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SAT Math : Plane Geometry

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Example Questions

Example Question #321 : Plane Geometry

A 12x16 rectangle is inscribed in a circle. What is the area of the circle?

Possible Answers:

100π

10π

90π

50π

120π

Correct answer:

100π

Explanation:

Explanation: Visualizing the rectangle inside the circle (corners touching the circumference of the circle and the center of the rectangle is the center of the circle) you will see that the rectangle can be divided into 8 congruent right triangles, with the hypotenuse as the radius of the circle. Calculating the radius you divide each side of the rectangle by two for the sides of each right triangle (giving 6 and 8). The hypotenuse (by pythagorean theorem or just knowing right triangle sets) the hypotenuse is give as 10. Area of a circle is given by πr². 10² is 100, so 100π is the area.

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Example Question #322 : Plane Geometry

A circle is inscribed in a square whose side is 6 in. What is the difference in area between the square and the circle, rounded to the nearest square inch?

Possible Answers:

$4\ in^{2}$

$8\ in^{2}$

$12\ in^{2}$

$14\ in^{2}$

$11\ in^{2}$

Correct answer:

$8\ in^{2}$

Explanation:

The circle is inscribed in a square when it is drawn within the square so as to touch in as many places as possible. This means that the side of the square is the same as the diameter of the circle.

Let $\pi =3.14$

$A_{square}= s^{2} = (6)^{2} = 36 in^{2}$

$A_{circle}= \pi r^{2}=3.14\cdot3^{2}=3.14\cdot9=28.26 in^{2}$

So the approximate difference is in area $8\ \textup{in}^{2}$

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Example Question #323 : Plane Geometry

Two equal circles are cut out of a rectangular sheet of paper with the dimensions 10 by 20. The circles were made to have the greatest possible diameter. What is the approximate area of the paper after the two circles have been cut out?

Figure_2

Possible Answers:

Correct answer:

Explanation:

The length of 20 represents the diameters of both circles. Each circle has a diameter of 10 and since radius is half of the diameter, each circle has a radius of 5. The area of a circle is A = πr² . The area of one circle is 25π. The area of both circles is 50π. The area of the rectangle is (10)(20) = 200. 200 - 50π gives you the area of the paper after the two circles have been cut out. π is about 3.14, so 200 – 50(3.14) = 43.

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Example Question #81 : Circles

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:

275π ft²

525π ft²

325π ft²

175π 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 #81 : Plane Geometry

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

Possible Answers:

$20\pi$

$25\pi$

$4\pi$

$10\pi$

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:

$1296\pi \ ft^{3}$

$336\pi \ ft^{3}$

None of the other answers are correct.

$504\pi \ ft^{3}$

$1936\pi \ ft^{3}$

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 #6 : 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:

$\pi$

$4\pi$

$2$

$2\pi$

$4$

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 #85 : Circles

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

Possible Answers:

16π

64π

8π

32π

12π

Correct answer:

16π

Explanation:

Circarea

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Example Question #31 : How To Find The Area Of A Circle

There are two circles, one with a circumference of $80\pi$ and the other has a circumference of $40\pi$ . What is the ratio of the larger circle's area to the smaller circle's area?

Possible Answers:

$2\pi$

$4\pi$

Correct answer:

Explanation:

The circumference of a circle is equal to the diameter of the circle times $\pi$ . The diameter is equal to twice the radius so:

$C=2\pi r$

The radius of the first can be solved as follows:

$80\pi=2\pi r_{1}$

$r{_{1}}=40$

Likewise for the second circle:

$40\pi=2\pi r_{2}$

$r_{2}=20$

The are of a circle is given by the following formula:

$A=\pi r^2$ .

The ratio of the larger area to the smaller area can be found as follows:

$\frac{A_{1}}{A_{2}}=\frac{\pi r_{1}^{2}}{\pi r_{2}^{2}}=\frac{\pi (40)^2}{\pi(20)^2}=\frac{1600\pi}{400\pi}$

Cancelling out $\pi$ and dividing gives the correct answer of

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Example Question #32 : 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:

$16x^{2}$

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

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

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

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

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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SAT Math : Plane Geometry

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #321 : Plane Geometry

Example Question #322 : Plane Geometry

Example Question #323 : Plane Geometry

Example Question #81 : Circles

Example Question #81 : Plane Geometry

Example Question #81 : Circles

Example Question #6 : Area Of A Circle

Example Question #85 : Circles

Example Question #31 : How To Find The Area Of A Circle

Example Question #32 : How To Find The Area Of A Circle

All SAT Math Resources

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