Math / Geometry

Question6. a) Decompose this polygon so that its area can be calculated. All measurements are in centimeters. b) Calculate its area. Organize your work so that it can be followed by others. (c) 2020 Zearn. Licensed to you pursuant to Zearn's Terms of Use.
This work is a derivative of Open Up Resources' 6-8 Math curriculum, which is available to download for free at openupresources.org and used under the CC BY 4.0 license.

Studdy Solution

STEP 1

What is this asking? We need to figure out the area of this heart shape by cutting it up into easier shapes and adding up their areas! Watch out! Don't forget to include the units in your final answer, and make sure you're adding the areas of the right pieces!

STEP 2

1. Slice and Dice
2. Area of the Triangles
3. Area of the Rectangle
4. Total Area

STEP 3

Let's slice this heart into shapes we know how to handle.
We can make two triangles on top and one rectangle below!
Imagine drawing a horizontal line across the heart, right where the triangles meet the rectangle.
This gives us our three shapes!

STEP 4

Remember, the area of a triangle is $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .
Looking at our heart, the base of each triangle is half of the total width of the heart, which is $\frac{1}{2} \cdot 12 \text{ cm} = 6 \text{ cm}$ .

STEP 5

The height of each triangle is $\textbf{3 cm}$ .
So, the area of one triangle is $\frac{1}{2} \cdot 6 \text{ cm} \cdot 3 \text{ cm} = 9 \text{ cm}^2$ .

STEP 6

Since we have two identical triangles, their combined area is $2 \cdot 9 \text{ cm}^2 = \textbf{18 cm}^2$ .
Awesome!

STEP 7

The area of a rectangle is just length times width.
The width of our rectangle is the same as the total width of the heart, which is $\textbf{12 cm}$ .

STEP 8

The height of the rectangle is $\textbf{6 cm}$ .
So, the area of the rectangle is $12 \text{ cm} \cdot 6 \text{ cm} = \textbf{72 cm}^2$ .
Fantastic!

STEP 9

Now, let's add the areas of the two triangles and the rectangle to get the total area of the heart: $18 \text{ cm}^2 + 72 \text{ cm}^2 = \textbf{90 cm}^2$ .
We did it!

STEP 10

The total area of the heart-shaped polygon is $\textbf{90 cm}^2$ .

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Studdy Solution

STEP 1

What is this asking? We need to figure out the area of this heart shape by cutting it up into easier shapes and adding up their areas! Watch out! Don't forget to include the units in your final answer, and make sure you're adding the areas of the right pieces!

STEP 2

1. Slice and Dice
2. Area of the Triangles
3. Area of the Rectangle
4. Total Area

STEP 3

Let's slice this heart into shapes we know how to handle.
We can make two triangles on top and one rectangle below!
Imagine drawing a horizontal line across the heart, right where the triangles meet the rectangle.
This gives us our three shapes!

STEP 4

Remember, the area of a triangle is $\frac{1}{2} \cdot \text{base} \cdot \text{height}$ .
Looking at our heart, the base of each triangle is half of the total width of the heart, which is $\frac{1}{2} \cdot 12 \text{ cm} = 6 \text{ cm}$ .

STEP 5

The height of each triangle is $\textbf{3 cm}$ .
So, the area of one triangle is $\frac{1}{2} \cdot 6 \text{ cm} \cdot 3 \text{ cm} = 9 \text{ cm}^2$ .

STEP 6

Since we have two identical triangles, their combined area is $2 \cdot 9 \text{ cm}^2 = \textbf{18 cm}^2$ .
Awesome!

STEP 7

The area of a rectangle is just length times width.
The width of our rectangle is the same as the total width of the heart, which is $\textbf{12 cm}$ .

STEP 8

The height of the rectangle is $\textbf{6 cm}$ .
So, the area of the rectangle is $12 \text{ cm} \cdot 6 \text{ cm} = \textbf{72 cm}^2$ .
Fantastic!

STEP 9

Now, let's add the areas of the two triangles and the rectangle to get the total area of the heart: $18 \text{ cm}^2 + 72 \text{ cm}^2 = \textbf{90 cm}^2$ .
We did it!

STEP 10

The total area of the heart-shaped polygon is $\textbf{90 cm}^2$ .

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Studdy Solution

STEP 1

What is this asking? We need to figure out the area of this heart shape by cutting it up into easier shapes and adding up their areas! Watch out! Don't forget to include the units in your final answer, and make sure you're adding the areas of the right pieces!

STEP 2

1. Slice and Dice2. Area of the Triangles3. Area of the Rectangle4. Total Area

STEP 3

Let's **slice** this heart into shapes we know how to handle.We can make **two triangles** on top and **one rectangle** below!Imagine drawing a horizontal line across the heart, right where the triangles meet the rectangle.This gives us our three shapes!

STEP 4

STEP 5

The **height** of each triangle is 3 cm\textbf{3 cm}3 cm.So, the area of *one* triangle is 12⋅6 cm⋅3 cm=9 cm2\frac{1}{2} \cdot 6 \text{ cm} \cdot 3 \text{ cm} = 9 \text{ cm}^221​⋅6 cm⋅3 cm=9 cm2.

STEP 6

Since we have **two** identical triangles, their combined area is 2⋅9 cm2=18 cm22 \cdot 9 \text{ cm}^2 = \textbf{18 cm}^22⋅9 cm2=18 cm2.Awesome!

STEP 7

The area of a rectangle is just length times width.The **width** of our rectangle is the same as the total width of the heart, which is 12 cm\textbf{12 cm}12 cm.

STEP 8

The **height** of the rectangle is 6 cm\textbf{6 cm}6 cm.So, the area of the rectangle is 12 cm⋅6 cm=72 cm212 \text{ cm} \cdot 6 \text{ cm} = \textbf{72 cm}^212 cm⋅6 cm=72 cm2.Fantastic!

STEP 9

Now, let's add the areas of the two triangles and the rectangle to get the **total area** of the heart: 18 cm2+72 cm2=90 cm218 \text{ cm}^2 + 72 \text{ cm}^2 = \textbf{90 cm}^218 cm2+72 cm2=90 cm2.We did it!

STEP 10

The total area of the heart-shaped polygon is 90 cm2\textbf{90 cm}^290 cm2.

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1. Slice and Dice
2. Area of the Triangles
3. Area of the Rectangle
4. Total Area

Let's slice this heart into shapes we know how to handle.
We can make two triangles on top and one rectangle below!
Imagine drawing a horizontal line across the heart, right where the triangles meet the rectangle.
This gives us our three shapes!

The height of each triangle is $\textbf{3 cm}$ .
So, the area of one triangle is $\frac{1}{2} \cdot 6 \text{ cm} \cdot 3 \text{ cm} = 9 \text{ cm}^2$ .

Since we have two identical triangles, their combined area is $2 \cdot 9 \text{ cm}^2 = \textbf{18 cm}^2$ .
Awesome!

The area of a rectangle is just length times width.
The width of our rectangle is the same as the total width of the heart, which is $\textbf{12 cm}$ .

The height of the rectangle is $\textbf{6 cm}$ .
So, the area of the rectangle is $12 \text{ cm} \cdot 6 \text{ cm} = \textbf{72 cm}^2$ .
Fantastic!

Now, let's add the areas of the two triangles and the rectangle to get the total area of the heart: $18 \text{ cm}^2 + 72 \text{ cm}^2 = \textbf{90 cm}^2$ .
We did it!

The total area of the heart-shaped polygon is $\textbf{90 cm}^2$ .