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Chapter 9: Perimeter and Area of Polygons Circumference and Area of Circles

9.1 Perimeter of Polygons ___________________ is the distance around a polygon. The perimeter of any polygon can be found by __________________ the lengths of all its sides. In polygons where several sides have the same length, as in a square or a rectangle, those congruent sides can be combined. As a result, we can come up with formulas such as: Perimeter of a square = _______, where s is the length of a side

(since ___ + ___ + ___ + ___ = ____ ) or

Perimeter of a rectangle = _____ + _____, where l = length and w = width What formula for perimeter could you write for a regular heptagon or regular decagon? **Since perimeter is the sum of the lengths of sides, it is measured the same way the sides are: in linear units.

Ex: the perimeter of a polygon might be 14 inches or 20 miles.

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You try: 1. Find the perimeter of each of the following: a) a square with a side of 17 inches b) a rectangle that has a length of 12.3cm and a width of 7.8cm c) a rhombus with a side of 2½ feet d) a triangle with sides 7.09, 3.46 and 6.124 millimeters e) an isosceles trapezoid with bases of 32 and 45 and congruent legs of 27 f) a right triangle with legs 8 and 15 centimeters g) a regular nonagon with sides of 8.23 feet h) a rectangle with consecutive sides of 17 inches and 2 feet

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i) a square with a diagonal equal to 4√2 j) a rhombus with diagonals of 6 and 8 inches 2. Find the perimeter of each of the following in terms of an algebraic expression: a) an equilateral triangle with a side of 4x b) a rectangle that has a length of 2x+7 and a width of x+3 c) a regular pentagon with a side 3x-2 d) a parallelogram with consecutive sides 3x and x+1 e) an isosceles triangle with congruent legs 5x-4 and base of x+5

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3. Find the length of a side of each of the following given its perimeter. (some answers may be left as an algebraic expression) a) an equilateral triangle with a perimeter of 324.75 centimeters b) a regular hexagon with a perimeter of 24x+66 c) a square with a perimeter of 183½ feet d) find the width if the perimeter of a rectangle is 44

4. The perimeter of a regular polygon measures 84 cm. If one of the sides measures 14 cm, what is the name of the polygon?

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9.2 Understanding Area of the Polygon While perimeter measures the distance around a polygon, ____________ measures the amount of space within the polygon. **Here’s an episode of Cyberchase for Real that explains the relationship between Area and Perimeter. It’s called the Dumas Diamond and can be found on the PBS site: http://pbskids.org/cyberchase/videos/cyberchase-the-dumas-diamond/ Area measures the number of ______________ units to fill up a particular polygon. For example, the rectangle below can be divided, as shown, into 2 rows of 3 small squares. Hence, the area is 6 square units.

**Area is ALWAYS measured in square units, square feet, square inches, square centimeters, etc.

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9.3 Area of a Rectangle In the rectangle below, the base has a length of 3 units and the height has a length of 2 units:

While we could choose to simply count the number of squares, realize that the number of squares is 3 x 2, or 6 square units. As a result:

Area of a Rectangle = _____________ x ____________

Or

Area of a Rectangle = _________ but most commonly known as

Area of a Rectangle = _______________ x ______________ Or

Area of a Rectangle = _______________

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You try: 1. Find the area of a rectangle that has a length of 12 inches and a width of 7 inches. 2. Find the area of a rectangle that has a length of 23.8 cm and a width of 14.65 cm 3. A rectangular plot of land has an area of 3130 square feet. If the width is 40 feet, find the length of the rectangular plot. 4. Find the area of throw rug that has the dimensions 3½’ by 5½’. 5. The length of a rectangle is 8 more than its width. Represent the area of the rectangle as an algebraic expression.

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6. Represent the area of a rectangle as an algebraic expression if the length and width are represented by x+10 and x-3, respectively. 7. The area of a rectangle is represented by x2+5x-14. What are the dimensions of the rectangle if the area is 112 sq. units.

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9.4 Area of a Square A _________________ is a kind of rectangle. To find the area of a square, we can use the same formula that we used to find the area of any rectangle: length times width. But in a square, the length and the width are the same number. As a result:

Area of a Square = ______________

or Area of a Square = _________

Since there is a definite relationship between the diagonal of a square and the length of its leg, we can also say that:

Area of a Square = _________________ or

Area of a Square = ______________

So to find the area of a square with a diagonal of 4, it’s ½(__2) or ½(____) or _____ square units. Remember to follow PEMDAS and to square “d” before you multiply it by ½.

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You try: 1. Find the area of a square given the length of its side. a) 6 in b) 9.2 cm c) ¾ yd. d) 3x e) x+5 2. Find the area of a square given the length of its diagonal. a) 8 in b) 7.8 cm c) ½ yd. d) 14x

e) 6√2 ft.

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3. Find the length of the square’s side when its area is given. a) 225 in2 b) 6.76 cm2

c) 42¼ yd2 d) 121x2 e) x2+2x+1 4. Find the length of the square’s diagonal when its area is given. a) 72 in2 b) 13.52 cm2 c) 1.62 u2 d) 162x2 e) 10 ft2

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9.5 Area of a Parallelogram We said that the area of a rectangle was length times width, since that formula would give us the number of square units within the rectangle. Let’s take a rectangle, and play with it a bit: See the diagonal line inserted into the rectangle? It cuts off a right triangle. If we take that right triangle, and move it to the left of the rectangle, look at what we get:

The area—the amount inside the shape—hasn’t changed. As a result, we can say that the area of the resulting parallelogram is the same as the area of the original rectangle: Area of a Parallelogram = ______________ x ________________

or Area of a Parallelogram = _____________

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Remember, of course, that the base and height must be _______________________. That means, unlike in a rectangle, you’ll need to have an altitude in order to find the area of a typical parallelogram. But that’s not the only formula for area of a parallelogram. Another formula is based on the idea that the sin(90°) = 1. That formula is

Area of a Parallelogram = ________________________,

where a and b are ___________________ ________________ and ∠C is the ___________________ _________________ between them. To find the area of a parallelogram with sides of 12 and 14, and the included angle of 24°, we evaluate (____)(____)sin_____° and get 168(0.4067) or approximately ___________ square units. 12 14 If you use the parallelogram’s obtuse angle, what happens to its area?

24°

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You try: 1. Find the area of a parallelogram given its base and height, respectively. a) 12 & 7 b) 10.5 & 3.9 c) x+8 & x d) x-2 & x+5 2. Find the area of a parallelogram given its consecutive sides and included angle. Round final answers to the nearest hundredths place. a) 15, 8, & 38° b) 8.2, 14 & 57° c) 19, 25 & 150° d) 4.6, 7 & 110°

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3. The area of a parallelogram is 2718 sq. units. Find the altitude if the base of the parallelogram is 36. 4. The area of a parallelogram is 125 in2. Find the acute angle, to the nearest minute, if the consecutive sides are 14 and 20 inches.

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9.6 Area of a Triangle The standard version of the formula to find the area of a triangle is:

Area of a Triangle = ___ ____________ x ___________ or

Area of a Triangle = _______________ (Do you remember why? It’s because you can insert a diagonal into any parallelogram, forming two congruent triangles. Each has half the area of the original parallelogram.) Remember that there were two formulas for the parallelogram. Since a triangle is half of a parallelogram, we can also cut the other area formula in half and get:

Area of a Triangle = __________________________,

where _____ and ______ are consecutive sides, and ______ is the included angle between them. To find the area of a triangle with sides of 6 and 10, connected by a 30° angle, it’s ½(____)(____)sin______°, or _______ square units.

½(base)(height)

½(base)(height)

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There is a third formula to find the area of a triangle called _____________________ __________________. This formula will allow you to find the area of any triangle in which you’ve been given all three sides:

Area of a Triangle = ______________________ where a, b, and c are the _______________ and s is the ______________________. (Semi-perimeter is ___________ the perimeter.) For example: A triangle has sides of 8, 12, and 16. The perimeter is ______, and semi-perimeter is _______. So the area is √_____(____ - ____)(____ - ____)(____ - ____) = √____(____)(____)(____) = √________ = 12√15 square units or ≈ 46.48 square units

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Just when you thought you knew all there was to know about the area of a triangle, along comes yet another formula. This formula is the result of using the formula for area of a triangle “ ½absinC” and the fact that the “sin60°= √3/2”. You know that the angles of an equilateral triangle each measure 60°. Put all that information together, and you get the formula:

Area of an Equilateral Triangle = _________________ or

Area of an Equilateral Triangle = _________________ So to find the area of an equilateral triangle with sides of 6, we substitute in: ____2√3 = ______√3 = ______√3 square units. 4 4

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You try: 1. Find the area of a triangle when the base and height are given, respectively. a) 28 & 13 b) 17.8 & 8 c) 15.3 & 9.8 2. Find the altitude of a triangle when the area is 190 cm2 and the base is 16 cm. 3. Two consecutive sides of a triangle are 22 and 18. Find the area, to the nearest tenth, when the included angle is 40°. 4. Two consecutive sides of a triangle are 12 and 14. Find the measure of the included angle, to the nearest degree, if the area is 67 in2.

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5. Find the area of a triangle if the sides are 8, 11 and 13. a) Simplify the radical answer. b) Round the decimal answer to the nearest thousandths place. 6. a) In simplest radical terms, find the area of an equilateral triangle with a side of 14 inches. b) Convert your radical answer in part a to a decimal answer rounded to the nearest hundredths place. 7. Find the length of a side of an equilateral triangle when its area is 81√3 u2.

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9.7 Area of a Trapezoid Remember a trapezoid is a quadrilateral with one pair of parallel sides, called the _________________. Since the trapezoid has one long base and one short base we average the bases and multiply that average by the height: Area of a Trapezoid = ___ __________ (_________ + ________)

or Area of a Trapezoid = _____________________

If our trapezoid had bases of 12 and 18, with an altitude of 10, its area would be ½(____)(____ + _____) or ____(_____) or ________ units2. Remember that the “altitude” or “height” is a line perpendicular to the bases. In some, but not all, cases it may be one of the legs. There’s another formula for area of a trapezoid. Remember that we defined the __________________ of a trapezoid as a line parallel to the bases, midway between them. The length of the median is the average of the lengths of the bases. As a result, we can also say

Area of a Trapezoid = (_____________)(_____________) or

Area of a Trapezoid = ____________

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You try: 1. The bases of a trapezoid are 16 and 28. Find the area when the altitude is a) 12 b) 7 c) 9.3 d) 6½ e) 11¾ 2. Find the area of a trapezoid when the height and median are a) 3 & 9 b) 12.6 & 3.5 c) 7.03 & 15

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3. The area of a trapezoid is 156.24 in2. Find its altitude when the median is 16.8 inches. 4. The area of a trapezoid 360 cm2. Find the length of the shorter base when the trapezoid’s height is 16 cm and the longer base is 25 cm.

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9.8 Area of a Rhombus A rhombus, as you’ll recall, is a parallelogram with congruent sides. As a result, we can use any of the parallelogram formulas to find the area of a rhombus. But the fact that a rhombus has perpendicular diagonals gives us another formula:

Area of a Rhombus = ___(_____________)(_____________) or

Area of a Rhombus = _________________ A rhombus with diagonals of 20 and 26 would have an area of ½(_____)(_____) or ____________ units2.

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You try: 1. Find the area of a rhombus given the following diagonal lengths. a) 13 & 15 b) 11.2 & 16.4 c) 26 & 28 2. Find the length of the longer diagonal if the shorter diagonal is 10 feet and the area of the rhombus is 70 square feet. 3 The length of a side of a rhombus is 17 cm. One diagonal is 16 cm. Find the area of the rhombus. 4. The length of one side of a rhombus is 32 mm. If an angle of the rhombus is 24°, what is the area to the nearest hundredth millimeter?

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9.9 Area Formulas Reference Guide Rectangle A = (base)(height) A = bh A = (length)(width) A = lw Square A = side² A = s2

A = ½(diagonal2) A = ½(d2) Parallelogram A = base x height A = bh A = abSinC

§ a & b are the lengths of 2 consecutive sides § C is the included angle

Triangle A = ½(base)( height) A = ½ bh A = ½abSinC

§ a & b are the lengths of 2 consecutive sides § C is the included angle

A = √s(s-a)(s-b)(s-c) (this is called Heron’s Formula) § a,b, & c are the sides § s = semi-perimeter = ½(a + b + c) § used when only the 3 sides are known

Equilateral Triangle A = s²√3 s = side 4 Trapezoid A = ½h(base1 + base2 ) A = ½ h(b1+b2) A = height x median A = hm

§ the median is the average of the bases Rhombus A = ½ (diagonal1)(diagonal2) A =½(d1)(d2) * any parallelogram formula can also be used

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9.10 Area Practice Problems

1. Find the area of a rectangle whose diagonal is 13 cm & whose width is 12 cm. 2. Find the area of a square whose perimeter is 26 cm. 3. Find the area of a triangle to the nearest tenth whose sides are 11, 12, & 13. 4. Find the area of a parallelogram ABCD to the nearest hundredth if AB = 12, AD = 19, & m∠A = 63°. 5. The area of a trapezoid is 144 in2 and its bases are 20 & 12. Find the length of the altitude.

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6. Find the area of an equilateral triangle whose perimeter is 48 cm. 7. Find the area of a rhombus whose diagonals measure 8.2 & 4 centimeters. 8. Find the area of a triangle if one of its angles contains 30° & the lengths of the sides including this angle are 18 & 15 inches:

9. Find the area of a square whose diagonal is 26 cm

10. The height of a triangle is 10 cm. Find the base of the triangle if its area is 95 cm².

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11. Find the area of triangle ABC, to the nearest hundredth, if AB = 14.6 cm, AC = 12.9 cm and m∠A = 46°. 12. If the area of a rhombus is 189 square yards and one diagonal is 18 yards, find the second diagonal. 13. Find the area of a right triangle whose hypotenuse is √65 inches and one leg is 4 inches. 14. A trapezoid has an area of 520 cm2 and a median of 16 cm. Find the altitude

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15. The area of a trapezoid is 58 square feet. If the height is 4 feet and the shorter base is 10 feet, find the length the longer base. 16. The area of a parallelogram is 92 in2. If the consecutive sides measure 14 and 17 inches, find the degree measure of their included angle to the nearest minute. 17. Express, in terms of x, the area of a square if the length of one side is represented by: x – 13 18. Express, in terms of x, the area of a parallelogram if the length of the base is represented by x – 5 and the height is represent by 2x + 1.

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19. The area of a parallelogram is 30 square feet and its base is 10 feet. Find the length of the altitude. 20. Find the length of a side of a square if the area is 144 square inches. 21. Find the perimeter of a square which has an area equal to that of a rectangle which has a length of 16 cm and width of 4 cm. 22. The area of a parallelogram is equal to the area of a square whose side is 8. The base of the parallelogram is represented by x + 1 and its altitude is represented by x – 11. Find the base and altitude of the parallelogram.

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23. Find the area of an isosceles right triangle if its hypotenuse is 18 inches. 24. In a triangle the sides are represented by 3x-1, 2x+5 and x+4. Find the area of the triangle if its perimeter is 32.

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9.11 Circumference of a Circle As you know, we can find the perimeter of a polygon by adding the lengths of each of its sides. But when it comes to circles, can’t take a ruler and measure the length of a curved shape. Instead of talking about perimeter, we find the _________________ of a circle. (“Circum” means “around” in Latin). The formula for circumference is

C = ___________, where r = the length of a radius.

Or, since one diameter is equal to two radii, we can say C = ___________, where d = the length of the diameter.

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9.12 Area of a Circle The formula for Area of a Circle is:

Area of a Circle = ___________ where r = the length of the radius. **There are a variety of mnemonic devices to help you remember which formula is which. Probably the easiest way to remember that Area is always measured in SQUARE units, and it’s the one that includes “r squared.” In the Circle Reference Guide (9.14) you will also see two lesser known area formulas for the circle. You try: 1. The diameter of a circle is 14. Find the circumference and area: a) in terms of pi b) using pi as 22/7 c) using the pi button on your calculator

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2. To the nearest hundredth, find the circumference and area of a circle if the radius is 2.5 cm. 3. The area of a circle is 196π sq. units. Find: a) the radius of the circle b) the diameter of the circle c) the circumference of the circle in terms of pi 4. Represent as algebraic expressions and in terms of pi, the circumference and the area of a circle whose radius is represented by x+5.

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9.13 Circle Reference Guide

The diameter of the circle is double its radius.

d = 2r

The radius of a circle is half its diameter.

r = ½d

Circumference is the distance around a circle.

Where C = circumference

Formulas: C = πd d = diameter

or r = radius

C = 2πr and π ≈ 3.14 ≈ 22/7

Area is the space inside the circle.

Formulas: A = πr2 Where A = area

or r = radius

A = ½rC d = diameter

or C = circumference

A = ¼πd2 and π ≈ 3.14 ≈ 22/7

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9.14 Area Ratios When we spoke about similar polygons, we said that the ratio of their corresponding sides or the ______________ ________ ___________________, was the same as the ratio of their corresponding altitudes, angle bisectors, medians and perimeters.

What we did NOT include was the ratio of their areas. Because area is the product of two factors, the rule changes with area.

**The ratio of the areas of two similar polygons is the ______________ of the ratio of their corresponding sides.

So if two similar polygons have sides in the ratio 2:3, the ratio of their areas is 4:9.

Example:

rectangle I rectangle II

Ratio of Similitude is 2:6 = 3:9 = ____:____ Ratio of Areas is 6:54 = 1:9 = (1:3)2

Ratio of Perimeters is 10:30 = ____:____ Ratio of Areas = (Ratio of Similitude)2

2 3

6

9

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&

These ratios also apply to circles!

* the corresponding radii, diameters and Circumferences will have the same

“ratio of similitude”. C = r C = d C’ r’ C’ d’

*The ratio of Circle Areas will be the (Ratio Of Similitude)2 A = (r)² A = (d)² A = (C)² A’ (r’)² A’ (d’)² A’ (C’)²

You try: 1. Find the area ratios of 2 similar polygons when the ratio of their corresponding sides are given. a) 1:3 b) 4:7 c) 3:5 d) 8:1 e) x:y

&

&

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2. Two corresponding sides of 2 similar polygons are 8 & 12, find the ratio of the areas in simplest fraction form. 3. Find the ratio of similitude of 2 similar polygons when the ratio of their areas are given. a) 81:4 b) 25:36 c) 16:81 d) 1:121 e) a2:b2

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4. The area ratios of two rectangles is 4:25. What is the ratio of their perimeters? 5. The diameters of two circles are 9 and 16. a) What is the ratio of their radii? b) What is the ratio of their circumferences? c) What is the ratio of their areas?

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9.15 Relationship with the Square and the Circle

When a circle is inscribed in a square, the length of the circle’s diameter is equal to the length of the side of the square.

16cm

Square’s side is 16 cm,

Circle’s diameter is 16 cm.

Circle’s radius is 8 cm.

You try:

1. A circle is inscribed in a square. Find the length of the diameter and the radius of the circle when the square’s side measures:

a) 4 ft b) 25.77 cm c) 36½ in d) 8x units

2. A circle is inscribed in a square. Find the length of the side of the square when the circle’s diameter measures:

a) 13.98 mm b) 17 in c) 23¼ ft d) (x-3) units

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3. A circle is inscribed in a square. Find the length of the side of the square when the circle’s radius measures:

a) 16 in b) 12.86 mm c) 14½ cm d) (2x+5) units

When a square is inscribed in a circle (or a circle is circumscribed about a square), the length of the square’s diagonal is equal to the measure of the circle’s diameter.

Square’s diagonal is 12 cm.

Circle’s diameter is 12 cm.

Circle’s radius is 6 cm.

You try:

1. A square is inscribed in a circle. Find the length of the diameter and the radius of the circle when the square’s diagonal measures:

a) 3 ft b) 24 cm c) 18½ in d) 14x units

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2. A square is inscribed in a circle. Find the length of the square’s diagonal when the circle’s diameter measures:

a) 10.7 cm b) 17½ in c) 3.5 mm d) (3x+4) units

3. A circle is circumscribed about a square. Find the length of the diagonal of the square when the circle’s radius measures:

a) 7½ in b) 9.75 mm c) 14 cm d) (x-6) units

4. A square is inscribed in a circle. If the side of the square measures 10 units, what is the measure of the circle’s diameter?

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5. A circle is circumscribed about a square. If the side of the square measures 6 units, what is the measure of the circle’s radius?

6. A square is inscribed in a circle. If the circle’s diameter measures 8√2 units, what is the measure of the square’s diagonal? What is the measure of the length of the side of the square?

7. A circle is circumscribed about a square. If the circle’s radius measures 7√2 units, what is the measure of the square’s diagonal? What is the measure of the length of the side of the square?

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9.16 The Equilateral Triangle and its Circles

• The length of the radius (R) of a circle circumscribed about an equilateral triangle is equal to two-thirds of the length of the height (altitude) of the triangle.

• R = 2 x height 3

• The length of the radius (r) of a circle inscribed in an equilateral triangle is equal to one-third of the length of the altitude of the triangle.

• r = 1 x height 3

• The length of the radius (R) of a circle circumscribed about an equilateral triangle is twice the length of the radius (r) of the circle inscribed in the equilateral triangle.

• R = 2r

�

Height Radius

�

radius

Height

�

radius

Radius

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You Try:

1. Find the length of the radius of the circle inscribed in an equilateral ∆ the length of whose altitude is 15.

2. Find the length of the radius of the circle circumscribed about an equilateral ∆ the length of whose altitude is 18.

3. Find the length of the radius of the circle inscribed in an equilateral ∆ the length of whose side is 18.

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4. Find the length of the radius of the circle circumscribed about an equilateral ∆the length of whose side is 18.

5. The length of the radius of a circle circumscribed about an equilateral ∆ is 12. Find the length of the altitude of the ∆.

6. Find, in radical form, the length of the radius of a circle circumscribed about and equilateral ∆ the length of whose side is 24.

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9.17 Shaded Area

Sometimes it is possible to find the area of a shape composed of two or more other shapes. Very often we shade the area in question, and use addition or subtraction to find the new area. As a basic rule of thumb: if you would have to use tape to connect the original figures, you’ll need to ADD them. If it would take scissors to cut a piece or pieces away, you’ll have to SUBTRACT.

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9.18 Regular Polygon Area

A Regular Polygon is both __________________ and ____________________. - Examples of regular polygons: a square, a STOP sign, the Pentagon in Washington D.C. The ______________________ OF A REGULAR POLYGON is the radius of the circumscribed circle. In the diagram above, the purple line in the hexagon is a radius of the circumscribed circle, yet touches the vertex of the polygon. That makes it the RADIUS of the polygon. On the other hand, the ______________________ of a circle is the radius of the inscribed circle. In the diagram above, the purple line is the APOTHEM.

⋅

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The apothem of a regular polygon is the __________________ ________________ of the side. It hits the midpoint of the side, and forms right angles with the side. A ___________________ ANGLE of a polygon is one with its vertex at the center of the polygon, and whose sides are consecutive radii. To find the measure of a central angle, the formula is:

Central Angle = _______

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How to find the area of any regular polygon: EX: Find the area of a regular hexagon whose sides measure 12 cm. ***ALWAYS follow these 5 steps:

1. Find the central angle using the formula 360/n.

2. Draw an isosceles triangle. Draw the apothem to create the right triangle, cutting both the base and the central angle in half. 3. Use the right triangle and Trig to find the length of the apothem. 4. Find the perimeter: (number of sides) x (length of each side) 5. Find the area using the formula A = ½ apothem*perimeter

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You Try: 1. Find the area of a regular decagon whose sides are 18 cm.

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2. Find the area of a regular 12-sided polygon whose sides are 16 cm.

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3. Find the area of a regular octagon whose sides are 14 cm.

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4. Find the area of a regular 16-sided polygon whose sides are 40 yd.

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5. Find the area of a regular pentagon whose sides are 9 cm.

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6. Find the area of a regular nonagon whose apothem is 4 inches.

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7. Find the area of a regular 20-sided polygon whose radius is 100 in.

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8. Find the area of a regular nonagon whose radius is 12 cm.