LESSON 6.3 – FINDING PERIMETER AND AREA IN THE COORDINATE PLANE CONCEPT: DISTANCE IN THE COORDINATE PLANE EQ: HOW DO WE FIND AREA & PERIMETER IN THE COORDINATE PLANE? (G.GPE.7) VOCABULARY: DISTANCE FORMULA, POLYGON, AREA, PERIMETER 6.2.2: CALCULATING PERIMETER AND AREA
1
THINK-PAIR-SHARE • Think back to the distance formula and when you used it. Take a minute and write down everything you remember about using the distance formula. • With your partner, compare your notes to see if you missed anything. • Wait to be called on and then share your answers with the class.
6.2.2: CALCULATING PERIMETER AND AREA
2
INTRODUCTION In the previous lesson, the distance formula was used to find the distance between two given points. In this lesson, the distance formula will be applied to perimeter and area problems. A polygon is a two-dimensional figure formed by three or more segments. We will use the distance formula to find the perimeter, or the sum of the lengths of all the sides of a polygon, and the area, the number of square units inside of a polygon, such as finding the amount of carpeting needed for a room. Be sure to use the appropriate units (inches, feet, yards, etc.) with your answers.
6.2.2: CALCULATING PERIMETER AND AREA
3
AREA OF A PARALLELOGRAM • A parallelogram includes shapes such as squares, rectangles, rhombuses. • The area of a parallelogram is found using the formula: Area = 𝐛𝐡 (𝒃𝒂𝒔𝒆 ∗ 𝒉𝒆𝒊𝒈𝒉𝒕) • The length of the base and height are found using the distance formula. • The final answer must include the appropriate label (units², feet², inches², meters², centimeters², etc.)
6.2.2: CALCULATING PERIMETER AND AREA
4
GUIDED PRACTICE, EXAMPLE 1
Parallelogram ABCD has vertices A (-5, 4), B (3, 4), C (5, -1), and D (-3, -1). Calculate the area and perimeter of parallelogram ABCD.
6.2.2: CALCULATING PERIMETER AND AREA
5
EXAMPLE 1, CONTINUED We need to find the length of all four sides before we can find the area and the perimeter. So we will use the distance formula: 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐
𝑨 −𝟓, 𝟒
𝑩 𝟑, 𝟒
𝑪 𝟓, −𝟏
𝟑 − −𝟓
𝟐
+ 𝟒−𝟒
𝟐
=
𝟑+𝟓
𝟐
+ 𝟒−𝟒
𝟐
=
𝟖
• 𝑨𝑩 =
𝟐
+ 𝟎
𝑫(−𝟑, −𝟏)
𝟐
= 𝟔𝟒 + 𝟎 = 𝟔𝟒 = 𝟖 • The length of 𝑨𝑩 is 8 units
6.2.2: CALCULATING PERIMETER AND AREA
6
EXAMPLE 1, CONTINUED 𝒙𝟐 − 𝒙𝟏
Distance formula:
𝑨 −𝟓, 𝟒 • 𝑩𝑪 = =
+ 𝒚𝟐 − 𝒚𝟏
𝑩 𝟑, 𝟒
𝟓−𝟑 𝟐
𝟐
𝟐
𝟐
+ −𝟏 − 𝟒
+ −𝟓
𝑪 𝟓, −𝟏
𝟐
𝑫(−𝟑, −𝟏)
𝟐
𝟐
= 𝟒 + 𝟐𝟓 = 𝟐𝟗 = 𝟓. 𝟑𝟗 • The length of 𝑩𝑪 is 5.39 units
6.2.2: CALCULATING PERIMETER AND AREA
7
EXAMPLE 1, CONTINUED 𝒙𝟐 − 𝒙𝟏
Distance formula:
𝑨 −𝟓, 𝟒 • 𝑪𝑫 =
𝟐
𝑩 𝟑, 𝟒
−𝟑 − 𝟓
𝟐
=
−𝟑 − 𝟓
=
−𝟖
𝟐
+ 𝒚𝟐 − 𝒚𝟏
𝑪 𝟓, −𝟏
+ −𝟏 − −𝟏
𝟐
𝟐
𝟐
+ −𝟏 + 𝟏
+ 𝟎
𝟐
𝑫(−𝟑, −𝟏)
𝟐
= 𝟔𝟒 + 𝟎
= 𝟔𝟒 = 𝟖 • The length of 𝑪𝑫 is 8 units
6.2.2: CALCULATING PERIMETER AND AREA
8
EXAMPLE 1, CONTINUED 𝒙𝟐 − 𝒙𝟏
Distance formula:
𝑨 −𝟓, 𝟒
𝟐
+ 𝒚𝟐 − 𝒚𝟏
𝑩 𝟑, 𝟒
𝑪 𝟓, −𝟏
−𝟑 − −𝟓
𝟐
+ −𝟏 − 𝟒
𝟐
=
−𝟑 + 𝟓
𝟐
+ −𝟏 − 𝟒
𝟐
=
𝟐
• 𝑨𝑫 =
𝟐
+ −𝟓
𝟐
𝑫(−𝟑, −𝟏)
𝟐
= 𝟒 + 𝟐𝟓
= 𝟐𝟗 = 𝟓. 𝟑𝟗 • The length of 𝑨𝑫 is 5.39 units
6.2.2: CALCULATING PERIMETER AND AREA
9
EXAMPLE 1, CONTINUED 𝐴𝐵 = 8 units 𝐵𝐶 = 5.39 units 𝐶𝐷 = 8 units 𝐴𝐷 = 5.39 units • Find the perimeter by adding up all the sides: 𝑷 = 𝟖 + 𝟓. 𝟑𝟗 + 𝟖 + 𝟓. 𝟑𝟗 = 𝟐𝟔. 𝟕𝟖 𝒖𝒏𝒊𝒕𝒔 Find the area by using the formula 𝑎𝑟𝑒𝑎 = 𝑏ℎ • 𝐴𝐵 or 𝐶𝐷 is the base and they are the same length so 𝑏 = 8 • The height can be found by drawing a perpendicular line straight up from D to side 𝐴𝐵 and down from B to side 𝐶𝐷. • You can do this by counting the units or using the distance formula • Finding the distance from D to the point (−3, 4) and the distance from B to the point where the perpendicular line touches 𝐶𝐷 at (4, −1)
6.2.2: CALCULATING PERIMETER AND AREA
10
EXAMPLE 1, CONTINUED Area of a Parallelogram = 𝒃𝒉 • Base = 8 units • Height = 5 units • Area of a parallelogram = 𝒃𝒉 = 𝟖 ∗ 𝟓
= 𝟒𝟎 𝒖𝒏𝒊𝒕𝒔𝟐
6.2.2: CALCULATING PERIMETER AND AREA
11
AREA OF A TRIANGLE • The area of a triangle is found by using the formula: 𝟏 𝟏 Area = 𝒃𝒉 ∗ 𝒃𝒂𝒔𝒆 ∗ 𝒉𝒆𝒊𝒈𝒉𝒕 𝟐
𝟐
• The height of a triangle is the perpendicular distance from a vertex to the base of the triangle.
• Determining the lengths of the base and the height is necessary if these lengths are not stated in the problem. • The final answer must include the appropriate label (units², feet², inches², meters², centimeters², etc.)
6.2.2: CALCULATING PERIMETER AND AREA
12
GUIDED PRACTICE, EXAMPLE 2
Triangle ABC has vertices A (2, 1), B (4, 5), and C (7, 1). Calculate the perimeter and area of triangle ABC.
6.2.2: CALCULATING PERIMETER AND AREA
13
EXAMPLE 2 We need to find the length of all three sides before we can find the area and the perimeter. So we will use the distance formula: 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐
𝑨 𝟐, 𝟏 • 𝑨𝑩 = =
𝟒−𝟐 𝟐
𝟐
𝟐
+ 𝟓−𝟏
+ 𝟒
𝑩 𝟒, 𝟓
𝑪(𝟕, 𝟏)
𝟐
𝟐
= 𝟒 + 𝟏𝟔
= 𝟐𝟎 = 𝟒. 𝟒𝟕 • The length of 𝑨𝑩 is 4.47 units
6.2.2: CALCULATING PERIMETER AND AREA
14
EXAMPLE 2, CONTINUED 𝒙𝟐 − 𝒙𝟏
Distance Formula:
𝑨 𝟐, 𝟏 • 𝑨𝑪 =
=
𝟕−𝟐
𝟓
𝟐
𝟐
+ 𝟏−𝟏
+ 𝟎
𝟐
+ 𝒚𝟐 − 𝒚𝟏
𝑩 𝟒, 𝟓
𝟐
𝑪 𝟕, 𝟏
𝟐
𝟐
= 𝟐𝟓 + 𝟎 = 𝟐𝟓 = 𝟓 • The length of 𝑨𝑪 is 5 units.
6.2.2: CALCULATING PERIMETER AND AREA
15
EXAMPLE 2, CONTINUED 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐 𝑨 𝟐, 𝟏 𝑩 𝟒, 𝟓 𝑪(𝟕, 𝟏)
Distance Formula:
• 𝑩𝑪 = =
𝟕−𝟒 𝟑
𝟐
𝟐
+ 𝟏−𝟓
+ −𝟒
𝟐
𝟐
= 𝟗 + 𝟏𝟔 = 𝟐𝟓 =5 • The length of 𝑩𝑪 is 5 units.
6.2.2: CALCULATING PERIMETER AND AREA
16
EXAMPLE 2, CONTINUED 𝑨𝑩 = 𝟒. 𝟒𝟕 𝐮𝐧𝐢𝐭𝐬
𝑨𝑪 = 𝟓 𝐮𝐧𝐢𝐭𝐬
𝑩𝑪 = 𝟓 𝐮𝐧𝐢𝐭𝐬
• Find the perimeter by adding up all the sides: 𝑷 = 𝟒. 𝟒𝟕 + 𝟓 + 𝟓 = 𝟏𝟒. 𝟒𝟕 𝒖𝒏𝒊𝒕𝒔 𝟏 𝟐
Find the area by using the formula 𝒂𝒓𝒆𝒂 = 𝒃𝒉 • 𝐴𝐶 is the base so 𝑏 = 5 • The height can be found by drawing a perpendicular line straight down from B to side 𝐴𝐶. • Then find the distance from B to the point where the perpendicular line touches 𝐴𝐶 at (4,1) • You can do this by counting the units or using the distance formula
6.2.2: CALCULATING PERIMETER AND AREA
17
EXAMPLE 2, CONTINUED 𝟏 𝟐
Area of a Triangle = 𝒃𝒉 • Base = 5 units • Height = the distance from 𝑩 𝟒, 𝟓 to 𝟒, 𝟏 = 4 𝟒−𝟒 𝟐+ 𝟏−𝟓 𝟐 = 𝟎 𝟐 + −𝟒 𝟐 = 𝟎 + 𝟏𝟔 = 𝟏𝟔 = 𝟒 𝟏 𝟐
• Area of a triangle = 𝒃𝒉 𝟏 = 𝟓 𝟒 𝟐 = 𝟏𝟎 𝒖𝒏𝒊𝒕𝒔𝟐
6.2.2: CALCULATING PERIMETER AND AREA
18
AREA OF A TRAPEZOID • The area of a trapezoid is found by using the formula: Area = 𝟏/𝟐 𝒃𝟏 + 𝒃𝟐 𝒉 • A trapezoid has a smaller base (𝒃𝟏 ) and a larger base (𝒃𝟐 ) . You will need to add both bases together in the area formula. • The height of a trapezoid is the perpendicular distance from a vertex to the base of the trapezoid. • The final answer must include the appropriate label (units², feet², inches², meters², centimeters², etc.)
6.2.2: CALCULATING PERIMETER AND AREA
19
GUIDED PRACTICE, EXAMPLE 3
Trapezoid EFGH has vertices E (-8, 2), F (-4, 2), G (-2, -2), and H (-10, -2). Calculate the perimeter and area of trapezoid EFGH.
6.2.2: CALCULATING PERIMETER AND AREA
20
EXAMPLE 3, CONTINUED We need to find the length of all four sides before we can find the area and the perimeter. So we will use the distance formula: 𝒙𝟐 − 𝒙𝟏 𝟐 + 𝒚𝟐 − 𝒚𝟏 𝟐
𝑬 −𝟖, 𝟐
𝑭 −𝟒, 𝟐
𝑮 −𝟐, −𝟐
−𝟒 − −𝟖
𝟐
+ 𝟐−𝟐
𝟐
=
−𝟒 + 𝟖
𝟐
+ 𝟐−𝟐
𝟐
=
𝟒
• 𝑬𝑭 =
𝟐
+ 𝟎
𝑯(−𝟏𝟎, −𝟐)
𝟐
= 𝟏𝟔 + 𝟎 = 𝟏𝟔 = 𝟒 • The length of 𝑬𝑭 is 4 units
6.2.2: CALCULATING PERIMETER AND AREA
21
EXAMPLE 3, CONTINUED 𝒙𝟐 − 𝒙𝟏
Distance formula:
𝑬 −𝟖, 𝟐
𝑭 −𝟒, 𝟐
𝟐
+ 𝒚𝟐 − 𝒚𝟏
𝑮 −𝟐, −𝟐
−𝟐 − −𝟒
𝟐
+ −𝟐 − 𝟐
𝟐
=
−𝟐 + 𝟒
𝟐
+ −𝟐 − 𝟐
𝟐
=
𝟐
• 𝑭𝑮 =
𝟐
+ −𝟒
𝟐
𝑯(−𝟏𝟎, −𝟐)
𝟐
= 𝟒 + 𝟏𝟔
= 𝟐𝟎 = 𝟒. 𝟒𝟕 • The length of 𝑭𝑮 is 4.47 units
6.2.2: CALCULATING PERIMETER AND AREA
22
EXAMPLE 3, CONTINUED 𝒙𝟐 − 𝒙𝟏
Distance formula:
𝑬 −𝟖, 𝟐 • 𝑮𝑯 =
𝟐
𝑭 −𝟒, 𝟐 −𝟏𝟎 − −𝟐
=
−𝟏𝟎 + 𝟐
=
−𝟖
𝟐
𝟐 𝟐
+ 𝟎
+ 𝒚𝟐 − 𝒚𝟏
𝟐
𝑮 −𝟐, −𝟐
+ −𝟐 − −𝟐 + −𝟐 + 𝟐
𝑯(−𝟏𝟎, −𝟐)
𝟐
𝟐
𝟐
= 𝟔𝟒 + 𝟎
= 𝟔𝟒 = 𝟖 • The length of 𝑮𝑯 is 8 units
6.2.2: CALCULATING PERIMETER AND AREA
23
EXAMPLE 3, CONTINUED 𝒙𝟐 − 𝒙𝟏
Distance formula:
𝑬 −𝟖, 𝟐
𝑭 −𝟒, 𝟐 −𝟏𝟎 − −𝟖
𝟐
=
−𝟏𝟎 + 𝟖
𝟐
=
−𝟐
• 𝑬𝑯 =
𝟐
𝟐
+ 𝒚𝟐 − 𝒚𝟏
𝑮 −𝟐, −𝟐
+ −𝟐 − 𝟐 + −𝟐 − 𝟐
+ −𝟒
𝟐
𝑯(−𝟏𝟎, −𝟐)
𝟐 𝟐
𝟐
= 𝟒 + 𝟏𝟔
= 𝟐𝟎 = 𝟒. 𝟒𝟕 • The length of 𝑬𝑯 is 4.47 units
6.2.2: CALCULATING PERIMETER AND AREA
24
EXAMPLE 3, CONTINUED 𝑬𝑭 = 𝟒 𝐮𝐧𝐢𝐭𝐬 𝑭𝑮 = 𝟒. 𝟒𝟕 𝐮𝐧𝐢𝐭𝐬 𝑮𝑯 = 𝟖 𝐮𝐧𝐢𝐭𝐬 𝑬𝑯 = 𝟒. 𝟒𝟕 𝐮𝐧𝐢𝐭𝐬 • Find the perimeter by adding up all the sides: 𝑷 = 𝟒 + 𝟒. 𝟒𝟕 + 𝟖 + 𝟒. 𝟒𝟕 = 𝟐𝟎. 𝟗𝟒 𝒖𝒏𝒊𝒕𝒔 •
Find the area by using the formula 𝒂𝒓𝒆𝒂 =
𝟏 𝟐
𝒃𝟏 + 𝒃𝟐 𝒉
• 𝑬𝑭 𝒊𝒔 𝒃𝟏 𝒂𝒏𝒅 𝑮𝑯 𝒊𝒔 𝒃𝟐 . So 𝒃𝟏 = 𝟒 𝒂𝒏𝒅 𝒃𝟐 = 𝟖. • The height can be found by drawing a perpendicular line straight down from E to side 𝐺𝐻 or from F to side 𝐺𝐻. • You can do this by counting the units or using the distance formula
6.2.2: CALCULATING PERIMETER AND AREA
25
EXAMPLE 3, CONTINUED Area of a Trapezoid =
𝟏 𝟐
𝒃𝟏 + 𝒃𝟐 𝒉
• 𝒃𝟏 = 𝟒 units • 𝒃𝟐 = 𝟖 units • Height = 4 units • Area of a trapezoid =
𝟏 𝟐
𝒃𝟏 + 𝒃𝟐 𝒉 𝟏 = (𝟒 + 𝟖)(𝟒) 𝟐
= 𝟐𝟒 𝒖𝒏𝒊𝒕𝒔𝟐
6.2.2: CALCULATING PERIMETER AND AREA
26
YOU TRY! Find the perimeter and area of rectangle JKLM. • Reminder: Perimeter = sum of all the sides Area = 𝒃𝒉 (𝒃𝒂𝒔𝒆 ∗ 𝒉𝒆𝒊𝒈𝒉𝒕)
6.2.2: CALCULATING PERIMETER AND AREA
27
3-2-1 3 – List three things you learned from this lesson. 2 – List two things you used in this lesson that you learned in previous lessons.
1 – Write one question you still have about area and perimeter of polygons.
6.2.2: CALCULATING PERIMETER AND AREA
28
