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ACT Math : Plane Geometry

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Example Questions

Example Question #93 : Plane Geometry

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

Possible Answers:

100π

10π

120π

50π

90π

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 #31 : 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:

$12\ in^{2}$

$4\ in^{2}$

$11\ in^{2}$

$8\ in^{2}$

$14\ in^{2}$

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 #43 : How To Find The Area Of A Circle

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 #564 : Plane Geometry

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 #21 : How To Find The Area Of A Circle

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

Possible Answers:

$20\pi$

$4\pi$

$25\pi$

$10\pi$

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 #101 : Geometry

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:

$504\pi \ ft^{3}$

$1296\pi \ ft^{3}$

$1936\pi \ ft^{3}$

None of the other answers are correct.

$336\pi \ ft^{3}$

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 #25 : Radius

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:

$\pi$

$4$

$2\pi$

$4\pi$

$2$

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 #31 : How To Find The Area Of A Circle

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

Possible Answers:

8π

32π

16π

12π

64π

Correct answer:

16π

Explanation:

Circarea

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Example Question #561 : Geometry

What is the area of a cirlce with a circumference of $24\pro\pi$ ?

Possible Answers:

$576\pi$

$36\pi$

$144\pi$

$24\pi$

$48\pi$

Correct answer:

$144\pi$

Explanation:

The formula for the circumference of a circle is $c = 2*\pi*r$ . Because we are given the circumference we substitute $24\pi$ for and solve for , yielding r = 12 .

Next, we need to plug in our value for into the formula for the area of a circle.

$A = \pi * r^{2}$ and get

$A = 144\pi$

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Example Question #31 : Radius

A circle has a cricumference of $16\pi$ . Given this information, find the area of the circle.

Possible Answers:

$64\pi$

$8\pi$

$16\pi$

$64\pi^2$

Correct answer:

$64\pi$

Explanation:

To find the area of a circle, we use the formula

$A=\pi r^2$

where r is the radius.

However, the problem does not give us the circles radius. In order to solve for the area we must find the radius using the circumference. Circumference of a circle follows the equation

$C=2\pi r$ ,

so since we know the circumference we can manipulate the equation and plug in our value to solve for radius.

$r=\frac{C}{2\pi}=\frac{16\pi}{2\pi}=8$ .

Now that we know the radius we simply plug it into the area formula to solve for our final answer.

$A=\pi (8)^2=64\pi$

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ACT Math : Plane Geometry

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #93 : Plane Geometry

Example Question #31 : Plane Geometry

Example Question #43 : How To Find The Area Of A Circle

Example Question #564 : Plane Geometry

Example Question #21 : How To Find The Area Of A Circle

Example Question #101 : Geometry

Example Question #25 : Radius

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

Example Question #561 : Geometry

Example Question #31 : Radius

All ACT Math Resources

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