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ISEE Upper Level Math : Plane Geometry

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Example Questions

Example Question #191 : Plane Geometry

Find the area of a circle with a diameter of 16in.

Possible Answers:

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

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

$128\pi \text{ in}^2$

$84\pi \text{ in}^2$

$96\pi \text{ in}^2$

Correct answer:

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

Explanation:

To find the area of a circle, we will use the following formula:

$A = \pi r^2$

where r is the radius of the circle.

Now, we know the diameter is 16in. We also know the diameter is two times the radius. Therefore, the radius is 8in. So, we get

$A = \pi \cdot (8\text{in})^2$

$A = \pi \cdot 64\text{in}^2$

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

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Example Question #51 : Circles

A circle has a diameter of 18cm. Find the area.

Possible Answers:

$324\pi \text{ cm}^2$

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

$124\pi \text{ cm}^2$

$81\pi \text{ cm}^2$

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

Correct answer:

$81\pi \text{ cm}^2$

Explanation:

To find the area of a circle, we will use the following formula:

$A = \pi \cdot r^2$

where r is the radius of the circle.

Now, we know the diameter of the circle is 18cm. We also know the diameter is two times the radius. Therefore, the radius is 9cm.

Knowing this, we can substitute into the formula. We get

$A = \pi \cdot (9\text{cm})^2$

$A = \pi \cdot 81\text{cm}^2$

$A = 81\pi \text{ cm}^2$

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Example Question #1 : Circumference Of A Circle

What is the circumference of a circle with a radius equal to ?

Possible Answers:

$14\pi$

$196\pi$

$7\pi$

$49\pi$

Correct answer:

$14\pi$

Explanation:

The circumference can be solved using the following equation:

$C=2r\pi=2(7)\pi=14\pi$

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Example Question #1 : Circumference Of A Circle

The radius of a circle is t^2 . Give the circumference of the circle in terms of .

Possible Answers:

$2\pi t^2$

$\pi t$

$4\pi t^2$

$\pi t^2$

$2\pi t$

Correct answer:

$2\pi t^2$

Explanation:

The circumference can be calculated as $Circumference =2\pi r$ , where is the radius of the circle.

$Circumference =2\pi r=2\pi t^2$

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Example Question #2 : How To Find Circumference

The area of a circle is 25t^2 . Give the circumference of the circle in terms of .

Let $\pi=3.14$ .

Possible Answers:

20t

17.71 t^2

17.71t

20t^2

15t

Correct answer:

17.71t

Explanation:

The area of a circle can be calculated as $Area=\pi r^2$ , where is the radius of the circle.

$Area=\pi r^2\Rightarrow 25t^2=\pi r^2\Rightarrow r^2=\frac{25t^2}{\pi}$

$\Rightarrow r=\frac{5t}{\sqrt{\pi}}=\frac{5t}{\sqrt{3.14}}\approx 2.82t$

The circumference can be calculated as $Circumference =2\pi r$ , where is the radius of the circle.

$Circumference =2\pi r\Rightarrow Circumference =2\times 3.14\times 2.82t=17.71t$

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Example Question #2 : Circumference Of A Circle

What is the circumference of a circle with a radius of ?

Possible Answers:

$8\pi$

$16\pi$

$28\pi$

$16\pi^{2}$

Correct answer:

$16\pi$

Explanation:

The circumference can be solved using the following equation:

$C=2r\pi$

Where represents the radius. Therefore, when we substitute our radius in we get:

$=2(8)\pi =16\pi$

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Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

The perimeter of a given rectangle is equal to the circumference of a given circle. The circle has radius inches; the width of the rectangle is $8 \pi$ inches. What is the length of the rectangle?

Possible Answers:

$(30 - 8 \pi )$ inches

$44 \pi$ inches

$14 \pi$ inches

$22 \pi$ inches

$7\pi$ inches

Correct answer:

$22 \pi$ inches

Explanation:

The circumference of a circle with radius inches is

$C = 2 \pi r = 2 \pi \cdot 30 = 60 \pi$ inches.

The perimeter of the rectangle is therefore $60 \pi$ inches. To find its length, substitute $W = 8\pi$ and $P = 60\pi$ into the equation and solve for :

2 L + 2W = P

$2 L + 2 \cdot 8 \pi = 60 \pi$

$2 L + 16 \pi = 60 \pi$

$2 L + 16 \pi - 16 \pi = 60 \pi - 16 \pi$

$2 L =44 \pi$

$2 L \div 2=44 \pi \div 2$

$L = 22\pi$ inches

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Example Question #191 : Geometry

Sector

The area of the shaded region in the above diagram is $72 \pi$ . Give the circumference of the circle.

Possible Answers:

$8 \pi$

$72 \pi$

$36 \pi$

$18 \pi$

Correct answer:

$36 \pi$

Explanation:

The shaded region is an $80 ^{\circ }$ sector; this represents

$\frac{80}{360} = \frac{80 \div 40 }{360\div 40} = \frac{2}{9}$

of the circle. The area of the region is therefore

$A = \frac{2}{9} \cdot \pi r^{2}$ ,

where is the radius. Setting $A = 72 \pi$ and solving for :

$\frac{2}{9} \cdot \pi r^{2} = 72 \pi$

$\frac{9}{2 \pi} \cdot \frac{2}{9} \cdot \pi r^{2} =\frac{9}{2 \pi} \cdot 72 \pi$

$r^{2} =324$

$r = \sqrt{324 } =18$

The circumference of a circle is its radius multiplied by $2 \pi$ , or

$C = 2 \pi r = 2 \pi \cdot 18 = 36 \pi$

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Example Question #1 : How To Find Circumference

A circle has a radius of 5 miles, what is its circumference?

Possible Answers:

314 mi

$25 \pi mi$

125 mi

$10\pi mi$

Correct answer:

$10\pi mi$

Explanation:

A circle has a radius of 5 miles, what is its circumference?

To find circumference, use the following formula:

$Circumference=2\pi r$

Where r is our radius.

Plug in 5 miles to get our answer:

$Circumference=2\pi (5mi)=10 \pi mi$

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Example Question #2 : How To Find Circumference

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the circumference of the stamping surface?

Possible Answers:

Cannot be determined from the information provided.

9.25cm

$7 \pi cm$

12.25 cm

Correct answer:

$7 \pi cm$

Explanation:

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the circumference of the stamping surface?

Find circumference via the following:

$C=2 \pi r$

Where r is our radius, which is 3.5 cm.

$C=2(3.5cm) \pi =7\pi cm$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Plane Geometry

Study concepts, example questions & explanations for ISEE Upper Level Math

All ISEE Upper Level Math Resources

Example Questions

Example Question #191 : Plane Geometry

Example Question #51 : Circles

Example Question #1 : Circumference Of A Circle

Example Question #1 : Circumference Of A Circle

Example Question #2 : How To Find Circumference

Example Question #2 : Circumference Of A Circle

Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

Example Question #191 : Geometry

Example Question #1 : How To Find Circumference

Example Question #2 : How To Find Circumference

All ISEE Upper Level Math Resources

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