Area of Rectangles 2 Return to Table of Contents 3 Slide 7 / 152 - PDF document

Slide 1 / 152 Slide 2 / 152 Geometry Area of Figures 2015-10-27 www.njctl.org Slide 3 / 152 Slide 4 / 152 Table of Contents Throughout this unit, the Standards for Mathematical Practice are used. Click on the topic to go to that section Area of Rectangles MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. Area of Triangles MP3: Construct viable arguments and critique the reasoning of Law of Sines others. MP4: Model with mathematics. Area of Parallelograms MP5: Use appropriate tools strategically. MP6: Attend to precision. Area of Regular Polygons MP7: Look for & make use of structure. Area of Circles & Sectors MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Area of Other Quadrilaterals Practice" Pull-tabs (e.g. a blank one is shown to the right on Area & Perimeter of Figures in the Coordinate Plane this slide) with a reference to the standards used. PARCC Sample Questions If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. Slide 5 / 152 Slide 6 / 152 Area of a Rectangle The area of a rectangle is defined to be the number of squares of area "1" that can fit within it. In the below drawing, 6 unit squares fit within the below rectangle of height 2 and based 3. Area of Rectangles 2 Return to Table of Contents 3

Slide 7 / 152 Slide 8 / 152 Area of a Rectangle Area of a Rectangle In general, the area of a rectangle is equal to its base times its height. This can also be referred to as its length times its width. Sometimes, the dimensions will not be given, so you will need to calculate them before calculating the area. A rectangle = length x width (lw) = base x height (bh) Since a rectangle is a quadrilateral with 4 right angles, 2 right triangles can be formed when drawing one of its diagonals. Therefore, Pythagorean Theorem can be a helpful formula. Another helpful formula is the perimeter formula for a rectangle: P = 2 l + 2 w There might also be questions asking you about the population density h of a town, state, or country. It is the ratio that represents the number of people living per square mile and can be found by dividing the total population by the total area. b Slide 9 / 152 Slide 10 / 152 Example 1 What is the area of a rectangle that has a length of 8.4 The diagonal of a rectangle is 34 feet and its length is 14 feet more cm and a width of 3.7 cm? than its width. Find the length, width, and area of the rectangle. x + 14 We know that the width is unknown, so let's call it "x". x Therefore, the length will be "x + 14". 34 ft Using the Pythagorean Theorem & our Algebra skills, we can solve for x. x 2 + (x + 14) 2 = 34 2 click x 2 + x 2 + 28x + 196 = 1,156 click 2x 2 + 28x - 980 = 0 Since x is the length of a side, it click 2(x 2 + 14x - 480) = 0 has to be positive. Therefore, our final answer is click 2(x + 30)(x - 16) = 0 x = 16 ft = width click x + 30 = 0 or x - 16 = 0 length = 30 ft click Area = 480 ft 2 x = -30 or x = 16 click click Slide 11 / 152 Slide 12 / 152 2 Televisions, are advertised using the length of the 3 The population density is the amount of people living per diagonal. For example, a 26" TV could have a length of square mile. If the town of Geometryville is a rectangular 24" and a width of 10", as shown below. town that has a length of 24 miles and a width of 13 miles, and its population is 2,500 people, what is the 24" population density of the town? 10" 26" What is the area of an 80" TV if the length is 69.3"?

Slide 13 / 152 Slide 14 / 152 4 The diagonal of a rectangle is 10 feet and its width is 2 5 The diagonal of a rectangle is 10 feet and its width is 2 feet less than its length. What is the length of the feet less than its length. What is the width of the rectangle? rectangle? A 4 feet A 4 feet B 6 feet B 6 feet C 8 feet C 8 feet D 9 feet D 9 feet Slide 15 / 152 Slide 16 / 152 6 The diagonal of a rectangle is 10 feet and its width is 2 feet less than its length. What is the area of the rectangle? A 80 ft 2 B 60 ft 2 Area of Triangles C 48 ft 2 D 24 ft 2 Return to Table of Contents Slide 17 / 152 Slide 18 / 152 Area of a Right Triangle Area of Any Triangle Now, let's find the area formula for an arbitrary scalene triangle ABC Clearly, the area of each of the below right triangles is equal to half with base "b" and height "h." Keep in mind that the height is always the area of the rectangle they comprise. measured perpendicular to the base that is opposite the vertex. Since the area of the rectangle is bh, the area of each right triangle is 1/2 bh. A h A Δ = 1/2 bh h C B b b

Slide 19 / 152 Slide 20 / 152 Area of Any Triangle Area of Any Triangle Area ΔADC = 1/2 b'h Let's draw right triangle ACD such that the new triangle plus the click original triangle form a larger right triangle ABD. Area ΔADB = 1/2 (b' + b) h click ΔADC is a right Δ with base "b' " and height "h". Area ΔABC = Area ΔADB - Area ΔADC ΔADB is a right Δ with base "b' + b" and height "h". click Area ΔABC = 1/2 (b' + b) h - 1/2 b'h click Area ΔABC = 1/2 b'h + 1/2bh - 1/2b'h click A Area ΔABC = 1/2 bh A click h h D C B b' b D b' C B b Slide 21 / 152 Slide 22 / 152 Area of Any Triangle Area of Any Triangle So, the area of any triangle, not just right triangles, is given by : While the formula is the same for any triangle: Area Δ = 1/2 bh For triangles that are not right triangles, the height is usually not Area Δ = 1/2 bh directly given. A Instead, you may be given the lengths of one or more sides and the measures of one or more angles. A h For instance, in this case 8 C B how would you find the 5 b height...and then the area? 40 ° C B 4 Slide 23 / 152 Slide 24 / 152 Area of Any Triangle Area of Any Triangle Draw an altitude from the vertex to the base, creating a right triangle sin θ = opposite / hypotenuse = opp / hyp with the same height as our original triangle. click opp = (hyp)(sin θ) ΔABD has the same height as our original triangle ΔABC. click h = (8)(sin 40°) click And both triangles share a side, AB, and an angle, ∠ B. h = (8)(0.64) click Which trig function would allow us to find the value of "h"? h = 5.1 units A click A Now that you have the base h 8 and the height of 5 h 5 8 ΔABC, how would you find the 40 ° area? 40 ° D C B 4 D C B 4

Slide 25 / 152 Slide 26 / 152 Area of Any Triangle Area of Any Triangle Examining our solution, we can see that the area of the triangle is given by half the product of two sides multiplied by the sine of the A = 1/2 bh angle between them. A = 1/2 (4)(8)(sin 40°) In this case, the sides were of length 4 and 8 and the angle is 40°. A = 1/2 (4)(8)(0.64) A = 10.2 units 2 A = 1/2 bh A A A = 1/2 (4)(8)(sin 40°) A = 1/2 (4)(8)(0.64) A = 10.2 units 2 h 5 8 h 5 8 40 ° 40 ° D C B D C B 4 4 Slide 27 / 152 Slide 28 / 152 Example Area of Any Triangle Replacing the numbers by labeling the angles with upper case letters and the sides opposite them with matching lower case letters Find the area of ΔABC. yields this formula. In this case, the altitude you draw will be within the triangle. A Δ = 1/2 ac sinB But, there's nothing special about those sides and that angle, so it will A also be true that: A Δ = 1/2 ab sinC 4 8 Or A Δ = 1/2 bc sinA A 50 ° c b The area of a triangle is C B 9 equal to the product of any two sides and the sine C B of the included angle. a Slide 29 / 152 Slide 30 / 152 7 Find the area of ΔDEF. Round your answer to the 8 Find the area of Δ GHI . Round your answer to the nearest hundredth. nearest hundredth. E H 16 in. 8 in. 28° I G 34° 9 in. F D 12 in.

Slide 31 / 152 Slide 32 / 152 9 Find the area of ΔJKL. Round your answer to the 10 Find the area of ΔPQR. Round your answer to the nearest hundredth. nearest hundredth. K Q 15 in. 9 in. 52° 32° R P J L 14 in. 6 in. Slide 33 / 152 Slide 34 / 152 Law of Sines We just learned that we can find the area of any triangle with any of these three formulas: A Δ = 1/2 ac sinB A Δ = 1/2 ab sinC A Δ = 1/2 bc sinA Law of Sines Since the area of a triangle will be the same regardless of which formula we use, these three formulas must be equal. Setting them equal to one another will give us A a general relationship between the sides and the c b angles of a triangle. Return to Table of Contents C B a Slide 35 / 152 Slide 36 / 152 Law of Sines Law of Sines This relationship between the sides and angles of a triangle will be true for all triangles. A Δ = 1/2 ac sinB = 1/2 ab sinC = 1/2 bc sinA a b c = = sinA sinB sinC Let's look at one pair at a time and simplify. OR 1/2 ac sinB = 1/2 ab sinC AND 1/2 ab sinC = 1/2 bc sinA sinA sinB sinC = = a b c ac sinB = ab sinC ab sinC = bc sinA click click c sinB = b sinC a sinC = c sinA click click A c / sinC = b / sinB a / sinA = c / sinC c b click click a b c = = sinA sinB sinC C B click a