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Common Core: 8th Grade Math : Geometry

Study concepts, example questions & explanations for Common Core: 8th Grade Math

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All Common Core: 8th Grade Math Resources

7 Diagnostic Tests 75 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #26 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.

Possible Answers:

$7.9\textup{ units}$

$9.5\textup{ units}$

$8.6\textup{ units}$

$8.2\textup{ units}$

$9.1\textup{ units}$

Correct answer:

$8.6\textup{ units}$

Explanation:

Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:

10 2

Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:

a^2+b^2=c^2

In this equation:

$a \textup{ and }b=\textup{legs}$

$c=\textup{hypotenuse}$

We can count the number of units on the coordinate plane that were used to create the legs of our drawn triangle. Afterwards, we can use the Pythagorean Theorem to solve for the length of the hypotenuse, or the original diagonal line.

In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:

7^2+5^2=c^2

49+25=c^2

74=c^2

$\sqrt{74}=\sqrt{c^2}$

8.6=c

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Example Question #161 : Geometry

Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.

Possible Answers:

$10.8\textup{ units}$

$10.3\textup{ units}$

$10.5\textup{ units}$

$11.1\textup{ units}$

$11.3\textup{ units}$

Correct answer:

$10.8\textup{ units}$

Explanation:

Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:

11 2

Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:

a^2+b^2=c^2

In this equation:

$a \textup{ and }b=\textup{legs}$

$c=\textup{hypotenuse}$

In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:

4^2+10^2=c^2

16+100=c^2

116=c^2

$\sqrt{116}=\sqrt{c^2}$

10.8=c

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Example Question #21 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

Use the Pythagorean Theorem to calculate the length of the line shown on the provided coordinate plane. Round the answer to the nearest tenth.

Possible Answers:

$20.7\textup{ units}$

$19.3\textup{ units}$

$19.6\textup{ units}$

$20.1\textup{ units}$

$19.9\textup{ units}$

Correct answer:

$20.1\textup{ units}$

Explanation:

Notice that the diagonal line from the problem could be the hypotenuse of a right triangle. If we add two more lines, then we can create a closed figure in the shape of a triangle:

12 1

Let's use the Pythagorean Theorem to calculate the length of the line that represents the hypotenuse of a right triangle. The Pythagorean Theorem states that for right triangles:

a^2+b^2=c^2

In this equation:

$a \textup{ and }b=\textup{legs}$

$c=\textup{hypotenuse}$

In order to solve for this problem we want to substitute in the known side lengths for the triangle's legs:

18^2+9^2=c^2

324+81=c^2

405=c^2

$\sqrt{405}=\sqrt{c^2}$

20.1=c

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Example Question #52 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Susie walks north from her house to a park that is 30 meters away. Once she arrives at the park, she turns and walks west for 80 meters to a bench to feed some pigeons. She then walks north for another 30 meters to a concession stand. If Susie returns home in a straight line from the concession stand, how far will she walk from the concession stand to her house, in meters?

Possible Answers:

100

200

Correct answer:

100

Explanation:

Susie walks 30 meters north, then 80 meters west, then 30 meters north again. Thus, she walks 60 meters north and 80 meters west. These two directions are 90 degrees away from one another.

At this point, construct a right triangle with one leg that measures 60 meters and a second leg that is 80 meters.

You can save time by using the 3:4:5 common triangle. 60 and 80 are $3\cdot 20$ and $4\cdot 20$ , respectively, making the hypotenuse equal to $5\cdot 20=100$ .

We can solve for the length of the missing hypotenuse by applying the Pythagorean theorem:

$a^{2}+b^{2}=c^{2}$

Substitute the following known values into the formula and solve for the missing hypotenuse: side .

$a= 60,\ b= 80,\ c= ?$

$(60)^{2}+(80)^{2}=c^{2}$

$3600+6400=c^{2}$

$10,000=c^{2}$

c=100

Susie will walk 100 meters to reach her house.

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Example Question #622 : Geometry

Chestnut wood has a density of about $0.45\frac{g}{cm^3}$ . A right circular cone made out of chestnut wood has a height of three meters, and a base with a radius of two meters. What is its mass in kilograms (nearest whole kilogram)?

Possible Answers:

$5,755 \; \textrm{kg}$

$5,955 \; \textrm{kg}$

$6,055 \; \textrm{kg}$

$5,655 \; \textrm{kg}$

$5,855 \; \textrm{kg}$

Correct answer:

$5,655 \; \textrm{kg}$

Explanation:

First, convert the dimensions to cubic centimeters by multiplying by 100 : the cone has height 300cm , and its base has radius 200cm .

$3m*\frac{100cm}{1m}=300cm\ \ \2m*\frac{100cm}{1m}=200cm$

Its volume is found by using the formula and the converted height and radius.

$V=\frac{1}{3} \pi r^{2}h$

$V=\frac{1}{3} \pi (200cm)^{2} ( 300cm )\approx 12,566,371 \; \textrm{cm}^{3}$

Now multiply this by $0.45\frac{g}{cm^3}$ to get the mass.

$m=V*\rho$

$12,566,371cm^3*0.45\frac{g}{cm^3} \approx 5,654,867g$

Finally, convert the answer to kilograms.

$5654867g *\frac{1kg}{1,000g} \approx 5,655kg$

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Example Question #1 : How To Find The Volume Of A Cone

A cone has the height of 4 meters and the circular base area of 4 square meters. If we want to fill out the cone with water (density = $1000\frac{Kg}{m^3}$ ), what is the mass of required water (nearest whole kilogram)?

Possible Answers:

10,000

5333

6333

4333

2133

Correct answer:

5333

Explanation:

The volume of a cone is:

$Volume=\frac{1}{3}\pi r^2h$

where is the radius of the circular base, and is the height (the perpendicular distance from the base to the vertex).

As the circular base area is $\pi r^2$ , so we can rewrite the volume formula as follows:

$Volume =\frac{A\times h}{3}$

where is the circular base area and known in this problem. So we can write:

$Volume =\frac{A\times h}{3}=\frac{4\times 4}{3}=\frac{16}{3}\approx 5.333 m^3$

We know that density is defined as mass per unit volume or:

$\rho =\frac{m}{v}$

Where $\rho$ is the density; is the mass and is the volume. So we get:

$m=\rho v=1000\times 5.333=5333 Kg$

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Example Question #162 : Geometry

The vertical height (or altitude) of a right cone is . The radius of the circular base of the cone is . Find the volume of the cone in terms of .

Possible Answers:

$\frac{2}{3}\pi t^2$

$\frac{1}{3}\pi t^2$

$\pi t^3$

$\frac{1}{3}\pi t^3$

$\frac{2}{3}\pi t^3$

Correct answer:

$\frac{2}{3}\pi t^3$

Explanation:

The volume of a cone is:

$Volume=\frac{1}3{}\pi r^2h$

where is the radius of the circular base, and is the height (the perpendicular distance from the base to the vertex).

$Volume=\frac{1}3{}\pi r^2h =\frac{1}3{}\pi t^2\times 2t=\frac{2}{3}\pi t^3$

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Example Question #163 : Geometry

A right cone has a volume of $8\pi$ , a height of and a radius of the circular base of . Find .

Possible Answers:

$\sqrt{2}$

$2\sqrt[3]{2}$

$\sqrt[3]{3}$

$\sqrt[3]{2}$

Correct answer:

$\sqrt[3]{2}$

Explanation:

The volume of a cone is given by:

$Volume=\frac{1}{3}\pi r^2h$

where is the radius of the circular base, and is the height; the perpendicular distance from the base to the vertex. Substitute the known values in the formula:

$Volume=\frac{1}{3}\pi r^2h\Rightarrow 8\pi=\frac{1}{3}\pi\times (2t)^2\times 3t$

$\Rightarrow 8\pi=4\pi t^3\Rightarrow t^3=2\Rightarrow t=\sqrt[3]{2}$

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Example Question #4 : How To Find The Volume Of A Cone

A cone has a diameter of $12\:m$ and a height of $4\:m$ . In cubic meters, what is the volume of this cone?

Possible Answers:

$54\pi\:m^3$

$36\pi\:m^3$

$48\pi\:m^3$

$144\pi\:m^3$

Correct answer:

$48\pi\:m^3$

Explanation:

First, divide the diameter in half to find the radius.

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

$\frac{12}{2}=6\:m$

Now, use the formula to find the volume of the cone.

$\text{Volume}=\frac{1}{3}\pi r^2 h$

$\text{Volume}=\frac{1}{3}\pi \times 6^2\times4=\frac{1}{3}\pi \times 36\times4=\frac{144}{3}\pi=48\pi\:m^3$

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Example Question #164 : Geometry

The height of a cylinder is 3 inches and the radius of the circular end of the cylinder is 3 inches. Give the volume and surface area of the cylinder.

Possible Answers:

V=74.78 in^3, A=103.04in^2

V=84.78 in^3, A=103.04in^2

V=84.78 in^3, A=123.04in^2

V=74.78 in^3, A=113.10in^2

V=84.82 in^3, A=113.10in^2

Correct answer:

V=84.82 in^3, A=113.10in^2

Explanation:

The volume of a cylinder is found by multiplying the area of one end of the cylinder (base) by its height or:

$Volume=\pi r^2h$

where is the radius of the circular end of the cylinder and is the height of the cylinder. So we can write:

$Volume=\pi r^2h=\pi \times 3^2\times 3=84.82in^3$

The surface area of the cylinder is given by:

$A=2\pi r^2+2\pi rh$

where is the surface area of the cylinder, is the radius of the cylinder and is the height of the cylinder. So we can write:

$A=2\pi r^2+2\pi rh$

$A=2\pi (3)^2+2\pi \times 3\times 3$

$A=18\pi+18\pi$

$A=36\pi$

A=113.10

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All Common Core: 8th Grade Math Resources

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Common Core: 8th Grade Math : Geometry

Study concepts, example questions & explanations for Common Core: 8th Grade Math

All Common Core: 8th Grade Math Resources

Example Questions

Example Question #26 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

Example Question #161 : Geometry

Example Question #21 : Apply The Pythagorean Theorem To Find The Distance Between Two Points In A Coordinate System: Ccss.Math.Content.8.G.B.8

Example Question #52 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #622 : Geometry

Example Question #1 : How To Find The Volume Of A Cone

Example Question #162 : Geometry

Example Question #163 : Geometry

Example Question #4 : How To Find The Volume Of A Cone

Example Question #164 : Geometry

All Common Core: 8th Grade Math Resources

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