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Lesson #10 Geometric Figures
When working with geometric figures, it is very important to know and understand the vocabulary. Usually a problem will be described in words, and one or two key words will make the difference in whether you correctly answer the question or not. Look for these words in a problem description, and make sure that you follow through on its meaning by using the appropriate formula (see the list on the next page).
For extra help, pay close attention to dimensions and units. One dimension gives a length (in units like inches, feet, miles, or meters). Lengths are associated with line segments or sides of a figure. Points at the ends of a line segment are called endpoints.
Two dimensions (length by width or length by height, for example) determine an area, the size of the region enclosed by the figure. Area is always measured in square units (like square feet, written as ft2). In contrast, the perimeter of a two-dimensional figure is the distance around the outside of a figure. Add the lengths of all sides to get the perimeter. The units for perimeter are always the same as the units for length. The corner points of a figure are called vertices (one point is a vertex). To help distinguish between area and perimeter, think of a farmer's pasture. The pasture is usually fenced (to keep the cows in). The area is the size of the pasture and the perimeter is the total length of fence.
In three dimensions (an object with length, height and width), we use volume for the size of the region inside and surface area for the size of the surface (outside) the object. The units for volume are cubic units (like cubic feet, written as ft3). Surface area really is an area measurement and has square units. For an example in three dimensions, think of a cardboard box. The volume is the size of the box -- how much it can hold. The surface area is the amount of cardboard used for all of the sides.
Example 1. Find the area and perimeter of the rectangle 9 cm long by 2 cm wide.
Use the formulas on the following page.
Area = lw = (9cm)(2cm) = 18 cm2.
Perimeter = 2l + 2w = 2(9cm) + 2(2cm) = 18 cm + 4 cm = 22 cm.
Example 2. Find the area of the triangle with a base of 12 inches and a height of 7 inches.
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Example 3.
The width of a rectangle is 5cm and the perimeter is 42 cm. What is the length of the rectangle? What is its area?|
Perimeter = 2l + 2w |
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Area and Volume Formulas |
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Triangle area The sum of all 3 angles is always 180°. |
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Right Triangle (contains 90° angle) area Pythagorean Theorem: (c is the side opposite the right angle) |
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Square Area A = s2 Perimeter P = 4s |
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Rectangle area A = lw perimeter P = 2l + 2w |
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Circle area circumference C = 2p r r = radius 2r = diameter |
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Parallelogram (opposite sides are parallel) area A = bh perimeter P = 2a + 2b |
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Sphere (radius r) volume surface area SA = 4p r2 |
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Rectangular Solid volume V = lwh surface area diagonal |
Example 4. Find the volume and surface area of the given box.
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Use length = x, width = 3, and height = 1 to fill into the volume and surface area formulas. Volume = lwh = x(1)(3) = 3x (in cubic meters) Surface Area = 2lw + 2lh + 2wh = 2x(3) + 2x(1) + 2(3)(1) |
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Exercises
For exercises 1-5, find the area and perimeter of each figure. For exercises 6-10, find the area only.
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11) If the perimeter of the rectangle ABCD is 20 feet, what is its length?
12) If the perimeter of a rectangle is 30 cm and its length is 12 cm, what is its width?
13) A box is made 7 inches wide by 10 inches long and 6 inches high.
(a) What is the volume of the box?
(b) What is the total surface area of the box?
(c) What would the surface area be if the box has no top?
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Exercises 14-16 refer to the box pictured at right. |
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14) What is the volume of the box? 15) In terms of x, what is the surface area of the box? 16) If the surface area is 62 in2, what is x? What is the volume of the box in this case? |
Right Triangles
Many geometry problems involve the use of the Pythagorean Theorem for right triangles. This is one of the most famous mathematical formulas of all time (a2 + b2 = c2). You will need to be familiar with its use and know how to use it properly. First of all, be aware that the Pythagorean Theorem applies only to right triangles (that is, triangles with a 90° angle). The two sides of the right triangle that form the 90° angle are called legs. These are always the two shorter sides. The longest of the three sides, called the hypotenuse, is always opposite the 90° angle. In the formula, a2 + b2 = c2, the numbers a and b represent the lengths of the two legs, and c represents the length of the hypotenuse. Whatever letters you might use for the three sides, make sure that the longest of the three sides is by itself in the formula.
Example 5. Find the length of the hypotenuse in a right triangle with legs 3 and 4 feet.
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Use a2 + b2 = c2, with a = 3 and b = 4. We solve for c, the hypotenuse. (3)2 + (4) 2 = c2 |
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Example 6. Find the length of the third side in the right triangle.
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Again, use a2 + b2 = c2, but here c = 4 and b = 1. a2 +12 = 42 |
Note that we use the root symbol (
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Example 7. Find the length of the third side in the right triangle.
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a2 + b2 = c2 |
More Exercises
Find the length of the unknown side in each right triangle.
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23) A boat travels due west for 30 kilometers and then turns due south. If it stops 50 kilometers from its starting point, how far south did the boat travel?
24) How long is the diagonal of a square 3cm by 3cm?
25) How long is the diagonal of a rectangle 5 feet by 9 feet?
Check the answers to Lesson 10 exercises
Try the review sheet for lessons 6-10 to prepare for exam 2.
Go on to Lesson 11
