Circumference of a circle, Circumference of an ellipse Area of square/rectangle, triangle, parallelogram, trapezium, any other polygon, circle Volume of cube/cuboid, pyramid, cone, cylinder, sphere Perimeter What is perimeter? The perimeter is the distance or length around the outside of a shape. It is calculated by adding the lengths of all the sides together, ie in diagram (left): Perimeter = = 18 cm As perimeter is a length or distance it is measured in units of length, eg mm, cm, etc.

11 It may help if you think of it as how far you would have to walk, to go all the way round the edges of the shape. Also if you are working out the perimeter of a complicated shape, it may help to mark your starting point (see left). This way you can be sure that you add all the sides once and once only. What is circumference? The perimeter of a circle, and also an ellipse (or oval), is called its circumference. The perimeter of shapes with straight edges can be measured using a ruler and adding the sides together as explained above. However, it is not possible to accurately measure a curve using a straight ruler. A different method is used to work out the circumferences of circles and ellipses. Calculating the circumference of a circle. The circumference of a circle is calculated using the following formula: Circumference = 2 r r = the radius of the circle. This is sometimes written as: Circumference = d d = the diameter of the circle. NB The radius of the circle is the distance

12 from the centre to the edge (see diagram left). The diameter of the circle is the length of a line going from one edge to another, passing through the centre. This is the same as twice the radius. What is? (pi) is the number that represents the ratio between the radius and diameter of a circle to its circumference and area. It has an endless number of decimal places. Here it is shown rounded to 8 decimal places If you have a scientific calculator you will have a ' button on it. ' Example 1 Calculate the circumference of a circle whose radius is 3 cm. Using the formula: Circumference = 2 In the example, r = 3. So: r Circumference = 2 x x 3 = = cm (4 dp) For more help with rounding decimals and decimal

13 places see the sub-topic Decimals in the menu to the left of the screen. Circumference of an ellipse. The circumference of an ellipse is difficult to calculate exactly. If you require the formula for the best approximation of this, look in a mathematics study dictionary. One is recommended in the 'Resources You Can Use' for this topic. Area What is area? The area is the surface space contained within the edges of a 2-D shape (see coloured area of shape left). Area is measured as the number of squares of a particular unit, eg mm 2, cm 2, m 2 etc.

14 Calculating the area of a square or rectangle. The shape, left, has been divided into squares. Each square is 1 cm 2 (1 cm x 1 cm). The area of the rectangle is calculated using the formula: Area = l x b l = length b = breadth. Here, l = 5 cm and b = 2 cm. So: Area = 5 x 2 = 10 cm 2 You can check the answer by counting how many 1 cm 2 squares there are within the shape. The same formula is used for the area of a square. As all the sides of a square are the same length, this will be a number multiplied by itself.

15 Example 2 What is the area of a square with sides 6 cm? Using the formula Area = l x b l = length b = breadth. Here, l and b both equal 6 cm. So: Area = 6 x 6 = 36 cm 2 This is the same as 6 2 (this is said as 'six squared' or 'six to the power two'). For help with powers see the sub topic Powers and Roots in the menu to the left of the screen. Calculating the area of a triangle. The area of a triangle is calculated using the formula: Area = ½ x b x h b = the length of the base of triangle h = the perpendicular height of triangle see the diagram left. Example 3 What is the area of a triangle with a base length of 5 cm and perpendicular height of 3 cm? Using the formula: Area = ½ x b x h In the example, b = 5 and h = 3. So: Area = ½ x 5 x 3

16 = 7.5 cm 2 Calculating the area of a parallelogram. A parallelogram is a four-sided shape where both pairs of opposite sides are parallel. This means that they are always the same distance apart and can never meet however far they are extended. The area of a parallelogram is calculated using the formula: Area = b x h b = the length of the base of the parallelogram h = the perpendicular height of parallelogram see the diagram left. Example 4 What is the area of a parallelogram with base length 6 cm and perpendicular height 2 cm? Using the formula Area = b x h In the example, b = 6 and h = 2. So: Area = 6 x 2 = 12 cm 2 Calculating the area of a trapezium. A trapezium is a four-sided shape with one pair of parallel sides. The area of a trapezium is calculated using the

17 formula: Area = (b + p) x h 2 b = the length of the base of trapezium p = the length of the parallel edge h = the perpendicular height of trapezium. NB. The base of the trapezium need not be the bottom of the shape, but it should be one of the parallel sides. Therefore when calculating the area, (b + p) means add together the lengths of the two parallel sides. Example 5 What is the area of a trapezium with parallel sides of length 8 cm and 6 cm? Its perpendicular height is 3 cm. Using the formula: Area = (b + p) x h 2 In the example, b = 8, p = 6 and h = 3. So: Area = (8 + 6) x 3 2 = 14 x 3

18 2 = 21 cm 2 Calculating the area of other polygons. A polygon is a 2-D shape enclosed by three or more sides. Triangles, squares, rectangles etc are all polygons. Shapes with more than four sides, eg. pentagon (5 sides), hexagon (6 sides), octagon (8 sides) etc. are also polygons. Each of these have special rules for calculating their area, if you need to calculate one of these look in a mathematics study dictionary. One is recommended in the 'Resources You Can Use' for this topic. Calculating the area of a circle. The area of a circle is calculated using the following formula Area = r 2 r = the radius of the circle. This is sometimes written as: Area = 1 / 4 ( d 2 )

19 d = the diameter of the circle. is explained above in the section about calculating the circumference of a circle. Example 6 What is the area of a circle with a radius of 5cm? Using the formula Area = r 2 In the example, r = 5. So: Area = x (5) 2 = x 25 = = cm 2 (2 dp) For help with rounding decimals and decimal places see the sub topic Decimals in the menu to the left of the screen. Volume. What is volume? The volume, or capacity, of a 3-D shape is how much space is contained within the shape. It may help to think of the 3-D shape (called a solid) as a vessel that can be filled with a certain volume of liquid. Volume is measured as the number of cubes of a

20 particular unit, eg mm 3, cm 3, m 3 etc. Calculating the volume of a cube or cuboid. A cube is a solid shape with 6 square faces. A cuboid is a solid shape with 6 faces that are either all rectangles, or a mixture of rectangles and squares, see diagram left. Boxes are the most common examples of cubes and cuboids. The volume of a cuboid is calculated using the formula: Volume = l x b x w l = length b = breadth w = width. The volume of a cube is calculated in the same way, however, because all the faces are square each of these lengths will be the same. Example 7 What is the volume of a cube, where each square face has sides 4 cm by 4 cm? Using the formula: Volume = l x b x w l, b, w all equal 4 cm. So: Volume = 4 x 4 x 4 = 64 cm 3

21 NB. This is the same as 4 3 (this is said as 'four cubed' or 'four to the power three'). For help with using powers see the sub topic Powers and Roots in the menu to the left of the screen. Calculating the volume of a pyramid. A pyramid is a solid. All but one of its faces must be triangles that meet a point. The base of a pyramid can be a triangle but does not have to be. The most common base of a pyramid is a square. However the base can be any straight-sided shape, ie any polygon. The number of faces a pyramid has depends on the number of sides of the base. A square-based pyramid is shown left. The volume of a pyramid is calculated using the formula: Volume = 1 / 3 Ah A = area of the base h = perpendicular height. NB The perpendicular height of a pyramid is the distance from the apex (top point) straight down to the base (see h on the diagram left)

22 Example 8 A pyramid has a square base with an area of 9 cm 2 (ie. the lengths of the sides of the square are each 3 cm). It has a perpendicular height of 2 cm. What is its volume? Using the formula: Volume = 1 / 3 Ah In the example, A = 9 and h = 2. So: Volume = 1 / 3 x 9 x 2 = 6 cm 3 Calculating the volume of a cone. The volume of a cone is calculated using the same formula as for a pyramid: Volume = 1 / 3 Ah A = area of the base h = perpendicular height. The base of a cone is a circle, so: Area of the base = r 2 r = radius of the circle. Therefore the formula can also be written as: Volume = 1 / 3 r 2 h r = radius of the circular base h = perpendicular height is explained above in the section about calculating the circumference of a circle.

23 Example 9 What is the volume of a cone with perpendicular height of 5 cm? The radius of the circular base is 3 cm. Using the formula: Volume = 1 / 3 r 2 h In the example, r = 3 and h = 5. So: Volume = 1 / 3 x x (3) 2 x 5 = 1 / 3 x x 9 x 5 = = cm 3 (2 dp) For more help with rounding decimals and decimal places see the subtopic Decimals in the menu to the left of the screen. Calculating the volume of a cylinder. The most common cylinders are cans and tubes. The volume of a cylinder is calculated using the formula: Volume = r 2 h where r = radius of one circular end (both ends of a cylinder will have the same radius) h = height of the cylinder NB This is the same as multiplying the area of one of the circular faces by the height of the cylinder. is explained above in the section about calculating the circumference of a circle.

24 Example 10 What is the volume of a cylinder 4 cm high, whose circular face has a radius of 2 cm? Using the formula: Volume = r 2 h In the example, r = 2 and h = 4. So: Volume = x (2) 2 x 4 = x 4 x 4 = = cm 3 (2 dp) For more help with rounding decimals and decimal places see the subtopic Decimals in the menu to the left of the screen. Calculating the volume of a sphere. A ball is the most common type of sphere seen in everyday life. The volume of a sphere is calculated using the formula: volume = 4 / 3 r 3 r = the radius of the sphere. This is sometimes written as: Area = 1 / 6 d 3 d = the diameter of the sphere. NB The radius of a sphere is the distance from the centre of the sphere to the outer edge. The diameter of a sphere is the

25 distance from one edge to another, going through the centre. This is the same as twice the radius. is explained above in the section about calculating the circumference of a circle. Example 11 What is the volume of a sphere with a radius of 7 cm? Using the formula Volume = 4 / 3 r 3 In the example, r = 7. So: Volume = 4 / 3 x x (7) 3 Volume = 4 / 3 x x 343 = = cm 3 (2 dp) For more help with rounding decimals and decimal places see the sub topic Decimals in the menu to the left of the screen.