Area Formulas for Polygons & Circles

Area

Perimeter is 1 Dimensional -length

Area is 2 Dimensional- planar space

What is Area?

 the amount of space taken up by a 2D shape or surface. Units for area are square units such as:

 $cm^2,\ m^2,\ ft^2,\ in^2,\ mi^2,\ km^2.....$  

Area Formulas

Here they are all together. Lets look at each one to understand where it comes from and how to use it...

Squares & Rectangles

The area of a Rectangle is the product of length times width.  Sometimes we call these sides base and height but no matter what we call the sides, the Area is the product of those numbers and has square units...

Multiple Choice

What does area mean?

• the distance around a shape

• half the distance around a shape

• the amount of space outside a shape

• the amount of space inside a shape

Multiple Choice

How would you find the area of the pink region?

• Add 4 and 3

• Multiply 4 and 3

• Add 4 and 4

• Square 4

Multiple Choice

What is the total area of the pink region?

• 3 cm²

• 4 cm²

• 7 cm²

• 12 cm²

Multiple Choice

What is the area of the shaded figure? 

• 4 square units 
• 6 square units 
• 10 square units 
• 2 square units 

Multiple Choice

Find the area of the rectangle.

• 8 square units

• 9 square units

• 10 square units

• 12 square units

Multiple Choice

What is the area of rectangle?

• 24 cm²

• 24 cm

• 11 cm²

• 22 cm

Squares

SInce both sides are congruent in a square, the Area formula is simply

  $A=s^2$  

...we can think of Area as being the sum of all the small squares that would fill a closed planar space

Multiple Select

When calculating the area of a square,

• you only need to know the measurement of 1 side

• The area is side squared

• The area of a square is measured in square units

Multiple Choice

Find the area

• 16 ft²

• 64 ft²

• 32 ft²

• 8 ft²

Multiple Choice

What is the area of a square with a length of 4 cm?

• 16 cm

• 16 square cm

• 8 cm

• 8 square cm

Multiple Choice

Find the area of this square

• 100 in

• 10 square units

• 40 units

• 100 square units

Square Roots

We have seen and used square roots before, but now we can see that the square root of area is the side length of a square with that area....

Multiple Choice

If I know the area of a square, how do I find the side length?

• I square the area.
• I divide the area by 2.
• I take the square root of the area.
• I divide the area by 4.

Multiple Choice

The area of a square is 36 cm².  What is the length of one side?

• 6 cm
• 24 cm
• 9 cm
• 9 cm

Parallelograms & Rhombi (given altitude)

The formula for Parallelograms and Rhombi are the same as a rectangle. We can see that if you cut a triangle off of a parallelogram (along altitude h) and slide it over, you get the same area as a Rectangle (base x height)

Parallelograms & Rhombi

when the Height is perpendicular to the base 

 $A=base\ \times height$   

Multiple Choice

If we were finding the area of this parallelogram, which value would we NOT use?

• 2 m

• 2.5 m

• 1.8 m

Multiple Choice

What is the area of this parallelogram?

• 75 square cm

• 35 square cm

• 15 square cm

• 105 square cm

Trapezoid

MN is the median of the trapezoid.  It is the average of base-1 (ZY) and base-2 (AB).  

If we multiply the median by the height, we get the area of the Trapezoid, so...

Area of a Trapezoid

 $A=\frac{1}{2}\left(b1+b2\right)\times h$  

    $A=\frac{h\left(b_1+b_2\right)}{2}$ 

 h = height      b = base

Multiple Choice

Find the Area:

• 84 m²

• 168 m²

• 112 m²

• 56 m²

Multiple Choice

Find the Area.

• 91 ft²

• 77 ft²

• 78 ft²

• 105 ft²

Multiple Choice

Determine the area of the trapezoid.

• 160 cm²
• 304 cm²
• 80 cm²
• 152 cm²

Kite and Rhombus 

If we take the purple triangle and add it to the peach, then if we take the green triangle and add it to the yellow, we see that the total area we have found is half the area of the product (rectangle area) of the diagonals...so

Area of a 

Kite and Rhombus 

 $A=\frac{1}{2}\left(d1\times d2\right)$  

 $A=\frac{d_1d_2}{2}$  

Multiple Choice

What is the area of this rhombus? The distances shown refer to the colored segments only.

• 48 cm²

• 24 cm²

• 16 cm²

• 32cm²

Multiple Choice

Find the area of the kite in square centimeters.

• 24

• 48

• 12

• 67.5

Triangles

If you flip a triangle over it makes a rectangle or parallelogram so the triangles area is Half that of the parallelogram with the same base and height...

Notice that Height is perpendicular from the base to the opposite vertex, h is the Altitude.

Triangles

 $A=\frac{1}{2}\left(base\times altitude\right)$  

altitude (h) is always Perpendicular to base

Multiple Choice

How do you find the area of a triangle?

• base x height

• take half of the base times the height

• add up all of the sides

• try ask em

Multiple Choice

Find the area of a triangle with a height of 4 mm and a base of 6 mm?

• 5 mm squared

• 6 mm squared

• 12 mm squared

• 24 mm squared

Multiple Choice

Find the area of the triangle.

• 30 m²

• 60 m²

• 16 m

• 30 m

Multiple Choice

• 31.5 units²

• 63 units²

• 77 units²

• 38.5 units²

• 99 units²

Circles

If we take the radius of this circle and square it, we can see that the total area of the circle with radius r is more than 3 but less than 4 times the area r².

The Area of the circle is exactly pi or about 3.14 times r²

Circles

 $A=\pi\times r^2$  

Multiple Choice

What is the area of this circle? Find the radius first.  $A=\pi\cdot r^2\ \$  

• 163.2

• 15.3

• 124.2

• 113.04

Multiple Choice

A clock face has a diameter of 10 inches. What is the approximate area of the clock face in square inches? Find the radius first.  $A=\pi\cdot r^2\ \$  

• 314 in²

• 31.4 in²

• 7.85 in²

• 78.5 in²

Multiple Choice

Find the area of the circle with radius 3 cm below. $A=\pi\cdot r^2\ \$  

• 100 cm²

• 225 cm²

• 9 cm²

• 28.26 cm²

Multiple Choice

What is the formula for finding the area of a circle?

• A = π·d

• C = 2·π·r

• A = π·r²

• C = π·d

Regular Polygons

If an n-sided polygon (n-gon) is Regular, it can be divided into n number of congruent triangles whose height is called the apothem. . Multiply one triangle area by the number of triangle (n) and you get the total area...

Multiple Choice

A line from the center of a regular polygon at right angles to any of its sides.

• apothem

• radius

• permimeter

• area

Multiple Choice

Find the area

• 96 cm²
• 48 cm²
• 36 cm²
• 48 cm

Multiple Choice

Find the area of this regular polygon

• 81.8 m²

• 204.4 m²

• 408.8 m²

• 40.8 m²

Composite Areas

Can use additon or subtraction of the areas of polygons and circles. Just like we did for compound perimeters, we need to find all the pieces then combine (add or subtract) as needed...

In this example we add the rectangle to the triangle area

Composite Areas

In these examples we need to subtract to find the shaded(grey) area

1- Subt Circle from Square

2- Subt Triangle from Rectangle

3- Subtract square from Rect

4- Subtract half circle from rect

Multiple Choice

Find the area of the figure shown.

• 34 units²
• 960 units²
• 220 units²
• 480 units²

Multiple Choice

Find the area

• 150m²

• 120m²

• 126m²

• 200m²

Multiple Choice

Find the area of the polygon.

• 14 in²
• 68 in²
• 36 in²
• 24 in²

Solving for a variable

As with any equation we have learned, you must be able to rearrange and solve the equation for any variable.
We will now use the Area equations to solve for: base, height, apothem when given Area


So from A= bh we also get

 $b=\frac{A}{h}$        and       $h=\frac{A}{b}$  

Multiple Choice

If the Area of a Rectangular mat is 200 square inches and it has a length of 10in, what is its width?

• 2000 in

• 2 in

• 20 in

• 14.14 in

Multiple Choice

If you were give the Area to a square and wanted to know the side lenght (s) , which equation would correctly solve for s?

•  $s=\frac{A}{2}$  

•  $s=A^2$  

•  $s=2A$  

•  $s=\sqrt{A}$  

Multiple Choice

What is the length of the side of a square with an area of 16cm²

• 256 cm

• 8 cm

• 4 cm

• 32 cm

Multiple Choice

What is the solution for radius (r) given the Area of a Circle?

• $r=\left(\frac{A}{\pi}\right)^2$

• $r=\left(\frac{A}{2\pi}\right)^{ }$

• $r=\sqrt{\frac{A}{\pi}}$

• $r=\left(\frac{A}{\pi}\right)^{ }$

Multiple Choice

About 1/3 of all US irrigation of crop land utilizes Center pivot irrigation systems which make large circles each with an area of about 502, 654m². What is the radius of one of these circles?

• 400m

• 709m

• 226m

• 167,550m

Multiple Choice

A kite has an Area of  $250\ cm^2$  .  If the Long Diagonal is 25cm, what is the lenght of the short diagonal?

• 10cm

• 20cm

• 5cm

...next up:

Nets, Surface Area, Volume