Surface Area of Shapes 1

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Geometry

Surface Area and Volume

Surface Area of Shapes

Surface Area of Shapes 1

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Question 1 of 7

1. Question

Which of the following shows the formula for the surface area of a cylinder?

`\text(SA) = 2pi\text(r)^2 + 2pi\text(r)\text(h)`
`\text(SA) = pi\text(r)^2 + pi\text(r)\text(h)`
`\text(SA) = 2pi\text(r)^2\text(h) `
`\text(SA) = 2/3 pi\text(r)^2 + 2pi\text(r)\text(h)`

Hint

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A surface area is the total area of the outside faces of a figure.

First, we need to show the surfaces of the cylinder.

Since we are looking for the area, we can use the formula for Area of a Circle for the two circles


`\text(Area)`	`=`	`(pi times \text(r)^2) times 2`	Multiply by `2` since there are two circles
`\text(Area)`	`=`	`2 pi \text(r)^2`

Next, notice that the curved side creates a rectangle that has a length equal to the circumference of a circle

Hence, we can use the formula for the circumference of a circle as the value of length


`\text(Area)`	`=`	`\text(length) times \text(height)`	Area of a Rectangle formula
`\text(Area)`	`=`	`(2 pi \text(r)) times \text(h)`	Substitute the circumference formula to length
`\text(Area)`	`=`	`2pi\text(r)\text(h)`

Finally, add the two formula to get the formula for the Surface Area of a cylinder

`\text(SA)`

`=`

`2pi\text(r)^2+2pi\text(r)\text(h)`

`\text(SA)=2pi\text(r)^2+2pi\text(r)\text(h)`

Question 2 of 7

2. Question

What is the surface area of this cube?

Surface Area`=` (54)`cm^2`

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Surface Area of a Cube

`SA= 6 xx color(royalblue)(text(side))^2`

Labelling the given lengths

`color(royalblue)(\text(side)=3)`

Use the formula to find the surface area

`SA`	`=`	`6 xx color(royalblue)(text(side))^2`	Surface area of a cube formula

	`=`	`6 xx color(royalblue)(text(3))^2`	Plug in the known lengths

	`=`	`6 xx 9`	Simplify
	`=`	`54`
	`=`	`54 \ cm^2`

The given measurements are in centimetres, so the surface area is measured as centimetres squared

Volume`=54 \ cm^2`

Question 3 of 7

3. Question

What is the surface area of this cube?

Surface Area`=` (216)`m^2`

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Surface Area of a Cube

`SA= 6 xx color(royalblue)(text(side))^2`

Labelling the given lengths

`color(royalblue)(\text(side)=6)`

Use the formula to find the surface area

`SA`	`=`	`6 xx color(royalblue)(text(side))^2`	Surface area of a cube formula

	`=`	`6 xx color(royalblue)(text(6))^2`	Plug in the known lengths

	`=`	`6 xx 36`	Simplify
	`=`	`216`
	`=`	`216 \ m^2`

The given measurements are in metres, so the surface area is measured as metres squared

Volume`=216 \ m^2`

Question 4 of 7

4. Question

Find the surface area of the Sphere

Round your answer to `1` decimal place

Use `pi=3.141592654`

`\text(Surface Area )=` (15393.8, 15386, 15400) `\text(cm)^2`

Hint

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Correct

Fantastic!

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Surface Area of a Sphere

`\text(SA )=4 times pi times``\text(radius)^2`

Showing and Labelling the Surfaces

Use the formula to find the surface area of the sphere

Use `pi=3.141592654` See `pi` explained

`\text(SA)`	`=`	`4 times pi times``\text(radius)^2`	Surface Area formula
`\text(SA)`	`=`	`4 times pi times``35^2`	Plug in the known lengths
`\text(SA)`	`=`	`15393.804`
`\text(SA)`	`=`	`15393.8 \text(cm)^2`	Rounded to one decimal place

The given measurements are in centimetres, so the area is measured as square centimetres

`\text(SA)=15393.8 \text(cm)^2`

The answer will depend on which `pi` you use.

In this solution we used: `pi=3.141592654`.

Using	Answer
`pi=3.141592654`	`15393.8 cm^2`
`pi=3.14`	`15386 cm^2`
`pi=(22)/(7)`	`15400 cm^2`

Question 5 of 7

5. Question

What is the surface area of this Rectangular Prism?

Surface Area`=` (384)`cm^2`

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Surface Area of a Rectangular Prism

`SA= 2 xx (color(royalblue)(text(width)) xx color(green)(text(height)) + color(orange)(text(depth)) xx color(green)(text(height)) + color(royalblue)(text(width)) xx color(orange)(text(depth)))`

Labelling the given lengths

`color(royalblue)(text(width)=4)`

`color(green)(text(height)=12)`

`color(orange)(text(depth)=9)`

Use the formula to find the surface area

`SA`	`=`	`2 xx (color(royalblue)(text(width)) xx color(green)(text(height)) + color(orange)(text(depth)) xx color(green)(text(height)) + color(royalblue)(text(width)) xx color(orange)(text(depth)))`

	`=`	`2 xx (color(royalblue)(text(4)) xx color(green)(text(12)) + color(orange)(text(9)) xx color(green)(text(12)) + color(royalblue)(text(4)) xx color(orange)(text(9)))`

	`=`	`384`	Simplify

	`=`	`384 \ cm^2`

The given measurements are in centimetres, so the surface area is measured as centimetres squared

Volume`=384 \ cm^2`

Question 6 of 7

6. Question

Find the surface area of the Pyramid

The given measurements are in units

`\text(Surface Area )=` (360) `\text(cm)^2`

Hint

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Area of a Triangle Formula

`\text(Area )=1/2 times``\text(base)``times``\text(height)`

Area of a Square Formula

`\text(Area )=``\text(side)``times``\text(side)`

In a regular square, all sides are equal

Showing and Labelling the Surfaces

We need to add the areas of all the faces of the pyramid: the four triangles and the square base

Since the triangle sides are slanted, we need to find the slanted height (which is also the perpendicular height) of the triangles first.

Label the sides of the right triangle formed within the pyramid

Use the Pythagorean Theorem Formula to solve for `c`, which is equal to the `\text(height)`

`a^2``+``b^2`	`=`	`c^2`	Pythagoras’ Theorem Formula
`12^2``+``5^2`	`=`	`\text(h)^2`	Plug in the known lengths
`169`	`=`	`\text(h)^2`
`\text(height)`	`=`	`13 \text(cm)`	Take the square root of both sides

Next, solve for the area of the triangles using the Area of a Triangle formula

Note that there are four sides of the pyramid with the same lengths, so we will multiply this area by `4` for the surface area

`\text(Area)`_{`\text(triangles)`}	`=`	`1/2 times``\text(base)``times``\text(height)`

	`=`	`1/2 times``10``times``13``=``65 \text(cm)^2`

Now, solve for the area of the square base using the Area of a Square formula

`\text(Area)`_{`\text(square)`}	`=`	`\text(side)``times``\text(side)`
	`=`	`10``times``10``=``100 \text(cm)^2`

Finally, add all the areas to find the surface area of the figure

`\text(SA)`	`=`	`(4times``65``)+``100`	Plug in the areas
`\text(SA)`	`=`	`360 \text(cm)^2`

The given measurements are in centimetres, so the area is measured as square centimetres

`\text(SA)=360 \text(cm)^2`

Question 7 of 7

7. Question

Find the surface area of the open Cylinder

Round your answer to `2` decimal places

Use `pi=3.141592654`

`\text(Surface Area )=` (829.38, 828.96, 829.71) `\text(m)^2`

Hint

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Correct

Excellent!

Incorrect

Surface Area of an Open Cylinder

`2pi\text(r)\text(h)+pi\text(r)^2`

Showing and Labelling the Surfaces

We need to add the areas of all the faces of the figure: the curved face and the circular base

Since this is an Open Cylinder we will edit the original formula.

`\text(Surface Area of a Cylinder)`	`=`	`2pi\text(r)\text(h)+``2``pi\text(r)^2`
`\text(Surface Area of an Open Cylinder)`	`=`	`2pi\text(r)\text(h)+pi\text(r)^2`	Notice we eliminate the `2`

Now, recall that the radius is equal to half of the diameter

`\text(radius)`	`=`	`1/2 times ``12`

`\text(radius)`	`=`	`6`

Finally, use the formula to solve for the surface area of the open cylinder

Use `pi=3.141592654` See `pi` explained

`\text(SA)`	`=`	`2pi``\text(r)``\text(h)` `+pi``\text(r)^2`	Formula for the surface area
`\text(SA)`	`=`	`2 times pi times``6``times``19` `+pi times``6^2`	Plug in the known lengths
`\text(SA)`	`=`	`829.38046`
`\text(SA)`	`=`	`829.38 \text(m)^2`	Rounded to two decimal places

The given measurements are in metres, so the area is measured as square metres

`\text(SA)=829.38 \text(m)^2`

The answer will depend on which `pi` you use.

In this solution we used: `pi=3.141592654`.

Using	Answer
`pi=3.141592654`	`829.38 m^2`
`pi=3.14`	`828.96 m^2`
`pi=(22)/(7)`	`829.71 m^2`

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