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6th Grade Math : Geometry

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Example Questions

Example Question #1 : Geometry

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

Possible Answers:

$65.5\textup{ in}^2$ $\displaystyle 65.5\textup{ in}^2$

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

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

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

Correct answer:

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

Explanation:

In order to solve this problem, we need to recall the formula for area of a right triangle:

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

Now we can substitute in our side lengths from the question:.

$A=\frac{13\times9}{2}$ $\displaystyle A=\frac{13\times9}{2}$

$A=\frac{117}{2}$ $\displaystyle A=\frac{117}{2}$

$\displaystyle A=58.5^2$

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Example Question #2 : Geometry

What is the volume of the rectangular prism in the following figure?

Possible Answers:

$146.5\textup{ cm}^3$ $\displaystyle 146.5\textup{ cm}^3$

$140\textup{ cm}^3$ $\displaystyle 140\textup{ cm}^3$

$138.5\textup{ cm}^3$ $\displaystyle 138.5\textup{ cm}^3$

$144\textup{ cm}^3$ $\displaystyle 144\textup{ cm}^3$

Correct answer:

$140\textup{ cm}^3$ $\displaystyle 140\textup{ cm}^3$

Explanation:

The formula used to find volume of a rectangular prism is as follows:

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

Substitute our side lengths:

$A=3\frac{1}{2}\times4\times10$ $\displaystyle A=3\frac{1}{2}\times4\times10$

$A=140\textup{ cm}^3$ $\displaystyle A=140\textup{ cm}^3$

Remember, volume is always written with cubic units because volume is how many cubic units can fit inside of a figure.

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Example Question #3 : Geometry

Hydraulic fracturing is a process used by gas companies to rupture and collects pockets of gas trapped within pockets of shale rock. A particular shale fracking site is $\frac{3}{4}\textup{ miles}$ $\displaystyle \frac{3}{4}\textup{ miles}$ in length and occupies an area of $\frac{1}{2}\textup{ miles}^2$ $\displaystyle \frac{1}{2}\textup{ miles}^2$. How wide is this particular site?

Possible Answers:

$\frac{2}{3}\textup{ miles}$ $\displaystyle \frac{2}{3}\textup{ miles}$

$1\textup{ mile}$ $\displaystyle 1\textup{ mile}$

$\frac{1}{3}\textup{ miles}$ $\displaystyle \frac{1}{3}\textup{ miles}$

$2\textup{ miles}$ $\displaystyle 2\textup{ miles}$

Correct answer:

$\frac{2}{3}\textup{ miles}$ $\displaystyle \frac{2}{3}\textup{ miles}$

Explanation:

In order to solve this question, we need to first recall how to find the area of a rectangle.

$\text{Area}=\text{Length} \times \text{Width}$ $\displaystyle \text{Area}=\text{Length} \times \text{Width}$

Substitute in the given values in the equation and solve for $\text{Width}$ $\displaystyle \text{Width}$.

$\frac{1}{2}\ miles^2=\frac{3}{4}\ miles\times Width$ $\displaystyle \frac{1}{2}\ miles^2=\frac{3}{4}\ miles\times Width$

Divide both sides by $\frac{3}{4}\ miles$ $\displaystyle \frac{3}{4}\ miles$

$\frac{\frac{1}{2}\ miles^2}{\frac{3}{4}\ miles}=\frac{\frac{3}{4}\ miles \times Width}{\frac{3}{4}\ miles}$ $\displaystyle \frac{\frac{1}{2}\ miles^2}{\frac{3}{4}\ miles}=\frac{\frac{3}{4}\ miles \times Width}{\frac{3}{4}\ miles}$

Dividing by a fraction is the same as multiplying by its inverse or reciprocal.

Find the reciprocal of $\frac{3}{4}\ miles$ $\displaystyle \frac{3}{4}\ miles$

$\frac{3}{4}\ miles\rightarrow \frac{4}{3}\ miles$ $\displaystyle \frac{3}{4}\ miles\rightarrow \frac{4}{3}\ miles$

Simplify and rewrite.

$Width=\frac{4}{3} \times \frac{1}{2}$ $\displaystyle Width=\frac{4}{3} \times \frac{1}{2}$

Multiply and solve.

$Width=\frac{4}{6}$ $\displaystyle Width=\frac{4}{6}$

Reduce.

$Width=\frac{2}{3}\ miles$ $\displaystyle Width=\frac{2}{3}\ miles$

The width of the fracking site is

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6th Grade Math : Geometry

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

All 6th Grade Math Resources

Example Questions

Example Question #1 : Geometry

Example Question #2 : Geometry

Example Question #3 : Geometry

All 6th Grade Math Resources

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