Math6 Quarter 4 Week 2 │Problem Solving involving Volume

Summary

This educational video script delves into the concept of volume, explaining it as the measure of space occupied by solid figures. It covers the volume formulas for various shapes, including cubes, rectangular prisms, pyramids, cylinders, cones, and spheres. The script demonstrates how to calculate the volume of each shape with given dimensions, providing step-by-step examples for a rectangular prism, triangular prism, pyramid, cylinder, cone, and sphere. It also includes practical problems, such as calculating the volume of an ice cream cone and a water tank, and concludes with the volume of a cube-shaped chalkbox. The video aims to teach viewers how to solve volume problems in mathematics.

Takeaways

• 📏 Volume is the number of cubic units contained in a space figure.
• 📐 The formula for the volume of a cube is side³ or side × side × side.
• 📦 For a rectangular prism, the volume is length × width × height.
• 🗻 The volume of a square pyramid is (side² × height) / 3.
• 🔺 For a rectangular pyramid, the volume is (length × width × height) / 3.
• 🔵 A cylinder's volume is π × radius² × height.
• 🔺 The volume of a cone is (π × radius² × height) / 3.
• ⚪ The formula for the volume of a sphere is (4/3) × π × radius³.
• 📏 Example: The volume of a rectangular prism with dimensions 7 cm × 3 cm × 4 cm is 84 cubic cm.
• 📐 Example: The volume of a triangular prism with a base of 12 cm, a height of 16 cm, and a prism height of 30 cm is 2880 cubic cm.

Q & A

What is the definition of volume in mathematics?

Volume is the number of cubic units contained in a space figure.

What is the formula for finding the volume of a cube?

The formula for the volume of a cube is \( s^3 \) or side times side times side.

How do you find the volume of a rectangular prism?

The formula for the volume of a rectangular prism is length times width times height.

What is the formula for the volume of a square pyramid?

The formula for the volume of a square pyramid is \( s^2 \times \text{height} / 3 \).

How is the volume of a rectangular pyramid calculated?

The formula for the volume of a rectangular pyramid is length times width times height divided by 3.

What is the formula for finding the volume of a cylinder?

The formula for the volume of a cylinder is \( \pi \times \text{radius}^2 \times \text{height} \).

How do you calculate the volume of a cone?

The formula for the volume of a cone is \( \pi \times \text{radius}^2 \times \text{height} / 3 \).

What is the formula for the volume of a sphere?

The formula for the volume of a sphere is \( 4/3 \times \pi \times \text{radius}^3 \).

How do you find the volume of a rectangular prism with a length of 7 cm, width of 3 cm, and height of 4 cm?

The volume is found by multiplying the length, width, and height: \( 7 \times 3 \times 4 = 84 \) cubic centimeters.

How is the volume of a triangular prism calculated?

The formula for the volume of a triangular prism is base area times height of the prism. For a triangle base, the base area is \( \text{base} \times \text{height} / 2 \).

What is the volume of a triangular prism with a triangle base area of 96 square meters and a height of 30 meters?

The volume is \( 96 \times 30 = 2880 \) cubic meters.

How is the volume of a rectangular pyramid with a length of 6 cm, width of 4 cm, and height of 5 cm calculated?

The volume is \( 6 \times 4 \times 5 / 3 = 40 \) cubic centimeters.

How do you find the volume of a cylinder with a radius of 2.2 cm and a height of 14.6 cm?

The volume is \( \pi \times 2.2^2 \times 14.6 \approx 221.88 \) cubic meters.

What is the volume of a cone with a radius of 6 cm and height of 10 cm?

The volume is \( \pi \times 6^2 \times 10 / 3 \approx 376.8 \) cubic centimeters.

How do you calculate the volume of a sphere with a diameter of 30 cm?

First, find the radius by dividing the diameter by 2, which is 15 cm. Then, use the formula \( 4/3 \times \pi \times 15^3 \approx 14130 \) cubic centimeters.

What is the volume of an ice cream cone with a diameter of 32 mm and height of 45 mm?

The volume is \( \pi \times 16^2 \times 45 / 3 \approx 12057.6 \) cubic millimeters.

How many liters of water can a water tank with an interior height of 10 meters and diameter of 6 meters hold if it is half full?

The tank's volume is \( \pi \times 3^2 \times 10 \approx 282.6 \) cubic meters, so half full it can hold approximately 141.3 cubic meters of water.

What is the volume of a pyramid with a base area of 84 square decimeters and height of 16 decimeters?

The volume is \( 84 \times 16 / 3 = 448 \) cubic decimeters.

How do you find the volume of a chalk box with each edge measuring 18 cm?

The volume is \( 18^3 = 5832 \) cubic centimeters.

Outlines

00:00

📚 Introduction to Volume Calculations

The script introduces the concept of volume in mathematics, explaining it as the number of cubic units contained within a space. It presents formulas for calculating the volume of various solid figures, including cubes, rectangular prisms, pyramids (both square and rectangular), cylinders, cones, and spheres. The script then demonstrates the application of these formulas through examples, starting with a rectangular prism with dimensions of 7 cm in length, 3 cm in width, and 4 cm in height, calculating its volume to be 84 cubic centimeters. It continues with the calculation of a triangular prism, a rectangular pyramid, a cylinder, a cone, and a sphere, providing step-by-step instructions and results for each.

05:01

🍦 Practical Applications of Volume Calculations

This paragraph extends the discussion on volume calculations to practical scenarios. It addresses how to calculate the volume of an ice cream cone given its diameter and height, resulting in a volume of 12057.6 cubic millimeters. It then explores the capacity of a water tank with a diameter of 6 meters and a height of 10 meters, determining how much water it can hold when half full, which is 141.2 cubic meters. The script also calculates the volume of a pyramid with a base area of 84 square decimeters and a height of 16 decimeters, finding it to be 448 cubic decimeters. Lastly, it computes the volume of a cube-shaped chalkbox with an edge length of 18 centimeters, which equals 5832 cubic centimeters. The paragraph concludes by emphasizing the importance of remembering these volume formulas for solving various mathematical problems.

Keywords

💡Volume

Volume refers to the amount of space that a substance or object occupies, typically measured in cubic units. In the context of the video, volume is the central theme as it discusses the calculation of space contained within various geometric shapes. For instance, the script explains how to calculate the volume of a cube, rectangular prism, and other solids, emphasizing the importance of volume in mathematics and its applications.

💡Cubic Units

Cubic units are units of measurement used to express volume, such as cubic centimeters or cubic meters. The script mentions cubic units when introducing the concept of volume and when providing formulas for calculating the volume of different shapes, indicating the necessity of these units in quantifying three-dimensional space.

💡Formula

A formula in mathematics is a concise way of expressing information symbolically, often involving numbers, symbols, and mathematical operations. The video script provides various formulas for calculating the volume of geometric shapes, such as 's^3' for a cube and 'length × width × height' for a rectangular prism, demonstrating the use of formulas in solving mathematical problems related to volume.

💡Cube

A cube is a three-dimensional shape with six equal square faces. The script uses the cube as an example of a geometric solid whose volume is calculated using the formula 's^3', where 's' is the length of one side. This illustrates the concept of volume calculation for a simple yet fundamental shape.

💡Rectangular Prism

A rectangular prism is a three-dimensional shape with six rectangular faces, sometimes referred to as a rectangular solid. The script explains how to find its volume using the formula 'length × width × height', showcasing a common geometric shape and its volume calculation method.

💡Pyramid

A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a single point called the apex. The video script distinguishes between square and rectangular pyramids, providing formulas for their volumes, such as 's^2 × h / 3' for a square pyramid, and emphasizing the unique geometry of these shapes.

💡Cylinder

A cylinder is a geometric shape with parallel and congruent circular bases connected by a curved surface. The script describes the volume of a cylinder using the formula 'πr^2 × h', where 'r' is the radius and 'h' is the height, highlighting the role of pi in calculating the volume of this shape.

💡Cone

A cone is a geometric shape with a circular base and a single vertex, tapering smoothly from the base to the vertex. The script uses the formula 'πr^2 × h / 3' to calculate the volume of a cone, demonstrating how the volume differs from that of a cylinder due to the division by 3.

💡Sphere

A sphere is a perfectly symmetrical three-dimensional shape where all points on the surface are equidistant from the center. The video script explains the volume of a sphere with the formula '4/3 × πr^3', showcasing the unique formula that accounts for the spherical shape's volume.

💡Pi (π)

Pi, represented by the Greek letter 'π', is a mathematical constant approximately equal to 3.14159. It is used in the formulas for calculating the volume of circular and spherical shapes, such as cylinders and spheres. The script mentions π in the context of these formulas, emphasizing its importance in geometry and trigonometry.

💡Problem Solving

Problem solving in mathematics involves applying knowledge and techniques to find solutions to complex questions. The script presents several problems, such as calculating the volume of an ice cream cone and a water tank, demonstrating the practical application of volume formulas in solving real-world problems.

Highlights

Introduction to the concept of volume as the number of cubic units contained in a space.

Volume formulas for different solid figures: cube, rectangular prism, square pyramid, rectangular pyramid, cylinder, cone, and sphere.

Calculation of the volume of a rectangular prism with dimensions 7 cm x 3 cm x 4 cm, resulting in 84 cubic centimeters.

Explanation of the volume formula for a triangular prism, involving the area of the triangular base and the height of the prism.

Volume calculation of a triangular prism with a base of 12, height of the triangle 16, and prism height 30, yielding 2880 cubic meters.

Rectangular pyramid volume formula application with dimensions 6 cm x 4 cm x 5 cm, resulting in 40 cubic centimeters.

Cylinder volume calculation using the formula with pi, radius, and height, resulting in 221.88 cubic meters.

Cone volume calculation with the formula involving pi, radius, height, and division by 3, resulting in 376.8 cubic centimeters.

Sphere volume calculation using the formula with pi, radius cubed, and division by 3, resulting in 14,130 cubic centimeters.

Problem-solving example: Calculating the volume of an ice cream cone with a diameter of 32 mm and height of 45 mm.

Calculating the volume of water a tank can hold when half-full, given its interior height and diameter.

Volume calculation of a pyramid with a base area of 84 square decimeters and a height of 16 decimeters.

Determining the volume of a chalkbox with each edge measuring 18 centimeters using the cube formula.

Emphasizing the importance of remembering volume formulas for various geometric shapes in problem-solving.

Encouragement for new viewers to subscribe and enable notifications for more math and science content.

Transcripts

[Music]

our subject

is mathematics 6.

our last topic is volume when we say

volume

it is the number of cubic units

contained in a space

figure these are the following formula

of volume for the different solid figure

for cube we have s

raised to 3 or side times side times

rectangular prism the formula is length

times weight times height

four pyramid for square pyramid we have

s squared times height divided by three

for rectangular pyramid

length times width times height divided

by 3

for cylinder pi radius squared times

height

for cone pi radius squared times height

divided by 3

and for sphere 4 pi

radius cubed divided by 3.

now let us find the volume of the

following figure

this figure is rectangular prism with

length of 7 centimeter width is 3

centimeter and height is 4 centimeter

the formula in finding the volume of

rectangular prism

is length times weight times height we

will substitute the value for us to find

the volume

so length is 7 centimeter width is 3

centimeter then

height is 4 centimeter so we will

multiply 7 times 3 times 4

is equal to 84 cubic centimeter so the

volume

of the rectangular prism is 84 cubic

centimeter

now let us find the volume of triangular

prism

the formula is base times height divided

by 2

times height general formula in finding

the volume of a prism is base times

height

since the base of this figure is

triangle we will find its base area with

the formula

of base times height divided by 2. the

first

h is for the height of a triangle and

the second h

is for the height of the prism

so we will substitute so the base of the

triangle

is 12 and the height of the triangle is

16

then divided by 2 the height of the

prism is

30 so 12 times 16 divided by 2

is equal to 96 then copy 30.

96 times 30 is equal to 2880

so the volume of this triangular prism

is 2880

cubic meter next is

rectangular pyramid the formula is

length times width times height divided

by 3

so the length is 6 width is 4 and height

is 5 so 6 times 4 times 5

is equal to 120 then divided by 3

is equal to 40. so the volume of this

rectangular pyramid is 40 cubic

centimeters

next is cylinder the formula is pi

radius squared times

height so let us substitute uh the value

of pi

is 3.14 radius is 2.2 then height

is 14.6 so following the order of

operations

exponent so 2.2 times 2.2 is equal to

4.84

then we will multiply 3.14 times 4.84

times 14.6 so it is equal to

221.88 thousand

four hundred ninety six hundred thousand

so let us uh round off or estimate this

a number to the nearest

hundredths so the answer is 221.88

cubic meters so since akata binance 8 i4

so remain among young 8 so the volume of

the cylinder is 221.88 cubic

meters next is cone

the formula is pi range squared times

height divided by

3. so let's substitute pi is 3.14

radius is six then height is ten then

divided by

three so first is six times six is equal

to

thirty six then multiply three point

fourteen times thirty six

times ten is equal to one thousand one

hundred

four then divided by 3 the answer is

376.8

so the volume of this cone is 376.8

cubic centimeters

then sphere the formula is 4 pi

radius cube divided by 3.

so substitute the value so pi is 3.14

then we have 15 since the given is

diameter

then the half of the diameter is the

rejuve so 30 divided by 2 is

15. so um

15 cubed so 15 times 15 times 15 is

equivalent to three thousand three

hundred

seventy five then multiply the three

four times three point fourteen times

three thousand three hundred seventy

five

is equal to forty two thousand three

hundred ninety

divided by three so the answer is

fourteen thousand one hundred

thirty so the volume of the sphere is

fourteen thousand one hundred thirty

cubic centimeters

now let's have a problem solving first

is an ice cream cone has a diameter of

32 millimeters

and the height of 45 millimeters what is

its

volume so we have the diameter then

height so the formula is pi radius

squared times

i divided by three so since given

diameter

and dagnassa formula i rejoice in young

diameter

since angliameter po i

twice of rejoice then radius is half of

the diameter

so substitute pi is 3.14

radius is 16. so 32 divided by 2 is 16.

then height is 45. so 16 times 16 is

equal to 256

then multiply 3.14 times 256 times 45

is equal to 36 172.8 divided by 3

so the volume of the ice cream cone is

12057.6

cubic millimeters second problem

a water tank has an interior height of

10 meters and a diameter of 6 meters

how many liters of water can it hold

half

full so the formula is pi radius squared

times height

so again given on diameter so we will

divide six

by two so angry is not an i3

so we have three point fourteen times

three squared times

ten so three times three is equal to

nine then multiply three point fourteen

times

nine times ten is equal to 282.6 cubic

meters so since again ups a problem i

how many liters of water can it hold

half full so kalahati lung

so i'm 282.6 cubic meters

so we will divide 282.6

divided by two so the answer is

one hundred forty one point three cubic

meters so on volume

water tank half full i141

141.2 cubic meters

a pyramid has a height of 16 decimeter

its base is 84 square decimeter

what is its volume so the formula

is base times height divided by three so

this is the general formula for pyramid

so since given a young base angle

and general formula so base is

84 then height is 16 multiply 84 times

16 is equal to

1344 divided by 3

the volume of the pyramid is 448 cubic

decimeters a chock box measures 18

centimeter

on each edge what is the volume of the

box so the formula

is s cube or side time side

times side so we have 18 raised to three

so we will multiply 18 by itself three

times

so 18 times 18 times 18 is equal to

5832 so the volume of the chalkbox is

5832 cubic centimeters

always remember this formula in solving

volume

so we have four cube rectangular prism

square pyramid rectangular pyramid

cylinder

cone and sphere

that's the end of our discussion for

today i hope you learned how to solve

volume thank you for watching if you are

new in our channel

please subscribe and click the

notification bell for more

math and science videos