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Two or more geometric shapes can make up a plane figure. There are no standard procedures for finding the area or perimeter of the figure in these cases. However, breaking down the figure into known geometric shapes helps reveal that information. This lesson will use real-life problems to discuss calculating the area and perimeter of figures with different shapes.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

## Finding the Area of a Pool

Tearrik began training at a newly opened gym in his neighborhood called Little Muscles. In the swimming area there are three pools. One is a circular pool with an area of square meters. Another is a rectangular pool with an area of square meters.

The shape of the third pool surprised Tearrik a little. It looks like the combination of half the circular pool with the rectangular pool.

What is the area of the third pool?

Discussion

## Composite Figures

A composite figure is a plane figure made of two or more two-dimensional figures combined in a way that creates a new figure. For example, the following figure is made of a rectangle, a triangle, and a semicircle.
The perimeter of such a figure is the total distance around it, just remember not to count the same side twice. Additionally, the area of a compound figure is found by combining up the areas of all the basic figures that make up it.
From the definition, composite figures can also have holes inside. For instance, consider a figure that is a combination of a square and a circle, where the circle is removed from the inside of the square, creating a hole.
In this case, the area of the composite figure is the difference between the area of the square and the area of the circle. The perimeter coincides with the perimeter of the square.
Example

## Area and Perimeter of the Gym's Logo

The wrestling room of the gym where Tearrik works out has a circular carpet with the gym's logo inside.

a What is the area of the logo? Answer in square meters rounded to one decimal place.
b What is the perimeter of the logo? Round the answer to the nearest integer.

### Hint

a Decompose the logo into familiar geometric shapes such as triangles, rectangles, trapezoids, and semicircles. Find the area of the involved figures and add them up.
b Use the Pythagorean Theorem to find the lengths of the slant sides. Notice that all these sides have the same length. The circumference of a semicircle is times the radius.

### Solution

a The gym's logo is a dumbbell formed by different geometric figures — a green rectangle, two purple trapezoids, two gray trapezoids, two gray rectangles, and two red semicircles. Thus, it is a composite figure.

The area of the logo is the sum of the areas of the involved figures. Then, start by recalling the formulas for finding the area of the corresponding figures.

Figure Area's Formula
Rectangle
Trapezoid
Semicircle

Next, find the area of each of the pieces that make up the logo.

Figure Dimensions (cm) Area
Green rectangle
Purple trapezoid

Gray trapezoid

Gray rectangle
Red semicircle
Since all the figures but the green rectangle appear twice in the logo, the corresponding areas must be multiplied by
Finally, choose the appropriate conversion factor to convert the area from square centimeters to square meters. Recall that square meter is the same as square centimeters.
Evaluate

Cross out common units

Cancel out common units

The gym's logo has an area of about square meters.
b The perimeter of the gym's logo, which is a composite figure, is the distance around the logo. From the given diagram, almost all side lengths can be clearly identified.
Notice that all the slant segments have the same length because each of them is the hypotenuse of a right triangle with legs measuring and centimeters. Therefore, their lengths can be found by using the Pythagorean Theorem.
Solve for
The length of each slant segment is centimeters. The length of each semicircle is times the radius. Since the radius is centimeters, the length of each semicircle is centimeters.
Finally, all the side lengths are added. There are six segments of centimeters, eight segments of centimeters, four segments of centimeters, four segments of centimeters, and two semicircles of centimeters.
Evaluate
The perimeter of the logo, rounded to the nearest integer, is centimeters.
Example

## Carpeting the Training Rooms

On the first floor of the gym, the weight room and the spinning room are connected by the locker room. The gym owner received many complaints about the current floor quality. The entire first floor, instead, will be covered using recycled yoga mats.

How many square meters of rubber matting does the gym owner have to buy?

### Hint

Add the areas of the two big rectangles and subtract the area of the locker room.

### Solution

Notice that the three rooms together form a composite figure that is shaped as two overlapping rectangles.

The area of the entire floor is then the sum of the areas of the big rectangles minus the area of the locker room. The area of the locker room must be subtracted because it is common for both rectangles.

Room Figure Dimensions (ft) Area
Spinning Upper Rectangle
Weight Lower Rectangle
Locker Small Rectangle
Finally, find the area of the entire floor.
The gym owner has to buy square feet of rubber matting to cover the entire floor.
Example

## Kettlebells

It is leg day and Tearrik wants to try a new exercise he discovered on social media — the single-kettlebell swing. Instead of the usual spherical shape, the kettleballs at this gym are polyhedrons.
The front view of the gym's kettlebells are regular octagons with a circular handle.
What is the area of the front view of the kettlebell? Round to the nearest integer.

### Hint

The area of a regular polygon is half the perimeter times the apothem. The apothem is the distance from the center of the polygon to any of its sides. The handle is what is left when a small semicircle is removed from a large semicircle.

### Solution

The front view of the kettlebell can be separated into two parts — the body and the handle. The body is a regular octagon and the handle looks like half a ring. The kettlebell's front view is then a composite figure whose area is the sum of these two areas.

### Area of the Octagon

The area of a regular polygon is half the perimeter times the apothem.

The perimeter of a regular polygon is equal to the side length times the number of sides. In this case, the number of sides is and all of them have a length of centimeters.
The apothem of a regular polygon is the distance from the center of the polygon to any of its sides. From the given diagram, the apothem of the kettlebell's front view is centimeters.
Simplify
The area of the regular octagon is square centimeters.

### Area of the Handle

The handle has the shape of half a ring which is what is left when a small semicircle is removed from a large semicircle.
The inner semicircle has a diameter of centimeters so its radius is centimeters. The outer semicircle has a diameter of centimeters so its radius is centimeters. The area of a semicircle is half the product of and the radius squared.
Use this formula to find the area of the semicircles.
Outer
Inner
The area of the handle is the difference between the area of the outer semicircle and the area of the inner semicircle.
The area of the front view of the handle is square centimeters.

### Area of the Kettlebell's Front View

Finally, add the areas of the regular octagon and the area of the handle.
The area of the front view of the kettlebell is about square centimeters.
Example

## New Training Zone

Tearrik noticed a poster on the wall saying that the gym would soon open its second floor with a boxing room and a multipurpose room. The blueprint for the second floor is shown next.

Find the perimeter and area of the of the entire floor. Round the answer to the nearest integer. Every unit in the blueprint corresponds to meters.

### Hint

Use the coordinate plane to find the side lengths of the composite figure. The perimeter is the sum of the lengths of the exterior sides. Break down the figure into known figures and find the sum of their areas.

### Solution

The first step involves finding the perimeter of the entire floor. After that, the area of the floor will be calculated.

### Perimeter of The Floor

Start by identifying the coordinates of the vertices of the blueprint and the lengths of the horizontal and vertical sides. Let be the length of the slant side of the entrance.

The value of can be calculated by substituting and into the Distance Formula.
Evaluate right-hand side
The slant side is units long. The length of the curved wall at the back of the stage is the length of a semicircle whose radius is units. The length of a semicircle is times the radius.
The wall at the back of the stage is about units long. By adding the lengths of the exterior sides of the blueprint, the perimeter of the entire floor can be calculated.
The perimeter of the entire floor in the blueprint is units. Convert this to meters by using the fact that every unit in the blueprint is the same as meters.
The entire floor has a perimeter of about meters.

### Area of The Floor

Notice that the entire floor can be broken down into some known figures.

The area of the floor is the sum of the areas of the involved figures.

Place Figure Dimensions Area
Entrance and locker room Trapezoid

Seating area, hall, and boxing room Rectangle
Multipurpose room without the stage Rectangle
Stage Semicircle