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Common Core: 6th Grade Math : Grade 6

Study concepts, example questions & explanations for Common Core: 6th Grade Math

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All Common Core: 6th Grade Math Resources

6 Diagnostic Tests 186 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #502 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$58.5\textup{ in}^2$ $\displaystyle 58.5\textup{ in}^2$

$62.5\textup{ in}^2$ $\displaystyle 62.5\textup{ in}^2$

$66\textup{ in}^2$ $\displaystyle 66\textup{ in}^2$

$64\textup{ in}^2$ $\displaystyle 64\textup{ in}^2$

$60\textup{ in}^2$ $\displaystyle 60\textup{ in}^2$

Correct answer:

$64\textup{ in}^2$ $\displaystyle 64\textup{ in}^2$

Explanation:

There are several different ways to solve for the area of a right triangle. In this lesson, we will transform the right triangle into a rectangle, use the the simpler formula for area of a rectangle to solve for the new figure's area, and divide this area in half in order to solve for the area of the original figure.

First, let's transform the triangle into a rectangle:

4 4

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=16\times 8$ $\displaystyle A=16\times 8$

A=128 $\displaystyle A=128$

Last, notice that our triangle is exactly half the size of the rectangle that we made. This means that in order to solve for the area of the triangle we will need to take half of the area of the rectangle, or divide it by $\displaystyle 2$ .

$128\div 2= 64\textup{ in}^2$ $\displaystyle 128\div 2= 64\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

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Example Question #503 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$101.5\textup{ in}^2$ $\displaystyle 101.5\textup{ in}^2$

$104\textup{ in}^2$ $\displaystyle 104\textup{ in}^2$

$105.5\textup{ in}^2$ $\displaystyle 105.5\textup{ in}^2$

$102\textup{ in}^2$ $\displaystyle 102\textup{ in}^2$

$100\textup{ in}^2$ $\displaystyle 100\textup{ in}^2$

Correct answer:

$102\textup{ in}^2$ $\displaystyle 102\textup{ in}^2$

Explanation:

First, let's transform the triangle into a rectangle:

1 1

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=17\times 12$ $\displaystyle A=17\times 12$

A=204 $\displaystyle A=204$

$204\div 2= 102\textup{ in}^2$ $\displaystyle 204\div 2= 102\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

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Example Question #511 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$93.5\textup{ in}^2$ $\displaystyle 93.5\textup{ in}^2$

$95\textup{ in}^2$ $\displaystyle 95\textup{ in}^2$

$90.5\textup{ in}^2$ $\displaystyle 90.5\textup{ in}^2$

$96\textup{ in}^2$ $\displaystyle 96\textup{ in}^2$

$91.5\textup{ in}^2$ $\displaystyle 91.5\textup{ in}^2$

Correct answer:

$93.5\textup{ in}^2$ $\displaystyle 93.5\textup{ in}^2$

Explanation:

First, let's transform the triangle into a rectangle:

2 2

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=17\times 11$ $\displaystyle A=17\times 11$

A=187 $\displaystyle A=187$

$187\div 2= 93.5\textup{ in}^2$ $\displaystyle 187\div 2= 93.5\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

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Example Question #512 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$89\textup{ in}^2$ $\displaystyle 89\textup{ in}^2$

$85\textup{ in}^2$ $\displaystyle 85\textup{ in}^2$

$83.5\textup{ in}^2$ $\displaystyle 83.5\textup{ in}^2$

$81\textup{ in}^2$ $\displaystyle 81\textup{ in}^2$

$87.5\textup{ in}^2$ $\displaystyle 87.5\textup{ in}^2$

Correct answer:

$85\textup{ in}^2$ $\displaystyle 85\textup{ in}^2$

Explanation:

First, let's transform the triangle into a rectangle:

3 3

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=17\times 10$ $\displaystyle A=17\times 10$

A=170 $\displaystyle A=170$

$170\div 2= 85\textup{ in}^2$ $\displaystyle 170\div 2= 85\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

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Example Question #513 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$74\textup{ in}^2$ $\displaystyle 74\textup{ in}^2$

$76.5\textup{ in}^2$ $\displaystyle 76.5\textup{ in}^2$

$78.5\textup{ in}^2$ $\displaystyle 78.5\textup{ in}^2$

$72.5\textup{ in}^2$ $\displaystyle 72.5\textup{ in}^2$

$80\textup{ in}^2$ $\displaystyle 80\textup{ in}^2$

Correct answer:

$76.5\textup{ in}^2$ $\displaystyle 76.5\textup{ in}^2$

Explanation:

First, let's transform the triangle into a rectangle:

4 4

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=17\times 9$ $\displaystyle A=17\times 9$

A=153 $\displaystyle A=153$

$153\div 2= 76.5\textup{ in}^2$ $\displaystyle 153\div 2= 76.5\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

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Example Question #514 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$117.5\textup{ in}^2$ $\displaystyle 117.5\textup{ in}^2$

$114\textup{ in}^2$ $\displaystyle 114\textup{ in}^2$

$114.5\textup{ in}^2$ $\displaystyle 114.5\textup{ in}^2$

$119\textup{ in}^2$ $\displaystyle 119\textup{ in}^2$

$117\textup{ in}^2$ $\displaystyle 117\textup{ in}^2$

Correct answer:

$117\textup{ in}^2$ $\displaystyle 117\textup{ in}^2$

Explanation:

First, let's transform the triangle into a rectangle:

1 1

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=18\times 13$ $\displaystyle A=18\times 13$

A=234 $\displaystyle A=234$

$234\div 2= 117\textup{ in}^2$ $\displaystyle 234\div 2= 117\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

Report an Error

Example Question #515 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$104\textup{ in}^2$ $\displaystyle 104\textup{ in}^2$

$105.5\textup{ in}^2$ $\displaystyle 105.5\textup{ in}^2$

$102.5\textup{ in}^2$ $\displaystyle 102.5\textup{ in}^2$

$108\textup{ in}^2$ $\displaystyle 108\textup{ in}^2$

$106\textup{ in}^2$ $\displaystyle 106\textup{ in}^2$

Correct answer:

$108\textup{ in}^2$ $\displaystyle 108\textup{ in}^2$

Explanation:

First, let's transform the triangle into a rectangle:

2 2

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=18\times 12$ $\displaystyle A=18\times 12$

A=216 $\displaystyle A=216$

$216\div 2= 108\textup{ in}^2$ $\displaystyle 216\div 2= 108\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

Report an Error

Example Question #516 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$99\textup{ in}^2$ $\displaystyle 99\textup{ in}^2$

$104.5\textup{ in}^2$ $\displaystyle 104.5\textup{ in}^2$

$98.5\textup{ in}^2$ $\displaystyle 98.5\textup{ in}^2$

$103\textup{ in}^2$ $\displaystyle 103\textup{ in}^2$

$105\textup{ in}^2$ $\displaystyle 105\textup{ in}^2$

Correct answer:

$99\textup{ in}^2$ $\displaystyle 99\textup{ in}^2$

Explanation:

First, let's transform the triangle into a rectangle:

3 3

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=18\times 11$ $\displaystyle A=18\times 11$

A=198 $\displaystyle A=198$

$198\div 2= 99\textup{ in}^2$ $\displaystyle 198\div 2= 99\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

Report an Error

Example Question #517 : Plane Geometry

What is the area of the right triangle in the following figure?

Possible Answers:

$84.5\textup{ in}^2$ $\displaystyle 84.5\textup{ in}^2$

$86.5\textup{ in}^2$ $\displaystyle 86.5\textup{ in}^2$

$90\textup{ in}^2$ $\displaystyle 90\textup{ in}^2$

$85.5\textup{ in}^2$ $\displaystyle 85.5\textup{ in}^2$

$88\textup{ in}^2$ $\displaystyle 88\textup{ in}^2$

Correct answer:

$90\textup{ in}^2$ $\displaystyle 90\textup{ in}^2$

Explanation:

First, let's transform the triangle into a rectangle:

4 4

Second, let's remember that the formula for area of a rectangle is as follows:

$A=l\times w$ $\displaystyle A=l\times w$

Substitute in our side lengths.

$A=18\times 10$ $\displaystyle A=18\times 10$

A=180 $\displaystyle A=180$

$180\div 2= 90\textup{ in}^2$ $\displaystyle 180\div 2= 90\textup{ in}^2$

Thus, the area formula for a right triangle is as follows:

$A=\frac{1}{2}(l\times w)$ $\displaystyle A=\frac{1}{2}(l\times w)$ or $A=\frac{l\times w}{2}$ $\displaystyle A=\frac{l\times w}{2}$

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Example Question #1 : Solid Geometry

A large crate in the shape of a rectangular prism has dimensions 5 feet by 4 feet by 12 feet. Give its volume in cubic yards.

Possible Answers:

$90\textrm{ yd}^{3}$ $\displaystyle 90\textrm{ yd}^{3}$

$8 \frac{8}{9 } \textrm{ yd}^{3}$ $\displaystyle 8 \frac{8}{9 } \textrm{ yd}^{3}$

$80 \textrm{ yd}^{3}$ $\displaystyle 80 \textrm{ yd}^{3}$

$10 \textrm{ yd}^{3}$ $\displaystyle 10 \textrm{ yd}^{3}$

$7 \frac{2}{9 } \textrm{ yd}^{3}$ $\displaystyle 7 \frac{2}{9 } \textrm{ yd}^{3}$

Correct answer:

$8 \frac{8}{9 } \textrm{ yd}^{3}$ $\displaystyle 8 \frac{8}{9 } \textrm{ yd}^{3}$

Explanation:

Divide each dimension by 3 to convert feet to yards, then multiply the three dimensions together:

V = LWH $\displaystyle V = LWH$

$V = \frac{5}{3} \cdot \frac{4}{3} \cdot \frac{12}{3}$ $\displaystyle V = \frac{5}{3} \cdot \frac{4}{3} \cdot \frac{12}{3}$

$V = \frac{5}{3} \cdot \frac{4}{3} \cdot \frac{4}{1}$ $\displaystyle V = \frac{5}{3} \cdot \frac{4}{3} \cdot \frac{4}{1}$

$V = \frac{80}{9 } = 8 \frac{8}{9 } \textrm{ yd}^{3}$ $\displaystyle V = \frac{80}{9 } = 8 \frac{8}{9 } \textrm{ yd}^{3}$

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All Common Core: 6th Grade Math Resources

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Common Core: 6th Grade Math : Grade 6

Study concepts, example questions & explanations for Common Core: 6th Grade Math

All Common Core: 6th Grade Math Resources

Example Questions

Example Question #502 : Plane Geometry

Example Question #503 : Plane Geometry

Example Question #511 : Plane Geometry

Example Question #512 : Plane Geometry

Example Question #513 : Plane Geometry

Example Question #514 : Plane Geometry

Example Question #515 : Plane Geometry

Example Question #516 : Plane Geometry

Example Question #517 : Plane Geometry

Example Question #1 : Solid Geometry

All Common Core: 6th Grade Math Resources

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