Perimeter, Area, and Volume
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1 Perimeter, Area, and Volume Perimeter of Common Geometric Figures The perimeter of a geometric figure is defined as the distance around the outside of the figure. Perimeter is calculated by adding all of the side lengths of the given geometric figure. Example 1 Find the perimeter of the figure given below. Convert the length of 1 foot to 1 inches. Then, add all of the side lengths to obtain inches + 7 inches + 5 inches + 1 inches + 4 inches = 1 inches. Formulas for perimeter of common geometric figures are given here. PERIMETER OF A RECTANGLE The perimeter P of a rectangle of width W and length L is P = ( W) + ( L). P = L + W W = width L = length
2 PERIMETER OF A CIRCLE The perimeter of a circle is referred to as the circumference of the circle. The circumference C of a circle of radius r is C = r where r is equal to the radius and.14. C = r Example What is the circumference of a 0 inch diameter pizza? The radius of this pizza is 10 inches. The circumference is r, where r = 10 and.14. P = inches = 6.8 inches Example What is the perimeter of a soccer field that is 150 feet wide and 100 yards long? The units of the length and the width must be converted so that they are the same type. The width of 150 feet may be converted to yards by using methods from section feet 1 yard = 50 yards 1 feet The perimeter is (100 yards) + (50 yards) = 00 yards MATH NOTE Include all units when doing calculations with measurements. When calculating perimeter, all units must be of the same type.
3 Areas of Common Geometric Figures Often, one must use area formulas in technical applications. Before listing these formulas, it is necessary to define area in terms of square units. DEFINITION OF AREA A square unit of area consists of a square with dimensions 1 unit by 1 unit. The area of this square is 1 square unit which is often written as 1 unit. Example: 1 square inch or 1 in consists of a 1 inch by 1 inch square. 1 inch 1 inch Example 4 What is the area of the following geometric figure? inches inches Since this figure contains four 1-inch by 1-inch squares, the area is 4 in. Example 5 What is the area of a square with dimensions 1 foot by 1 foot? Since a square with the dimensions 1 unit by 1 unit has an area of 1 square unit, a 1-foot by 1-foot square has an area of 1 ft.
4 The formulas given here may be used to calculate the areas of common geometric figures. AREA OF A SQUARE The area A of a square of dimensions s units by s units is A = s square units. s units s units AREA OF A RECTANGLE The area A of a rectangle with a length of L units and a width of W units is A = L W square units. L units W units 1 AREA OF A TRIANGLE The area A of a triangle with a base of b units and a height of h units is A = b h square units. A = 1 b h square units
5 Example 6 What is the area of a rectangle with a width of 4 inches and a length of feet? In order to calculate the area, the dimensions must be of the same units. Convert 4 inches to feet, and use the area formula of A = L W. A = feet feet = 6 ft or 6 square feet. MATH NOTE Convert mixed units into a single unit before using area or volume formulas. Example: If the dimensions of a rectangle are 41 inches by 5 feet, convert 5 feet into inches or convert 41 inches into feet before calculating an area. Example 7 Calculate the area of the figure below. 9 ft 10 ft This figure consists of a rectangle with a width of 5 ft and a length of 10 ft and a triangle with a base of 10 feet and a height of 4 feet. The height of 4 feet is calculated by subtracting 5 feet from 9 feet. The area of the rectangle is 5 ft 10 ft = 50 square feet. The area of the triangle is 1 10 ft 4 ft = 0 square feet. The total area is 50 ft + 0 ft = 70 square feet or 70 ft.
6 AREA OF A CIRCLE The area A of a circle with a radius of r units is A = r square units. Note that = pi and is approximately equal to.14 and r is equal to half the diameter. A = r square units Example 8 What is the area of a 0 foot diameter circular swimming pool? Since the diameter is 0 feet, the radius r is half of 0 which is 15 feet. The area is calculated using the formula A = r where =.14 and r = 15 feet. A =.14 (15 ft) =.14 5 ft = square feet or ft Example 9 What are the areas of a 0 inch diameter pizza and a 10 inch diameter pizza? The radius of the 0 inch pizza is 10 inches and the radius of the 10 inch pizza is 5 inches. The areas are A =.14 (10 in) = 14 in A =.14 (5 in) = 78.5 in Note that the area for the 0 inch pizza was more than three times the area of the 10 pizza.
7 Volumes of Common Geometric Solids Volume is a measure of -dimensional space and is measured in cubic units. The definition of a cubic unit is given below. DEFINITION OF VOLUME A cubic unit of volume consists of a cube of dimensions 1 unit by 1 unit by 1 unit. The volume of this cube is 1 cubic unit which is often written as 1 unit. Example: 1 cubic inch = 1 in and consists of a 1 inch by 1 inch by 1 inch cube. Formulas for the volumes of common geometric figures are given here. VOLUME OF A CUBE The volume V of a cube with sides of length s is s cubic units. V = s cubic units or s unit Example 10 What is the volume of a cubical chest freezer that is feet wide, feet long, and 4 inches deep? Convert 4 inches to feet. The volume is ( ft) = 8 cubic feet or 8 ft.
8 VOLUME OF A RECTANGULAR SOLID The volume V of a rectangular solid with the dimensions L, W, and H is L W H cubic units. This is also written as L W H unit. V = L W H cubic units or unit VOLUME OF A SPHERE The volume V of a sphere with radius r is r cubic units. Note that V = 4 r cubic units or unit VOLUME OF A CIRCULAR CYLINDER The volume V of a circular cylinder is r H cubic units where r is the radius and H is the height. H V = r H cubic units or unit
9 Example 11 A soup can has a diameter of inches and a height of 4 inches. How many cubic inches of soup are able to fit in the can? Using the fact that 18 ounces are equivalent to 1 gallon and 1 in are equivalent to 1 gallon, convert this volume to fluid ounces. The volume of this can is r H and r = 1.5 inches, H = 4 inches, and.14. Volume =.14 (1.5 in) 4 in = 8.6 in. To convert this answer to fluid ounces, use methods from section in 1 gallon 18 ounces 1 1 in 1 gallon = = 15.7 ounces (rounded to the nearest tenth of an ounce) 1 Example 1 How many cubic inches of air will fit within a 1-foot diameter basketball? Round the answer to the nearest tenth of a cubic inch. Since we are looking for an answer in in, the 1 foot measurement should be converted to 1 inches. Since the diameter is 1 inches, the radius is 6 inches. 4 4 The volume formula is V = r where.14 and r = The volume is.14 (6 in) = = 904. in This rounds to 904. in. MATH REMINDER Always include units whenever you are working with measurements! Keep units consistent within a calculation. Example: If you wish to calculate the volume of a rectangular solid with the dimensions 4 feet by feet by 18 inches, convert the 18 inch measurement to 1.5 feet. Then, calculate the volume as 4 ft ft 1.5 ft = 1 ft.
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