HSPT Math : How to find the area of a figure

Study concepts, example questions & explanations for HSPT Math

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Example Questions

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Example Question #1 : How To Find The Area Of A Figure

What is the area of a circle with a radius of 10?

Possible Answers:

\dpi{100} 20\pi

\dpi{100} 10\pi

\dpi{100} 100\pi

\dpi{100} 100

Correct answer:

\dpi{100} 100\pi

Explanation:

The formula for the area of a circle is \dpi{100} \pi r^{2}

\dpi{100} \pi r^{2}= \pi (10)^{2}=100\pi

Example Question #1 : Geometry

What is the area of a circle with a radius of 7?

Possible Answers:

Correct answer:

Explanation:

To find the area of a circle you must plug the radius into  in the following equation 

In this case the radius is 7 so we plug it into  to get 

We then multiply it by pi to get our answer 

Example Question #3 : How To Find The Area Of A Figure

What is the area of a square with a side length of 8?

Possible Answers:

Correct answer:

Explanation:

To solve this question you must know the formula for the area of a rectangle.

The formula is 

In this case the rectangle is a square so we can plug in the side length for both base and height to yield 

Perform the multiplication to arrive at the answer of 

Example Question #2 : Geometry

Diagram_a

Refer to the above diagram, which shows an equilateral triangle with one vertex at the center of a circle and two vertices on the circle.

What percent of the circle (nearest whole number) is covered by the triangle?

Possible Answers:

14%

12%

10%

18%

It is impossible to tell without knowing the radius.

Correct answer:

14%

Explanation:

Let  be the radius of the circle. The area of the circle is 

 is also the sidelength of the equilateral triangle. The area of the triangle is 

The percent of the circle covered by the triangle is:

Example Question #3 : How To Find The Area Of A Figure

Julie wants to seed her rectangular lawn, which measures 265 feet by 215 feet. The grass seed she wants to use gets 400 square feet of coverage to the pound; a fifty-pound bag sells for $66.00, and a ten-pound bag sells for $20.00. What is the least amount of money Julie should expect to spend on grass seed?

Possible Answers:

Correct answer:

Explanation:

The area of Julie's lawn is  square feet. The amount of grass seed she needs is  pounds. This requires three fifty-pound bags, the most economical option since it is cheaper to buy a fifty-pound bag for $66 than five ten-pound bags for $100.00. Julie will spend 

Example Question #6 : Geometry

Thingy

Above is a figure that comprises a red square and a white rectangle. The ratio of the length of the white rectangle to the sidelength of the square is . What percent of the entire figure is red?

Possible Answers:

Correct answer:

Explanation:

To make this easier, we will assume that the rectangle has length 5 and the square has sidelength 3. Then the area of the entire figure is 

,

and the area of the square is 

The square, therefore, takes up

of the entire figure.

Example Question #56 : Quadrilaterals

Rectangles

Note: Figure NOT drawn to scale.

What percent of the above figure is white?

Possible Answers:

Correct answer:

Explanation:

The large rectangle has length 80 and width 40, and, consequently, area

.

The white region is a rectangle with length 30 and width 20, and, consequently, area 

.

The white region is 

of the large rectangle.

Example Question #6 : How To Find The Area Of A Figure

A square is 9 feet long on each side.  How many smaller squares, each 3 feet on a side can be cut out of the larger square?

Possible Answers:

Correct answer:

Explanation:

Each side can be divided into three 3-foot sections.  This gives a total of  squares.  Another way of looking at the problem is that the total area of the large square is 81 and each smaller square has an area of 9.  Dividing 81 by 9 gives the correct answer.

Example Question #3 : Geometry

Assume π = 3.14

A man would like to put a circular whirlpool in his backyard. He would like the whirlpool to be six feet wide. His backyard is 8 feet long by 7 feet wide. By state regulation, in order to put a whirlpool in a backyard space, the space must be 1.5 times bigger than the pool. Can the man legally install the whirlpool? 

Possible Answers:

Yes, because the area of the whirlpool is 28.26 square feet and 1.5 times its area would be less than the area of the backyard.

No, because the area of the whirlpool is 42.39 square feet and 1.5 times its area would be greater than the area of the backyard.

Yes, because the area of the whirlpool is 18.84 square feet and 1.5 times its area would be less than the area of the backyard.

No, because the area of the backyard is smaller than the area of the whirlpool. 

No, because the area of the backyard is 30 square feet and therefore the whirlpool is too big to meet the legal requirement.

Correct answer:

Yes, because the area of the whirlpool is 28.26 square feet and 1.5 times its area would be less than the area of the backyard.

Explanation:

If you answered that the whirlpool’s area is 18.84 feet and therefore fits, you are incorrect because 18.84 is the circumference of the whirlpool, not the area.

If you answered that the area of the whirlpool is 56.52 feet, you multiplied the area of the whirlpool by 1.5 and assumed that that was the correct area, not the legal limit.

If you answered that the area of the backyard was smaller than the area of the whirlpool, you did not calculate area correctly.

And if you thought the area of the backyard was 30 feet, you found the perimeter of the backyard, not the area.

The correct answer is that the area of the whirlpool is 28.26 feet and, when multiplied by 1.5 = 42.39, which is smaller than the area of the backyard, which is 56 square feet. 

Example Question #4 : Radius

A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?

 Act_math_01

Possible Answers:

8π-8

2π-4

4π-4

8π - 16

8π-4

Correct answer:

8π - 16

Explanation:

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2.  Thus, the radius of the circle is half of the diameter, or 2√2.  The area of the circle is then π(2√2)2, which equals 8π.  Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

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