Virginia SOL 8 - 2020 Edition
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5.05 Review: Area of polygons and circles
Lesson

Rectangles and squares

A rectangle has two pairs of equal opposite sides. A square is a special rectangle with four equal sides.

Using dimensions to find the area

While dividing rectangles into unit squares is effective for finding the area, it can be time consuming, especially for larger rectangles. The method of multiplying the dimensions of a rectangle is the quickest way to find its area. The formula for the area of a square is very similar.

Area of a rectangle

The area of a rectangle is given by

$\text{Area }=\text{length }\times\text{width }$Area =length ×width , or

$A=l\times w$A=l×w

Area of a square

The area of a square is given by

$\text{Area }=\text{side }\times\text{side }$Area =side ×side , or

$A=s\times s=s^2$A=s×s=s2

 

Worked examples

QUestion 1

Find the area of the rectangle below.

Think: We know the length and width of the rectangle, so we can use the formula to find the area. The dimensions are given in mm, so the area will be in mm2.

Do:

$\text{Area }$Area $=$= $\text{length }\times\text{width }$length ×width (Formula for the area of a rectangle)
  $=$= $5\times3$5×3 (Substitute the values for the length and width)
  $=$= $15$15 (Perform the multiplication to find the area)

So this rectangle has an area of $15$15 mm2.

Reflect: The length of a rectangle is often defined to be the longest side, which makes the width the shortest side. But since the area formula uses multiplication, the order does not matter. So even if we use another convention that defines the length to be always the horizontal side, the same formula will give the same area.

 

Question 2

Find the area of the square below.

Think: We know the side length of the square, so we can use the formula to find the area. The dimension is given in cm, so the area will be in cm2.

Do:

$\text{Area }$Area $=$= $\text{side }\times\text{side }$side ×side (Formula for the area of a square)
  $=$= $6\times6$6×6 (Substitute the value for the side)
  $=$= $36$36 (Perform the multiplication to find the area)

So this square has an area of $36$36 cm2.

Reflect: The area of a square involves the product of the side length with itself. Another way to write this is using exponential notation, $\text{Area }=\text{side }^2$Area =side 2. In this example the area would be $6^2=36$62=36 cm2.

 

Using the area to find a dimension

We have found that the area of a rectangle is given by the product of its length and width. If we already know the area, along with one of the dimensions, we can use this relationship to find the remaining dimension.

The rectangle below has an area of $28$28 km2, and a length of $7$7 km. How can we determine the width of the rectangle?

From the formula, we know that $\text{Area }=\text{length }\times\text{width }$Area =length ×width , which means that $28=7\times\text{width }$28=7×width . So we want to find the number that multiplies with $7$7 to give $28$28.

In other words, we can find the number of times $7$7 fits into $28$28. This is given by $\frac{28}{7}=4$287=4, so the width of the rectangle is $4$4 km.

 

Practice questions

Question 3

Find the area of the rectangle shown.

Question 4

Find the area of the square shown.

Question 5

Find the width of this rectangle if its area is $66$66 m2 and its length is $11$11 m.

 

Triangles

The area of a triangle is the amount of space that can fit within its outline. We could draw a grid of unit squares on top of a triangle and count the number of squares it contains, but this can be time consuming and inaccurate.

Instead, we can use the base $b$b and perpendicular height $h$h (or simply the height) of a triangle to easily calculate its area. The height is the perpendicular distance from the base to the opposite vertex. It can be inside or outside a triangle depending on which side we select as the base.

 

The relationship between triangles and rectangles

Identifying the base and perpendicular height of a triangle is the first step in determining its area. This is because its area is directly related to the area of a particular rectangle. The applet below shows how we can break apart a triangle and rearrange its parts. We can see that the area of the triangle is equal to half the area of the rectangle that has a length equal to the base of the triangle and a width equal to the height of the triangle.

 

 

Calculating the area of a triangle

In the applet below, we will experiment with changing the dimensions of a triangle to see what effect this has on its area. Try different types of triangles, or varying just the base. Which changes will influence the area? Which changes do not affect the area?

The following guide outlines the key features and concepts in the applet.

  1. Click $\editable{\text{Show grid}}$Show grid (you can hide it again with $\editable{\text{Hide grid}}$Hide grid). If we assume this is a square centimeter grid, the number of squares that can fit within the triangle is its area. You may wish to count the squares for one triangle and check your answer with the one printed on the applet.
  2. Click and drag the right vertex of the triangle left or right to change the length of its base. Click and drag the apex of the triangle up or down and left or right to change the length and position of its perpendicular height. This can also be done with the $b$b, $h$h and "Apex" sliders.
  3. The area of the triangle is being calculated with a formula as you change the dimensions of the triangle. Can you work out the formula?

 

Formula for the area of a triangle

By using the applet above, you may have noticed that the area of a triangle can be found by multiplying the length of its base with its perpendicular height and halving this product.

Area of a triangle

The area of a triangle is given by

$\text{Area }=\frac{1}{2}\times\text{base }\times\text{height }$Area =12×base ×height , or

$A=\frac{1}{2}\times b\times h$A=12×b×h

 

Worked examples

Question 6

Find the area of the right triangle below.

Think: We can identify the base as the side with length $5$5 mm, so that the perpendicular height is the side with length $8$8 mm. Each dimension is in mm, so the area will be in mm2.

Do: Use the area formula with the given side lengths.

$\text{Area }$Area $=$= $\frac{1}{2}\times\text{base }\times\text{height }$12×base ×height (Formula for area of triangle)
  $=$= $\frac{1}{2}\times5\times8$12×5×8 (Substitute the values for the base and the height)
  $=$= $20$20 mm2 (Perform the multiplication to find the area)

So this right triangle has an area of $20$20 mm2.

Reflect: Notice that we could have switched which side we called the base and which side we called the height, and we would still arrive at the same area for the triangle. Think about how the orientation of the triangle relates to how we choose to label the sides.

 

question 7

Find the area of the scalene triangle below.

Think: This triangle has a base of $6$6 cm and a height of $4$4 cm. The area of this triangle will be in cm2.

Do: Use the area formula with the given side lengths.

$\text{Area }$Area $=$= $\frac{1}{2}\times\text{base }\times\text{height }$12×base ×height (Formula for area of triangle)
  $=$= $\frac{1}{2}\times6\times4$12×6×4 (Substitute the values for the base and the height)
  $=$= $12$12 cm2 (Perform the multiplication to find the area)

So this scalene triangle has an area of $12$12 cm2.

Reflect: Can you picture a rectangle that has twice the area of this triangle?

 

Question 8

Find the area of the oblique triangle below.

Think: In this case, the perpendicular height lies outside the triangle, but we know that the same formula still applies. The base is $3$3 m and the height is $5$5 m, so the area will be in m2.

Do: Use the area formula with the given side lengths.

$\text{Area }$Area $=$= $\frac{1}{2}\times\text{base }\times\text{height }$12×base ×height (Formula for area of triangle)
  $=$= $\frac{1}{2}\times3\times5$12×3×5 (Substitute the values for the base and the height)
  $=$= $7.5$7.5 m2 (Perform the multiplication to find the area)

So this oblique triangle has an area of $7.5$7.5 m2.

Reflect: Although we could change the order of the multiplication and still get the same area, in this case it is more appropriate to label the side with length $3$3 m as the base. Notice that the length of $5$5 m is not related directly to any side, so it is a perpendicular height.

 

Using the area to find a dimension

We have found that the area of a triangle is given by half of the product of its base and height. If we already know the area, along with one of the dimensions, we can use this relationship to find the remaining dimension.

The triangle below has an area of $45$45 km2, and a base of $9$9 km. How can we determine the height of the triangle?

From the formula, we know that $\text{Area }=\frac{1}{2}\times\text{base }\times\text{height }$Area =12×base ×height , which means that $45=\frac{1}{2}\times9\times\text{height }$45=12×9×height . So we want to find the number that multiplies with $\frac{1}{2}$12 and $9$9 to give $45$45.

In other words, we can find the number of times $\frac{1}{2}\times9$12×9 fits into $45$45. This is given by

$\frac{45}{\frac{1}{2}\times9}$4512×9 $=$= $\frac{2\times45}{9}$2×459 (Multiply the numerator and denominator by $2$2)
  $=$= $\frac{90}{9}$909 (Perform the multiplication in the numerator)
  $=$= $10$10 (Simplify the fraction)

So the height of the triangle is $10$10 km.

 

Practice questions

Question 9

Find the area of the triangle shown.

Question 10

Find the value of $h$h if the area of this triangle is $48$48 m2.

Question 11

Lisa has purchased a rectangular piece of fabric measuring $6$6 m in length and $9$9 m in width.

What is the area of the largest triangular piece she can cut out from it?

 

Parallelograms

A parallelogram is a quadrilateral with two pairs of opposite sides parallel. It looks like a rectangle that has been pushed over.

A parallelogram is like a slanted rectangle.

We will make use of the base and perpendicular height of the parallelogram to find its area.

Notice that a rectangle is a type of parallelogram, but not all parallelograms are rectangles. Can you see why? Think of what each shape has in common and how they differ.

 

Parallelograms and rectangles

Parallelograms can be easily rearranged into rectangles. In the applet below, we can rearrange a parallelogram with a base $b$b and a perpendicular height $h$h into a rectangle.

The following guide outlines the key features and concepts in the applet.

  1. Click and drag the blue circle on the $\text{Slide }$Slide slider. This will rearrange the parallelogram into a rectangle.
  2. Click and drag the blue circle on the $\text{Slant }$Slant slider. Does this change the area of the shape?
  3. Click the button $\editable{\text{Change dimensions}}$Change dimensions. You can now adjust the base and height to make a new parallelogram. This can be done with the $b$b slider and $h$h slider or by dragging the vertices of the parallelogram.
  4. The area of the parallelogram is being calculated with a formula as you change its dimensions. What part of the calculation changes when you change a dimension? Can you work out the formula?
  5. Click the button $\editable{\text{Decompose }}$Decompose to see if the new parallelogram can also rearrange into a rectangle.
  6. Click the button $\editable{\text{Show grid}}$Show grid for a grid. Assume that this is a square centimeter grid. To remove it, click the button $\editable{\text{Hide grid}}$Hide grid.

 

Formula for the area of a parallelogram

By using the applet above, we can make the following observations:

  • Changing the slant of the parallelogram without changing the base and height did not affect its area. This means that the area of a parallelogram depends only upon its base and its perpendicular height, not the slanted height.
  • The base of the parallelogram is the same as the length of the rectangle.
  • The perpendicular height of the parallelogram is the same as the width of the rectangle.
  • As the area of a rectangle can be found with $\text{Area }=\text{length }\times\text{width }$Area =length ×width , then the area of a parallelogram can be found in a similar way.
Area of a parallelogram

The area of a parallelogram is given by

$\text{Area }=\text{base }\times\text{height }$Area =base ×height , or

$A=b\times h$A=b×h

Unlike a rectangle, there are generally no right angles in a parallelogram. But we should remember that the height and base are at right angles to each other when we work out the area of a parallelogram.

 

Worked examples

Question 12

Find the area of the parallelogram below.

Think: This parallelogram has a base of $6$6 cm and a height of $4$4 cm. We can rearrange it into a rectangle with length $6$6 cm and width $4$4 cm.

This rectangle has the same area as the parallelogram, which means we can find the area of the parallelogram by calculating the product of its base and height.

Do: We can use the given dimensions in the formula to find the area.

$\text{Area }$Area $=$= $\text{base }\times\text{height }$base ×height (Formula for the area of a parallelogram)
  $=$= $6\times4$6×4 (Substitute the values for the base and height)
  $=$= $24$24 (Perform the multiplication to find the area)

So the parallelogram has an area of $24$24 cm2.

 

Question 13

What is the area of this parallelogram?

Think: The base always refers to a side of the parallelogram, while the height is the perpendicular distance between two opposite sides. In this parallelogram the base is $12$12 m and the height is $17$17 m.

Do: We can use the given dimensions in the formula to find the area.

$\text{Area }$Area $=$= $\text{base }\times\text{height }$base ×height (Formula for the area of a parallelogram)
  $=$= $12\times19$12×19 (Substitute the values for the base and height)
  $=$= $228$228 (Perform the multiplication to find the area)

So the parallelogram has an area of $228$228 m2.

Reflect: Sometimes the height will be labeled within the parallelogram, and sometimes it will be convenient to indicate the height with a label outside the parallelogram.

 

Practice questions

Question 14

Complete the table to find the area of the parallelogram shown.

  1. Area $=$= base $\times$× height m2  
    Area $=$= $\editable{}\times\editable{}$× m2 (Fill in the values for the base and height)
    Area $=$= $\editable{}$ m2 (Complete the multiplication to find the area)

Question 15

Find the area of the parallelogram shown.

Question 16

Find the area of a parallelogram whose base is $15$15 cm and height is $7$7 cm.

 

Trapezoids

Trapezoids may have several different properties, but they are related to parallelograms.

The best part is, you can see for yourself how a trapezoid is exactly half of its related parallelogram or rectangle. Move the slider in this applet to see what happens. 

 

We can see from this applet that the area of a trapezoid is going to be half that of a parallelogram with the same height and a base which is the sum of the top and bottom of the trapezoid.

Area of trapezoid

For a trapezoid with height $h$h, top length $a$a and bottom length $b$b,

$A=\frac{1}{2}\left(a+b\right)h$A=12(a+b)h

 

Practice questions

QUESTION 17

Find the area of the trapezoid shown.

Question 18

Find the height $\left(h\right)$(h) if the area of the trapezoid shown is $36$36 cm2.

  1. Start by substituting the given values into the formula for the area of a trapezoid.

    $A=\frac{1}{2}\left(a+b\right)h$A=12(a+b)h

Did you know?

A trapezoid has only one pair of parallel sides, so $a$a cannot equal $b$b!

Circles

We already know that area is the space inside a 2D shape.  We can find the area of a circle, but we will need a special rule.  

Area of a Circle

$\text{Area of a circle}=\pi r^2$Area of a circle=πr2

When we are working with circle, we will be using pi. Whenever we work with $\pi$π, we should note that there is the full value of $\pi$π which we can get from our calculator, but there are also estimates such as $3.14$3.14 and $\frac{22}{7}$227. The question will often specify which value to use as they will give slightly different answers. 

Worked example

Question 19

Find the area of a circle with a diameter of $10$10 inches to two decimal places. Use the $\pi$π button on your calculator for the value of $\pi$π.

Think: We have been given the diameter instead of the radius, so we need to find the radius first by remembering that $r=\frac{d}{2}$r=d2. Then we can fill in the formula $A=\pi r^2$A=πr2.

$r=\frac{10}{2}$r=102 or $r=5$r=5

Do:

$A$A $=$= $\pi r^2$πr2

The formula for area of a circle

  $=$= $\pi\times5^2$π×52

Filling in the radius

  $=$= $\pi\times25$π×25

Evaluating the square

  $=$= $78.539816$78.539816...

Using a calculator to evaluate

  $=$= $78.54$78.54

Rounding correctly

 

 

Practice questions

QUESTION 20

Find the area of the circle shown, correct to one decimal place. Use the $\pi$π button on your calculator for the value of $\pi$π.

QUESTION 21

If the diameter of the circle is $8$8 cm, find its area correct to one decimal place. Use the $\pi$π button on your calculator for $\pi$π in your calculations.

QUESTION 22

If the radius of the circle is $5$5 cm, find its area, rounded to 2 decimal places. Use the $\pi$π button on your calculator for $\pi$π in your calculations.

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