Volume of Shapes 4 | VividMath

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Year 8

Surface Area and Volume

Volume of Shapes

Volume of Shapes 4

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Question 1 of 6

1. Question

Find the volume of the Pyramid

Round your answer to one decimal place

$\text(Volume )=$ (1326.6, 1324.9) $\text(cm)^3$

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Volume of a Pyramid

$\text(Volume )=1/3 times$ $\text(length)$

$times$ $\text(width)$

$times$ $\text(height)$

Labelling the given lengths

$\text(length)=18$

$\text(width)=11$

$\text(height)=?$

$\text(b (right triangle))=9$

$($ $18$

$divide2)$

$\text(c (right triangle))=22$

First, we need to find the height perpendicular to its base.

Label the sides of the right triangle formed within the pyramid

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

$a^2$ $+$ $b^2$	$=$	$c^2$	Pythagoras’ Theorem Formula
$\text(height)^2$ $+$ $9^2$	$=$	$22^2$	Plug in the known lengths
$\text(height)^2+81$	$=$	$484$	Evaluate $9^2$ and $22^2$
$\text(height)^2$	$=$	$403$	Subtract $81$ from both sides
$\text(height)$	$=$	$20.1 \text(cm)$	Take the square root of both sides

Next, find the area of the pyramid’s base, which is a rectangle

$\text(Area)$	$=$	$\text(length)$ $times$ $\text(width)$	Area of a Rectangle
	$=$	$18$ $times$ $11$	Plug in the known lengths
	$=$	$198 \text(cm)^2$

Finally, use the formula to find the volume

Note that $\text(area)$

$=$ $\text(length)$

$times$ $\text(width)$

$\text(Volume)$	$=$	$1/3 times$ $\text(length)$ $times$ $\text(width)$ $times$ $\text(height)$	Volume of a Pyramid
	$=$	$1/3 times$ $198$ $times$ $20.1$	Plug in the known lengths
	$=$	$1326.6 \text(cm)^3$

The given measurements are in centimetres, so the volume is measured as centimetres cubed

$\text(Volume)=1326.6 \text(cm)^3$

Question 2 of 6

2. Question

What is the volume of this cylinder?

Round your answer to

$2$ decimal places

Use

$pi~~3.14$

Volume $=$ (5281.24) $mm^3$

Correct

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Volume of a Cylinder

$V=pi xx color(royalblue)(\text(radius)^2) xx color(darkviolet)(\text(height))$

Labelling the given lengths

$color(darkviolet)(\text(height)=34)$

$color(royalblue)(\text(radius)=7)$

Use the formula to find the volume

$pi~~3.14$

$V$	$=$	$pi xx color(royalblue)(\text(radius)^2) xx color(darkviolet)(\text(height))$	Volume of a cylinder formula

	$=$	$3.14 xx color(royalblue)(7^2) xx color(darkviolet)(34)$	Plug in the known lengths

	$=$	$3.14 xx 49 xx 34$	Simplify

	$=$	$5,281.24 \ mm^3$	Rounded to 2 decimal places

The given measurements are in millimetres, so the volume is measured as millimetres cubed

Volume

$=5,281.24 \ mm^3$

Question 3 of 6

3. Question

What is the volume of this cone?

Round your answer to

$2$ decimal places

Use

$pi~~3.14$

Volume $=$ (15113.87) $cm^3$

Correct

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Incorrect

Volume of a Cone

$V=1/3 xx pi xx color(royalblue)(\text(radius)^2) xx color(darkviolet)(\text(height))$

Labelling the given lengths

$color(darkviolet)(\text(height)=40)$

$color(royalblue)(\text(radius)=19)$

Use the formula to find the volume

$pi~~3.14$

$V$	$=$	$1/3 xx pi xx color(royalblue)(\text(radius)^2) xx color(darkviolet)(\text(height))$	Volume of a cone formula

	$=$	$1/3 xx 3.14 xx color(royalblue)(19^2) xx color(darkviolet)(40)$	Plug in the known lengths

	$=$	$1/3 xx 3.14 xx 361 xx 40$	Simplify

	$=$	$15,113.86667$
	$=$	$15,113.87 \ cm^3$	Rounded to 2 decimal places

The given measurements are in centimetres, so the volume is measured as centimetres cubed

Volume

$=15,113.87 \ cm^3$

Question 4 of 6

4. Question

What is the volume of this Rectangular Pyramid?

Volume $=$ (50) $mm^3$

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Volume of a Rectangular Pyramid

$V=1/3 xx color(royalblue)(\text(length)) xx color(darkviolet)(\text(width))xx color(green)(\text(height))$

Labelling the given lengths

$color(royalblue)(\text(length)=5)$

$color(darkviolet)(\text(width)=3)$

$color(green)(\text(height)=10)$

Use the formula to find the volume

$V$	$=$	$1/3 xx color(royalblue)(\text(length)) xx color(darkviolet)(\text(width))xx color(green)(\text(height))$	Volume of a Rectangular Pyramid formula

	$=$	$1/3 xx color(royalblue)(\text(5)) xx color(darkviolet)(\text(3))xx color(green)(\text(10))$	Plug in the known lengths

	$=$	$50$
	$=$	$50 \ mm^3$

The given measurements are in millimetres, so the volume is measured as millimetres cubed

Volume

$=50 \ mm^3$

Question 5 of 6

5. Question

Find the volume of the Cone

Round your answer to

$1$ decimal place

Use

$pi=3.141592654$

$\text(Volume )=$ (5305.8, 5303.1, 5307.9) $\text(cm)^3$

Hint

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Volume of a Cone

$\text(Volume)=1/3 times pi times$ $\text(radius)^2$

$times$ $\text(height)$

Labelling the given lengths

$\text(radius)=?$

$\text(diameter)=23$

$\text(height)=?$

$\text(c (right triangle))=40$

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

$\text(radius)$	$=$	$1/2 times$ $23$

$\text(radius)$	$=$	$11.5$

Next, we need to find the height perpendicular to its base.

Label the sides of the right triangle formed within the pyramid

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

$a^2$ $+$ $b^2$	$=$	$c^2$	Pythagoras’ Theorem Formula
$\text(height)^2$ $+$ $11.5^2$	$=$	$40^2$	Plug in the known lengths
$\text(height)^2+11.5^2$	$=$	$40^2$	Evaluate $11.5^2$ and $40^2$
$\text(height)^2+132.25$	$=$	$1600$	Subtract $132.25$ from both sides
$sqrt(\text(height)^2)$	$=$	$sqrt(1467.75)$	Take the square root of both sides
$\text(height)$	$=$	$38.3112255 \text(cm)$

Finally, use the formula to find the volume

Use

$pi=3.141592654$ See

$pi$ explained

$\text(Volume)$	$=$	$1/3 times pi times$ $\text(radius)^2$ $times$ $\text(height)$	Volume of a Cone formula

	$=$	$1/3 times 3.141592654 times$ $11.5^2$ $times$ $38.3112255$	Plug in the known lengths

	$=$	$1/3 times 3.141592654 times 132.25 times 38.3112255$	Simplify

	$=$	$5305.79349$
	$=$	$5305.8 \text(cm)^3$	Rounded to one decimal place

The given measurements are in centimetres, so the volume is measured as centimetres cubed

$\text(Volume)=5305.8 \text(cm)^3$

The answer will depend on which

$pi$ you use.

In this solution we used:

$pi=3.141592654$ .

Using	Answer
$pi=3.141592654$	$5305.8 cm^3$
$pi=3.14$	$5303.1 cm^3$
$pi=(22)/(7)$	$5307.9 cm^3$

Question 6 of 6

6. Question

What is the volume of this hemisphere?

Round your answer to

$2$ decimal places

Use

$pi~~3.14$

Volume $=$ (1526.04) $cm^3$

Correct

Keep Going!

Incorrect

Volume of a Hemisphere

$V=2/3 xx pi xx color(royalblue)(\text(radius)^3)$

Labelling the given lengths

$color(royalblue)(\text(radius)=9)$

Use the formula to find the volume

$pi~~3.14$

$V$	$=$	$2/3 xx pi xx color(royalblue)(\text(radius)^3)$	Volume of a hemisphere formula

	$=$	$2/3 xx 3.14 xx color(royalblue)(9^3)$	Plug in the known lengths

	$=$	$2/3 xx 3.14 xx 729$	Simplify

	$=$	$1,526.04 \ cm^3$	Rounded to 2 decimal places

The given measurements are in centimetres, so the volume is measured as centimetres cubed

Volume

$=1,526.04 \ cm^3$

Quizzes

Volume of Shapes 1
Volume of Shapes 2
Volume of Shapes 3
Volume of Shapes 4
Volume of Composite Shapes 1
Volume of Composite Shapes 2
Surface Area of Shapes 1
Surface Area of Shapes 2
Surface Area of Shapes 3
Surface Area and Volume Mixed Review 1
Surface Area and Volume Mixed Review 2
Surface Area and Volume Mixed Review 3
Surface Area and Volume Mixed Review 4