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PSAT Math : Plane Geometry

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Example Question #41 : How To Find The Area Of A Circle

Screen_shot_2013-03-18_at_10.29.01_pm

Kate has a ring-shaped lawn which has an inner radius of 10 feet and an outer radius 25 feet. What is the area of her lawn?

Possible Answers:

325π ft²

525π ft²

175π ft²

275π ft²

125π ft²

Correct answer:

525π ft²

Explanation:

The area of an annulus is

$\pi (R^{2}-r^{2})$

where is the radius of the larger circle, and is the radius of the smaller circle.

$\pi (25^{2}-10^{2})$

$\pi (625-100)$

$525\pi$

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Example Question #31 : Radius

A 6 by 8 rectangle is inscribed in a circle. What is the area of the circle?

Possible Answers:

$20\pi$

$10\pi$

$4\pi$

$25\pi$

Correct answer:

$25\pi$

Explanation:

The image below shows the rectangle inscribed in the circle. Dividing the rectangle into two triangles allows us to find the diameter of the circle, which is equal to the length of the line we drew. Using a²+b²= c² we get 6²+ 8² = c². c² = 100, so c = 10. The area of a circle is $A=\pi r^2$ . Radius is half of the diameter of the circle (which we know is 10), so r = 5.

$A=\pi *5^2=25\pi$

Diagram_1

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Example Question #81 : Circles

A park wants to build a circular fountain with a walkway around it. The fountain will have a radius of 40 feet, and the walkway is to be 4 feet wide. If the walkway is to be poured at a depth of 1.5 feet, how many cubic feet of concrete must be mixed to make the walkway?

Possible Answers:

$1936\pi \ ft^{3}$

$1296\pi \ ft^{3}$

$336\pi \ ft^{3}$

None of the other answers are correct.

$504\pi \ ft^{3}$

Correct answer:

$504\pi \ ft^{3}$

Explanation:

The following diagram will help to explain the solution:

Foutain

We are searching for the surface area of the shaded region. We can multiply this by the depth (1.5 feet) to find the total volume of this area.

The radius of the outer circle is 44 feet. Therefore its area is 44²π = 1936π. The area of the inner circle is 40²π = 1600π. Therefore the area of the shaded area is 1936π – 1600π = 336π. The volume is 1.5 times this, or 504π.

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Example Question #1 : Area Of A Circle

How many times greater is the area of a circle with a radius of 4in., compared to a circle with a radius of 2in.?

Possible Answers:

$4$

$2$

$4\pi$

$2\pi$

$\pi$

Correct answer:

$4$

Explanation:

The area of a circle can be solved using the equation $A=\pi r^{2}$

The area of a circle with radius 4 is $\pi 4^{2}=16\pi$ while the area of a circle with radius 2 is $\pi 2^{2}=4\pi$ . $16\pi \div 4\pi =4$

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Example Question #81 : Circles

What is the area of a circle whose diameter is 8?

Possible Answers:

16π

8π

12π

32π

64π

Correct answer:

16π

Explanation:

Circarea

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Example Question #81 : Plane Geometry

What is the area, in square feet, of a circle with a circumference of $14\pi\ \text{feet}$ ?

Possible Answers:

$56\pi\ \text{ft}^2$

$49\pi\ \text{ft}^2$

$64\pi\ \text{ft}^2$

$92\pi\ \text{ft}^2$

$196\pi\ \text{ft}^2$

Correct answer:

$49\pi\ \text{ft}^2$

Explanation:

In order to find the area of a circle with a known circumference, first solve for the radius of the circle.

We know the circumference of a circle is equivalent to $d\pi$ , where $d=\text{diameter}$ .

$C=14\pi = d\pi$

$d = 14\ \text{ft}$

The radius of a circle is equal to half the diameter.

$r = \frac{d}{2}$

Therefore:

$r = \frac{14}{2}$

$r = 7\ \text{ft}$

The area of a circle is given by the equation $a = \pi r^{2}$ . Use the radius to solve for the area.

$a = \pi(7)^{2}$

$a = 49\pi\ \text{ft}^2$

The area of a circle with circumference $14\pi$ is $49\pi$ square feet.

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Example Question #85 : Circles

A square has a perimeter of 48 inches. What is the area, in square inches, of the largest circle that will fit entirely inside the square?

Possible Answers:

$48\pi\ \text{in}^2$

$24\pi\ \text{in}^2$

$72\pi\ \text{in}^2$

$64\pi\ \text{in}^2$

$36\pi\ \text{in}^2$

Correct answer:

$36\pi\ \text{in}^2$

Explanation:

A perimeter of a square is equal to the sum of the four equal sides:

P = 4s

Therefore, the length of one side of this square is 12:

48 = 4s

$s = 48\div4=12\ \text{in}$

We know the largest circle that can fit entirely inside the square will have a maximum diameter of 12 (the length of one side of the square).

$d=12\ \text{in}$

To find the area of this circle, we must find the radius by dividing the diameter by 2:

$r = \frac{d}{2}= \frac{12}{2}=6\ \text{in}$

The radius of the circle is 6. Using the formula for area, we find:

$a=\pi r^2$

$a=\pi (6)^2$

$a=36\pi\ \text{in}^2$

The area of the largest circle that will fit inside a square with a perimeter of 48 inches is $36\pi$ square inches.

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Example Question #41 : How To Find The Area Of A Circle

In the following diagram, the radius is given. What is area of the shaded region?

Circle_box

Possible Answers:

$64 x^{2} - 16\pi x^{2}$

$32 x^{2} -8\pi x^{2}$

$64 x^{2} -8\pi x^{2}$

$32 x^{2} - 16\pi x^{2}$

$16x^{2}$

Correct answer:

$32 x^{2} -8\pi x^{2}$

Explanation:

This question asks you to apply the concept of area in finding both the area of a circle and square. Since the cirlce is inscribed in the square, we know that its diameter (two times the radius) is the same length as one side of the square. Since we are given the radius, , we can find the area of both the circle and square.

Square:

This gives us the area for the entire square.

The bottom half of the square has area $\frac{1}{2} \cdot 64x^{}2 = 32x^{}2$ .

Now that we have this value, we must find the area that the circle occupies. The area of a circle is given by $\pi \cdot r^{}2$ .

So the area of this circle will be $\pi \cdot 16x^{}2$ .

The bottom half of the circle has half that area:

$\rightarrow \pi \cdot 8x^{}2$

Now that we have both our values, we can subtract the bottom half of the circle from the bottom half of the square to give us the shaded region:

$32 x^{}2 - \pi \cdot 8x^{}2$

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

A square has an area of 36 cm². A circle is inscribed and cut out. What is the area of the remaining shape? Use 3.14 to approximate π.

Possible Answers:

28.26 cm²

3.69 cm²

15.48 cm²

12.14 cm²

7.74 cm²

Correct answer:

7.74 cm²

Explanation:

We need to find the area of both the square and the circle and then subtract the two. Inscribed means draw within a figure so as to touch in as many places as possible. So the circle is drawn inside the square. The opposite is circumscribed, meaning drawn outside.

A_square = s² = 36 cm² so the side is 6 cm

6 cm is also the diameter of the circle and thus the radius is 3 cm

A circle = πr² = 3.14 * 3² = 28.28 cm²

The resulting difference is 7.74 cm²

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Example Question #81 : Plane Geometry

Gre10

In the square above, the radius of each half-circle is 6 inches. What is the area of the shaded region?

Possible Answers:

144 – 6π

144 – 36π

36 – 6π

144 – 9π

36 – 9π

Correct answer:

144 – 36π

Explanation:

We can find the area of the shaded region by subtracting the area of the semicircles, which is much easier to find. Two semi-circles are equivalent to one full circle. Thus we can just use the area formula, where r = 6:

π(6²) → 36π

Now we must subtract the area of the semi-circles from the total area of the square. Since we know that the radius also covers half of a side, 6(2) = 12 is the full length of a side of the square. Squaring this, 12² = 144. Subtracting the area of the circles, we get our final terms,

= 144 – 36π

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PSAT Math : Plane Geometry

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #41 : How To Find The Area Of A Circle

Example Question #31 : Radius

Example Question #81 : Circles

Example Question #1 : Area Of A Circle

Example Question #81 : Circles

Example Question #81 : Plane Geometry

Example Question #85 : Circles

Example Question #41 : How To Find The Area Of A Circle

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

Example Question #81 : Plane Geometry

All PSAT Math Resources

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