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Chapter 9—Basic Geometry

9-3 Finding Areas

 When you complete the work for this section, you should be able to do the following: Explain the meaning of area. Given the dimensions of a triangle or quadrilateral figure, calculate its area. Given the diameter or radius of a circle, calculate its area. Definition The area of a geometric figure is the number of squares of a given size that cover the surface of the figure.

The area of a plane figure is a measure of its surface. Generally speaking, the larger the dimensions of a plane figure, the larger its surface area.

Area is expressed in square units such as square inches, square feet, square miles, and so on. Units of area can also be abbreviated by using the square symbol. For example::

square inches = in2
square meters = m2
square miles = mi2

Area of Rectangles and Squares

You can always find the area of a rectangle or square by measuring and multiplying the lengths of two adjacent sides. For a rectangle, this means multiplying its length (l) times its width (w):

Equation The equation for the area of any rectangle is: A = lw Where: l is the length of the rectangle w is the width of the rectangle A is the area

Example

 Problem The sides of a certain rectangle measure 2 inches and 6 inches. What is the area of this rectangle? Procedure Cite the appropriate equation A = lw Assign the given values A = 2 · 6 Solve the equation A = 12 Solution The area of this rectangle is 12 square inches.

A square is really a certain type of rectangel where all sides happen to have the same length. Recall that you can find the area of a rectangle by multiplying the lengths of any two adjacent sides, namely l x w. For a square, however, the adjacent sides are equal. Suppose the sides of a square are equal to 3 inches. One side times its adjacent side = 3 x 3, or 32. This is were we get the simple equation for the area of a square:

Equation The equation for the area of a square   is: A = s2 Where: s is the length of the sides A is the area

Example

 Problem Each side of a certain square is 3 units long. What is the area of this square? Procedure Cite the appropriate equation A = s2 Assign the given values A = 32 Solve the equation A = 9 Solution The area of this square is  9 square units

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills: Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem. Rectangle Square

Area of a Triangle

When you get really good at geometry, you can prove that the area of a triangle is exactly equal to one-half the area of a rectangle that has the same dimensions for width and length--or base and height, as it is called for trangles.

Equation The equation for the area of any triangle is: A = ½bh Where: b is the length of the base of the triangle h is the height of the triangle A is the area

Example

 Problem The height of a certain triangle is 10 inches and the base is 4 inches. What is the area of this triangle? Procedure Cite the appropriate equation A = ½bh Assign the given values A = ½ · 4 · 10 Solve the equation A = 20 Solution The area is 20 square inches

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills: Note: Use these figures only for reference. The proportions do not necessarily match those cited in the problem.

Area of a Parallelogram

The equation for the area of a parallelogram has exactly the same form as the equation for the area of a rectangle--the product of two dimenions.

Equation The equation for the area of any parallelogram is: A = bh Where: b is the length of the base h is the height of the parallelogram A is the area

Example

 Problem What is the area of a parallelogram that has a height of 20 cm and a base of 80 cm? Procedure Cite the appropriate equation A = bh Assign the given values A = 80 · 20 Solve the equation A = 1600 Solution The area is 1600 square centimeters

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills: Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem.

Area of a Trapezoid

Equation The equation for the area of any trapezoid is: A = ½h(b1 + b2) Where: b1 and b2 are the lengths of the parallel sides of the  trapezoid h is the height of the trapezoid A is the area

Example

 Problem The parallel sides of a certain trapezoid measure 16 ft and 24 ft.  The height if 10 ft. What is the area? Procedure Cite the appropriate equation A = ½h(b1 + b2) Assign the given values A = ½ · 10 (16 + 24) Solve the equation A = ½ · 10 (40) A = 200 Solution The area is 200 square ft.

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills: Note: Use this figure only for reference. The proportions do not necessarily match those cited in the problem.

Area of a Circle

Equation The equation for the area of any circle is: A = pr2 Where: p = approximately 3.14 or 22/7 r is the radius of the circle A is the area of the circle

Example

 Problem Find the area of a circle that has a radius of 4 units. Procedure Cite the appropriate equation A = pr 2 Assign the given values A = p · 4 2 Solve the equation A = 3.14 · 16 A = 50.24 Solution The area of this circle is  50.24 square units

But Suppose you are given the diameter, rather than the radius, of a circle; and you need to find the area. The simplest approach is to find the radius by cutting the diameter in half. (Recall that the radius of a circle is equal to one-half the diameter). Once you know the radius, you can use the basic equation for finding the area.

Example

 Problem Find the area of a circle that has a diameter of 12 feet. Procedure Divide the diameter in half r = d/2 r = 12/2 = 6 Cite the appropriate equation A = pr 2 Assign the given values A = p · 62 Solve the equation and simplify A = 3.14 · 36 A = 113 Solution The area of this circle is  113 square units

Examples and Exercises

 Use these interactive examples and exercises to strengthen your understanding and build your skills:

Summary of Equations for the Area of Plane Figures

 Rectangle A = lw Square A = s2 Triangle A = ½bh Parallelogram A = bh Trapezoid A = ½h(b1 + b2) Circle A = pr2
These interactive examples and exercises give you a chance to test your understanding of finding the area of all these plane figures.

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