# High School: Geometry

### Modeling with Geometry HSG-MG.A.1

1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Like some of the best student-written metaphors around, this standard is all about students comparing things to other things. Of course, nothing really compares to the feelings of Geschpooklichkeit that we experience just thinking about it.

Perhaps even easier than the TV Guide crossword puzzle, students will find this standard to be as much about common sense as it is about geometry. Describing objects using the measures and properties of geometric shapes just means knowing what shapes look like and applying this knowledge to describe real-life objects.

Here are a few examples of tasks that students should be able to do:

• Recognize that the dice used in Monopoly are cubes, while the letter tiles in Scrabble are rectangular prisms
• Calculate the amount of floor left uncovered by the circular area rug in a rectangular living room
• Identify the person in the police line-up who robbed the convenience store based solely on the shadow the perp cast while fleeing the scene
• Describe a person's face as a perfect ellipse, like a circle that has been squished by a Thigh Master

It's really about helping students see geometry in the world around them. Angles, circles, squares, and spheres aren't hidden in the pages of a geometry textbook. They're everywhere we look.

If you want to get your students to start thinking a bit more abstractly, we suggest mentioning optical illusions and the Parisian lawn version of the Death Star. (It might just be a globe, but we'll need more evidence.)

Now toss a truncated cone (a lamp shade) over your head and we'll get this party started!

#### Drills

1. The perimeter of a square patio is 16 yards. How long is each side?

4 yd

In this case, the patio we're talking about takes the shape of a square. The formula for the perimeter of a square is P = 4s. All we need to do is substitute P = 16 in, and we should get s = 4 as our answer.

2. A fountain takes the shape of a half sphere with diameter 10 feet that sits, flat side down, in the center of a round brick-paved patio of diameter 100 feet. What is the area of the portion of the patio that is left uncovered?

2,475π ft2

The area of the entire patio is 2500π square feet, since A = πr2 = π(50)2 = 2500π. The base of the fountain has an area of 25π square feet, since, again, A = πr2 = π(5)2. The portion left uncovered will be the difference of the two numbers, or 2500π – 25π = 2475π square feet.

3. An equilateral triangle has a perimeter of 45 meters. To the nearest whole square meter, what is its area?

97 m2

With a perimeter of 45 m, each side of the equilateral triangle will be 15 m. To find its height, draw in the altitude, which will bisect the base and create a right triangle with the hypotenuse being another 15 m side. The short leg, which is half of the base, is 7.5 m. Using the Pythagorean theorem, we should end up with a height that is approximately 12.99 m. Use this height in the formula A = ½bh = (½)(15 m)(12.99 m) = 97.425 m2. To the nearest whole square meter, this will be 97 m2.

4. The perimeter of a square tabletop is 8 feet. What is the tabletop's area?

4 ft2

The formula for a square's perimeter is P = 4s. In this case, P = 8 ft, so 8 ft = 4s, which means s = 2 ft. The formula for its area is A = s2. Substituting in the side length, we end up with A = (2 ft)2 = 4 ft2.

5. A cylindrical tree trunk is 14 yards high from the ground up to the lowest branch, and it measures 4 yards around. What is the volume of wood in this section of the trunk?

56π yd3

The circumference is found by the formula C = 2πr and we can find the radius by substituting the "4 yards around" so that we have 4 = 2πr or 2π yd. Substitute this in the volume formula V = πr2h to get V = (π)(2π yd)2(14 yd) = 56π yd3.

6. A spherical ball is covered in a sheet of foil that has been cut so that there are no overlaps in the foil. The ball has a volume of 288π in3. What is the area of the sheet of foil?

144π in2

Using the formula for the volume of a sphere , and knowing that V = 288π in3 helps us find r = 6 in. Now what? We're looking for the area of the sheet that surrounds it. Sounds like surface area, so we can use the surface area formula for a sphere, SA = 4πr2, to find that SA = 144π in2.

7. A giant hollow bouncy ball has a diameter of 8 feet. What is the volume of air inside the ball?

The diameter of the sphere is 8 feet, giving a radius of 4 feet. The formula to find the volume of a sphere is , which gives us .

8. Standard-sized parking spots in Acme Parking Garage are 9 ft wide by 21 ft long and are separated by 4-inch wide paint lines. A one-quart bucket of paint covers 22 square feet on the rough concrete floor. How many buckets of paint will Joe, the Acme Parking Garage's line painter, need to purchase in order to paint the lines to mark 16 parking spots?

6 buckets

To mark off 16 spots, Joe will need to paint 17 lines. Each is 20 feet long and 4 inches wide (which is ⅓ of a foot). The total area he will paint is A = 17 × 21 ft × ⅓ ft = 119 ft2. Since one bucket covers 22 ft2, he will need five full buckets of paint plus part of a sixth one, so he will need to purchase six buckets to do the job. Hopefully they'll reimburse him.

9. The Statue of Freedom stands on a pedestal comprised of several tiers, the bottom of which is best described as an eight-pointed-star-ular prism, which from above looks something like this:

This star can be made using two squares, one rotated 45° at the center. The first square has side length 40 ft, and the other square has side length such that it covers half of the side of the first square where they intersect. What is the area of this star shape?

2,000 ft2

Since the area of a square equals the side length squared, the are of the first square will be (40 ft)2 = 1,600 ft2. The remainder of the shape is comprised of four right triangles, each with a hypotenuse of length 20 ft (since it is equal to half of the side of the first square). Since the second square is rotated 45°, we know that the angles where the squares intersect will also be 45°, meaning the two angles are congruent, as are the two sides of the triangle opposite them. This means each leg will have a length of  ft. The area of each of the triangles will be ½ × ( ft)2 = ½(200 ft2) = 100 ft2. Four of these triangles total 400 ft2, so the total of the big square plus the four triangles is 2,000 square feet.

10. A spherical water tank of radius 6 inches is filled with 6 inches of water. What is the volume of the water in the tank?