Area and volume: worksheet with answers

The perimeter of a rectangle is found using \(2l+2w\), where \(l\) is the length and \(w\) is the width.

Substitute the given values: \(2(18)+2(7)=36+14=50\). So the correct answer is 50.

Choice A doubles only the width: \(2(7)=14\). Choice B doubles only the length: \(2(18)=36\). Choice D treats all four sides as 18 units: \(4(18)=72\).

The area of a triangle is found using the formula \(A=\frac{1}{2}bh\), where \(b\) is the base and \(h\) is the perpendicular height.

Substitute the given values:

\(A=\frac{1}{2}(18)(14)\)

First multiply: \(18\times14=252\).

Then divide by 2: \(\frac{252}{2}=126\).

So, the area of the triangle is \(126\) square feet.

Use the formula for the area of a triangle: \(A=\frac{1}{2}bh\), where \(b\) is the base and \(h\) is the height.

Substitute the given values:

\(A=\frac{1}{2}(18)(14)\)

\(18 \times 14 = 252\), and \(\frac{252}{2}=126\).

So, the area of the triangle is \(126\) square feet.

The area of a rectangle is found using the formula \(A=l\times w\).

Substitute the given values: \(A=9\times 5=45\).

So, the area is \(45\) square feet. Choice C is correct.

Choice A comes from adding the dimensions: \(9+5=14\). Choice B comes from finding the perimeter: \(2(9+5)=28\). Choice D comes from incorrectly squaring the sum of the dimensions.

Use the prism volume formula \(V=Bh\).

Substitute the given values:

\(540=B(12)\)

Divide both sides by 12:

\(B=\frac{540}{12}=45\)

The area of the base is \(45\) square feet.

For a cylinder, the volume is equal to the area of the base multiplied by the height:

\(V=Bh\)

Substitute the given values:

\(1{,}260=70h\)

Divide both sides by \(70\):

\(h=\frac{1{,}260}{70}=18\)

Therefore, the height of the cylinder is \(18\) inches.

The volume of a cylinder can be found using \(V=Bh\), where \(V\) is the volume, \(B\) is the area of the base, and \(h\) is the height.

Substitute the given values:

\(1{,}080=45h\)

Divide both sides by \(45\):

\(h=\frac{1{,}080}{45}=24\)

So, the height of the cylinder is \(24\) inches.

Use the triangle area formula \(A=\frac{1}{2}bh\), where \(b\) is the base and \(h\) is the height.

Substitute the given values:

\(A=\frac{1}{2}(84)(26)\)

\(84\times 26=2184\), and \(\frac{2184}{2}=1092\).

So, the area of the triangle is \(1092\) square feet.

For similar figures, the ratio of the areas is the square of the ratio of the corresponding side lengths.

The side-length scale factor from \(JKLMNO\) to \(RSTUVW\) is \(4\), so the area scale factor is \(4^2=16\).

Since \(RSTUVW\) is the larger hexagon and has area \(180\), the area of \(JKLMNO\) is \(\frac{180}{16}=\frac{45}{4}=11.25\).

The volume of a sphere is \(V=\frac{4}{3}\pi r^{3}\). Since a hemisphere is half of a sphere, its volume is

\(V=\frac{1}{2}\left(\frac{4}{3}\pi r^{3}\right)=\frac{2}{3}\pi r^{3}\).

Substitute \(r=18\):

\(V=\frac{2}{3}\pi (18^{3})\).

Because \(18^{3}=5832\),

\(V=\frac{2}{3}\pi (5832)=3888\pi\).

Using \(\pi\approx 3.14\),

\(3888(3.14)\approx 12208\).

The closest answer choice is 12,200.

The volume of a sphere is \(V=\frac{4}{3}\pi r^{3}\). Because a hemisphere is half of a sphere, its volume is

\(V=\frac{1}{2}\left(\frac{4}{3}\pi r^{3}\right)=\frac{2}{3}\pi r^{3}\).

Substitute \(r=14\):

\(V=\frac{2}{3}\pi(14^{3})\).

Since \(14^{3}=2744\),

\(V\approx \frac{2}{3}\cdot 3.14 \cdot 2744\).

\(\frac{2}{3}\cdot 2744\approx 1829.3\), and

\(1829.3\times 3.14\approx 5744\).

The closest choice is 5,700.

For similar figures, areas scale by the square of the linear scale factor. Since the larger trapezoid has side lengths \(\frac{3}{2}\) times those of the smaller trapezoid, the area scale factor is \(\left(\frac{3}{2}\right)^2=\frac{9}{4}\).

Let \(x\) be the area of the smaller trapezoid. Then

\(x\cdot \frac{9}{4}=54\)

Solving gives

\(x=54\cdot \frac{4}{9}=24\).

So the correct answer is 24.