Area & Perimeter in the Coordinate Plane | Geometry | Eat Pi

Summary

This educational video script offers a clear guide on calculating the area and perimeter of shapes and polygons in the coordinate plane. The presenter explains the formula for a triangle's area as half the base times the height and demonstrates using a grid. For the perimeter, the Pythagorean theorem is applied to find the hypotenuse. The script then shifts to rectangles, introducing the distance formula to find the length and width when the grid isn't aligned, and uses it to calculate the area and perimeter. The explanation is practical, with step-by-step instructions and a friendly tone, making it accessible for viewers to grasp geometrical concepts.

Takeaways

• 📐 The area of a triangle can be found using the formula: \( \frac{1}{2} \times \text{base} \times \text{height} \).
• 📏 When shapes are aligned with the grid, it's easier to determine the base and height for calculating area.
• 📍 The base and height of a triangle in the coordinate plane can be measured directly from the grid.
• 🔢 For the given triangle example, the base is 7 units and the height is 5 units, resulting in an area of 17.5 square units.
• 🧩 To find the perimeter of a triangle, sum all its sides, including the hypotenuse found using the Pythagorean theorem.
• 📐 The Pythagorean theorem states that \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
• 🔑 The hypotenuse in the triangle example is approximately 8.6 units, calculated using the Pythagorean theorem with sides 7 and 5.
• 📏 The perimeter of the triangle is the sum of its sides, which is approximately 20.6 units.
• 📐 For shapes not aligned with the grid, like a rectangle, use the distance formula to find the length and width.
• 📏 The distance formula is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for finding the distance between two points.
• 🔢 The area of a rectangle is calculated by multiplying its length by its width, which are not necessarily aligned with the grid.
• 📐 The perimeter of a rectangle can be found by adding twice the length and twice the width.
• 📝 The video provides a step-by-step guide on using the distance formula and the Pythagorean theorem for calculating areas and perimeters in the coordinate plane.

Q & A

What is the formula to calculate the area of a triangle?

The formula to calculate the area of a triangle is one half the base times the height (A = 1/2 * base * height).

How can you determine the base and height of a triangle on a coordinate grid?

On a coordinate grid, the base can be determined by counting the horizontal spaces from one vertex to another, and the height is the vertical distance from the base to the opposite vertex.

What is the area of the triangle in the video if the base is 7 units and the height is 5 units?

The area of the triangle would be 17.5 square units (A = 1/2 * 7 * 5 = 17.5 units²).

How do you find the perimeter of a polygon when you know the lengths of its sides?

To find the perimeter, you add up the lengths of all the sides of the polygon.

What is the Pythagorean theorem used for in the context of the video?

The Pythagorean theorem (a² + b² = c²) is used to find the length of the hypotenuse of a right-angled triangle when the lengths of the other two sides are known.

How is the distance formula applied to find the length of a side of a rectangle on a coordinate grid?

The distance formula (√((x2 - x1)² + (y2 - y1)²)) is used to calculate the distance between two points, which can be the length or width of a rectangle if the points are its vertices.

What is the formula for the area of a rectangle?

The formula for the area of a rectangle is the length times the width (A = length * width).

How do you calculate the perimeter of a rectangle?

The perimeter of a rectangle is calculated by adding twice the length and twice the width (P = 2 * length + 2 * width).

What is the approximate perimeter of the rectangle in the video with a length of 12.7 units and a width of 4.2 units?

The approximate perimeter of the rectangle is 33.8 units (P = 2 * 12.7 + 2 * 4.2 ≈ 33.8 units).

What is the distance formula and how is it used to find the area of a shape on a coordinate grid?

The distance formula is used to calculate the distance between two points in a coordinate plane. It is used to find the lengths of sides of shapes, which can then be used to calculate the area of polygons by applying the appropriate area formula for the shape.

How can you find the area of a shape that is not aligned with the grid on a coordinate plane?

For shapes not aligned with the grid, you can use the distance formula to find the lengths of the sides of the shape, and then apply the area formula specific to that shape (e.g., for a rectangle, use length * width).

Outlines

00:00

📐 Geometry Basics: Perimeter and Area of Shapes

This paragraph introduces the concepts of finding the perimeter and area of shapes and polygons on a coordinate plane. The presenter uses a triangle as an example, explaining how to calculate its area with the formula (1/2) * base * height. They demonstrate how to measure the base and height directly from the grid and then apply the Pythagorean theorem to find the hypotenuse, which is necessary for calculating the perimeter. The process is detailed with specific numerical examples, making it easy to follow.

05:02

📏 Advanced Calculations: Area and Perimeter of a Rotated Rectangle

In this paragraph, the focus shifts to calculating the area and perimeter of a rectangle that is not aligned with the grid, requiring the use of the distance formula. The presenter explains how to find the length and width of the rectangle by applying the distance formula to the coordinates of the rectangle's vertices. They provide step-by-step instructions on how to calculate the length and width, including squaring and taking the square root of the sum of squared differences in x and y coordinates. After finding the dimensions, they calculate the area by multiplying the length by the width and the perimeter by adding twice the length and twice the width. The explanation includes rounding off to approximate values for practical purposes.

Keywords

💡Perimeter

Perimeter refers to the total distance around the outside of a two-dimensional shape. In the video, the concept is used to calculate the total length of the sides of a triangle and a rectangle in the coordinate plane, which is essential for understanding the properties of geometric shapes.

💡Area

Area is the amount of space enclosed within a two-dimensional shape and is typically measured in square units. The video explains how to find the area of a triangle using the formula (1/2) * base * height and for a rectangle using the distance formula, which is crucial for understanding the size of shapes in the coordinate plane.

💡Triangle

A triangle is a polygon with three edges and three vertices. The video script uses a triangle to demonstrate the calculation of area and perimeter, emphasizing the importance of identifying the base and height for area calculation and applying the Pythagorean theorem for finding the hypotenuse.

💡Base

In the context of geometry, the base of a triangle is one of its sides, typically the horizontal side when the triangle is positioned with respect to the coordinate plane. The script explains how the base's length is used in the area formula for a triangle.

💡Height

Height, in geometry, is the perpendicular distance from the base of a triangle to the opposite vertex. The script illustrates how to measure the height of a triangle to use in the area calculation formula.

💡Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The video uses this theorem to find the missing side of a triangle in the coordinate plane.

💡Hypotenuse

The hypotenuse is the longest side of a right-angled triangle, opposite the right angle. The script demonstrates how to calculate the length of the hypotenuse using the Pythagorean theorem, which is key to finding the perimeter of a triangle.

💡Rectangle

A rectangle is a quadrilateral with four right angles. The video explains how to calculate the area of a rectangle that is not aligned with the grid lines of the coordinate plane, using the distance formula instead of counting squares.

💡Distance Formula

The distance formula is used to calculate the distance between two points in the coordinate plane. It is given by the formula √((x2 - x1)² + (y2 - y1)²). The video script uses this formula to find the length and width of a rectangle for area and perimeter calculations.

💡Coordinate Plane

The coordinate plane is a two-dimensional plane where points are located by a pair of numerical coordinates, determining their position relative to two perpendicular axes. The script discusses finding the perimeter and area of shapes within this plane, emphasizing the grid's role in geometric calculations.

Highlights

Introduction to finding the perimeter and area of shapes and polygons in the coordinate plane.

Explanation of the formula for the area of a triangle: 1/2 * base * height.

Demonstration of calculating the area of a triangle using a grid for easy measurement.

Calculation of a triangle's area with base length of 7 units and height of 5 units.

Conversion of area calculation result to units squared.

Introduction to finding the perimeter of a triangle using the Pythagorean theorem.

Application of the Pythagorean theorem to find the hypotenuse of a triangle.

Calculation of the hypotenuse with sides of 7 and 5 units resulting in approximately 8.6 units.

Summation of all sides to find the perimeter of the triangle: 7 + 5 + 8.6 = 20.6 units.

Introduction to finding the area of a rectangle using the distance formula.

Explanation of the distance formula and its application to non-aligned shapes.

Calculation of the length of a rectangle using the distance between two points (3,8) and (-6,-1).

Finding the width of the rectangle using the distance between points (3,-8) and (6,-5).

Calculation of the rectangle's area using the formula: length * width.

Conversion of area to units squared and rounding to an approximate value.

Introduction to calculating the perimeter of a rectangle using the formula: 2 * (length + width).

Final calculation of the rectangle's perimeter as 33.8 units.

Encouragement for viewers to leave feedback and engage with the content.

Transcripts

what's up you freaking geniuses so in

this video i'm going to teach you how to

find the perimeter and area of shapes

and polygons in the coordinate plane all

right so first of all we have a triangle

right here right so the area

of a triangle of any triangle is equal

to one half the base times the height

okay now when your shapes are lined up

with the grid like this it's a little

bit easier to figure it out okay because

as you can see the base right here would

just be this horizontal side right so

the base how long is the base well it's

one two three four five six

seven so it's seven spaces long or seven

units long and then the height would be

right here right so the height would be

one two three four five okay so it's

five right there okay so if we wanted to

find

the area the area would be equal to one

half the base which is seven

times the height which is five

okay so it'd be equal to one half times

35

which is equal to

17.5

okay and whatever your units you're

using just put those there and then they

would be squared right so we'll just say

17.5 units squared

okay now in order to find the perimeter

we have to add up all the sides together

right so we already have these two and

if you notice we can actually use the

pythagorean theorem to figure out this

missing side right here the the

hypotenuse right here right this long

side so

the formula for that would be

a squared plus b squared is equal to c

squared okay where a and b right here

are your two shorter sides and then c is

your hypotenuse okay so first of all uh

we can plug in let's see we have a 7 and

a 5. those are our shorter sides right

so we can plug in a 7 here and a 5 here

okay so then we'll have

7 squared plus 5 squared is equal to c

squared okay now 7 squared is equal to

49

5 squared is 25

and that's equal to c squared okay now

49 plus 25 is equal to

well 49 plus 20 is 69 nice plus 5 is 74

okay so here we have

74 is equal to c squared

okay now to get rid of this exponent

right here the two right we have to take

the square root of that and what we do

to one side we do to the other right so

then here

these cancel out so then on this side

we're just left with c is equal to the

square root of 74 which is approximately

let me change my symbol to then

approximately

8.6

all right so this side right here is the

hypotenuse we're going to say it's about

8.6

okay now if we want to find the

perimeter now we can add up all the

sides together right so we're going to

have 7 plus 5

plus 8.6

and that's equal to

20.6 and if we have units here you know

you just write your units right there

right inches or feet miles whatever okay

so here's the perimeter

all right so here's our next example so

we have a rectangle right so the area of

a rectangle is

the length

times the width right so in order to

find the area here

since it's at an angle right it's not

lined up with the grid we can't just

count spaces like we did last time so in

this case we're going to have to use the

distance formula now if you don't

remember what that is

i'll write it right here so it's x of 2

minus x of 1

squared plus

y of 2 minus y of 1 squared okay so this

whole thing

is under a square root okay now if

you're not familiar with this i'll link

a video to that in the card above but

this is the formula you have to use in

order to find the distance between two

points okay so if we want to find this

side right here this length we have to

find the distance between these two

points right here right 3 comma 8 and

negative 6 comma negative 1.

okay so if we're going to use the

distance formula we have to label our

points here that we're going to use okay

so this one let's label this one as

x of 1 y of 1 and this one we'll label

as x of 2

y of 2 okay so now that we labeled our

two points right here now we can find

the distance between them okay by

plugging it in right here so we're going

to have the length is equal to the

square root of

all this junk right here right so x of 2

minus x sub 1 so we're going to have x

of 2 which is 3 minus x of 1 which is

negative 6. so 3 minus negative 6.

okay so

in parentheses we're going to have 3

minus

negative 6 and that's squared and then

we're going to add that to our other set

of parentheses right here y of 2 minus y

of 1. so y of 2 is 8

y of 1 is negative 1. so we're going to

have 8 minus

negative 1 and this is also squared okay

so then this is going to be equal to the

square root of now 3 minus negative 6

that's the same thing as 3 plus 6 right

so 3 plus 6 is equal to 9.

okay so we have 9 squared and then we're

going to add that to here so 8 minus

negative 1 so this is the same thing

again as 8 plus 1. so 8 plus 1 is equal

to 9 squared also right so

9 squared also okay so then here we're

going to have the square root of well 9

squared is equal to 81 right so we have

81 plus 81 which is equal to the square

root of

162

which if you plug into your calculator

you're gonna get approximately

12.7

okay so the length right the length

right here the distance right here is

equal to approximately

12.7

okay now we just have to find the width

right so again we're going to use the

distance formula to find the width okay

and to find this side now we're going to

have to use these two points right here

right 3 comma negative 8 and 6 comma

negative 5. okay so this one since it's

already labeled i'll just leave it as x

of 2 y 2 and then this one will be

x sub 1

y of one okay so these are the two

points we're using this time okay so now

that these are labeled now we can plug

them into our formula right here okay so

we're gonna have the square root of

remember it's x of two minus x of one

right so x of two is 3

x of 1 is 6. so we're going to have 3

minus 6 in our first set of parentheses

right here right 3 minus 6 and that's

squared and then we're going to add that

to this set of parentheses right here

y of 2 minus y of 1. so 8 minus five

okay

eight minus five and this is squared

also okay so then the width is equal to

the square root of three minus six

that's equal to negative three and this

is squared plus 8 minus 5 which is equal

to 3 and that's squared also so then we

have the square root of negative 3

squared is equal to 9

and 3 squared is equal to 9 okay so here

we have the square root of 18

which if you plug that into your

calculator you're going to get about

right approximately

4.2

okay so then the width over here is

approximately

4.2

all right and let's get rid of all this

crap in three two one boom all right

cool

now in order to find the area remember

the area is just the length times the

width right so here we have 12.7 times

4.2 so

12.7

times

4.2

and that's going to be approximately

again right because we're just rounding

everything

53.3

and those are units

squared all right don't forget those

right so that's the area right here now

to find the perimeter

remember we just have to add up all the

sides right and another way we could

write that is 2 times the length plus 2

times the width okay so then the

perimeter is going to be equal to two

times the length which is again

approximately 12.7

right plus two times the width which is

about 4.2

okay and then the perimeter uh so 2

times 12.7 that's equal to

25.4

plus 2 times 4.2 so then this is equal

to 8.4 right and if you add these up

you're going to get a nice grand total

of 33.8

units in length all right so then that's

your perimeter

so if you found the video helpful

definitely leave a thumbs up down below

and if you have any other questions or

want to see any other examples just let

me know in the comment section below

also there's a couple playlists attached

that i think you'll find helpful so

definitely check those out and i'll see

you there