1 7 assignment three dimensional figures

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Grade 7 (Virginia)

Course: grade 7 (virginia)   >   unit 7.

• Unit test Three-dimensional figures

• Math Article

Three Dimensional Shapes

In geometry, three-dimensional shapes or 3D shapes are solids that have three dimensions such as length, width and height. Whereas 2d shapes have only two dimensions, i.e. length and width. Examples of three-dimensional objects can be seen in our daily life such as cone-shaped ice cream, cubical box, a ball, etc. Students will come across different 3D shapes models in Maths.

Geometry is one of the practical sections of Mathematics that involves various shapes and sizes of different figures and their properties. Geometry can be divided into two types: plane and solid geometry . Plane geometry deals with flat shapes like lines, curves, polygons, etc., that can be drawn on a piece of paper. On the other hand, solid geometry involves objects of three-dimensional shapes such as cylinders, cubes, spheres, etc. In this article, we are going to learn different 3D shapes models in Maths such as cube, cuboid, cylinder, sphere and so on along with its definitions, properties, formulas and examples in detail.

Table of Contents:

• Three Dimensional Shapes Definition

Faces, Edges and Vertices

• 3D Shapes Model in Maths
• Surface Area and Volume Formulas
• Video Lessons
• Practice Questions

What are Three-Dimensional Shapes?

Shapes that can be measured in 3 directions are called three-dimensional shapes. These shapes are also called solids. Length, width, and height (or depth or thickness) are the three measurements of three-dimensional shapes. These are the part of three-dimensional geometry . They are different from 2D shapes because they have thickness. Several examples can be found in everyday life. Some of them are:

Solid Shapes in Maths

In Mathematics, the three-dimensional objects having depth, width and height are called solid shapes. Let us consider a few shapes to learn about them. You can find many examples of solid shapes around you, such as a mobile, notebook or almost everything you can see around is a solid shape.

Faces, Edges, and Vertices of Three Dimensional Shapes

Three-dimensional shapes have many attributes, such as vertices, faces, and edges. The flat surfaces of the 3D shapes are called faces. The line segment where two faces meet is called an edge. A vertex is a point where three edges meet.

Also, read: Vertices, Edges and Faces

List of Three-dimensional Shapes

The list of three-dimensional shapes are as follows:

Here, we are going to discuss the list of different three-dimensional shapes with their properties and the formulas of different 3D shapes.

A cube is a solid or three-dimensional shape which has 6 square faces. The cube has the following properties.

• All edges are equal

A cuboid is also called a rectangular prism, where the faces of the cuboid are a rectangle in shape. All the angles measure 90 degrees. The cuboid has

A prism is a 3D shape which consists of two equal ends, flat surfaces or faces, and also has identical cross-section across its length. Since the cross-section looks like a triangle, the prism is generally called a triangular prism. The prism does not have any curve. Also, a prism has

• 5 faces – 2 triangles and 3 rectangles

A pyramid a solid shape, whose outer faces are triangular and meet to a single point on the top. The pyramid base can be of any shape such as triangular, square, quadrilateral or in the shape of any polygon. The most commonly used type of a pyramid is the square pyramid, i.e., it has a square base and four triangular faces. Consider a square pyramid, it has

A cylinder is defined as a three-dimensional geometrical figure which has two circular bases connected by a curved surface. A cylinder has

• 2 flat faces – circles
• 1 curved face

A cone is a three-dimensional object or solid, which has a circular base and has a single vertex. The cone is a geometrical figure that decreases smoothly from the circular flat base to the top point called the apex. A cone has

• 1 flat face – circle

A sphere is a three-dimensional solid figure which is perfectly round in shapes and every point on its surface is equidistant from the point is called the center. The fixed distance from the center of the sphere is called a radius of the sphere. A sphere has

Three-dimensional Shapes related Articles

3d shapes model in maths project.

If you know what three-dimensional shapes are, it would be easy for you to build a 3 d shapes models in Maths such as projects  for constructing a house or a building. This would be easy for the students to make as they can measure the rooms easily. Rest all they need is cardboard, glue, scissors and art supplies to make it look exactly like a mini house or building.

Surface Area and Volume of 3D shapes

The two distinct measures used for defining the 3D shapes are:

• Surface Area

Surface Area  is defined as the total area of the surface of the three-dimensional object.  It is denoted as “SA”. The surface area is measured in terms of square units. The three different classifications of surface area are defined below. They are:

• Curved Surface Area (CSA) is the area of all the curved regions
• Lateral Surface Area (LSA) is the area of all the curved regions and all the flat surfaces excluding base areas
• Total Surface Area (TSA) is the area of all the surfaces including the base of a 3D object

Volume  is defined as the total space occupied by the three-dimensional shape or solid object. The volume is denoted as “V”. It is measured in terms of cubic units.

Learn more about the three-dimensional shapes with BYJU’S – The Learning App. Download the app today and start practice.

Video Lesson

Solid shapes.

Nets of Solid Shapes

To know about nets of solid shapes, watch the below video:.

Three Dimensional Shapes Examples (Solved problems)

Find the volume of a cube if its side length is 6 cm.

Given: Side length, a = 6 cm.

We know that the volume of cube = a 3  cubic units.

Hence, V = 6 3 = 216 cm 3

Hence, the volume of a cube is 216 cm 3 .

Example 2:  

Find the total surface area of a sphere, whose radius is 3 cm. Use (π = 3.14)

Given: Radius, r = 3 cm.

The formula to calculate the total surface area of a sphere is given by:

TSA of Sphere = 4πr² Square units

TSA of sphere = 4(3.14)(3) 2 cm 2

TSA of Sphere = 113.04 cm 2 .

Hence, the total surface area of a sphere is 113.04 cm 2 .

Example 3:  

Find the volume of a cuboid, whose dimensions is 4cm × 6 cm × 12 cm.

Given cuboid dimensions = 4cm × 6 cm × 12 cm

We know that the volume of a cuboid is lbh cubic units.

Hence, Volume of cuboid  = (4)(6)(12)  cm 3 .

V = 288 cm 3

Therefore, the volume of the cuboid is 288 cm 3 .

Practice Question on 3D Shapes Models in Maths

Solve the following problems on 3D shapes:

• Find the volume of a cylinder whose radius is 4 cm and height is 8 cm.
• Find the volume of a cone whose radius is 5 cm and height is 9 cm.
• Find the volume of a pyramid whose base area is 126 cm 2 and height is 10 cm.

Frequently Asked Questions on Three Dimensional Shapes

What are the three dimensional shapes, what are the different types of three dimensional shapes, is square a three dimensional shape, what is a three dimensional round shape object called, what are the examples of three dimensional shapes.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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Chapter 1, Lesson 7: Three-Dimensional Figures

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7.1.1: Figures in 1 and 2 Dimensions

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Learning Objectives

• Identify and define points, lines, line segments, rays, and planes.
• Classify angles as acute, right, obtuse, or straight.

Introduction

You use geometric terms in everyday language, often without thinking about it. For example, any time you say “walk along this line” or “watch out, this road quickly angles to the left” you are using geometric terms to make sense of the environment around you. You use these terms flexibly, and people generally know what you are talking about.

In the world of mathematics, each of these geometric terms has a specific definition. It is important to know these definitions, as well as how different figures are constructed, to become familiar with the language of geometry. Let’s start with a basic geometric figure: the plane.

Figures on a Plane

A plane is a flat surface that continues forever (or, in mathematical terms, infinitely) in every direction. It has two dimensions: length and width.

You can visualize a plane by placing a piece of paper on a table. Now imagine that the piece of paper stays perfectly flat and extends as far as you can see in two directions, left-to-right and front-to-back. This gigantic piece of paper gives you a sense of what a geometric plane is like: it continues infinitely in two directions. (Unlike the piece of paper example, though, a geometric plane has no height.)

A plane can contain a number of geometric figures. The most basic geometric idea is a point , which has no dimensions. A point is simply a location on the plane. It is represented by a dot. Three points that don’t lie in a straight line will determine a plane.

The image below shows four points, labeled A , B , C , and D .

Two points on a plane determine a line. A line is a one-dimensional figure that is made up of an infinite number of individual points placed side by side. In geometry, all lines are assumed to be straight; if they bend, they are called a curve. A line continues infinitely in two directions.

Below is line \(\ A B\) or, in geometric notation, \(\ \overleftrightarrow{A B}\). The arrows indicate that the line keeps going forever in the two directions. This line could also be called line \(\ BA\). While the order of the points does not matter for a line, it is customary to name the two points in alphabetical order.

The image below shows the points \(\ A\) and \(\ B\) and the line \(\ \overleftrightarrow{A B}\).

Name the line shown in red.

\(\ \overleftrightarrow{C F}\)

There are two more figures to consider. The section between any two points on a line is called a line segment . A line segment can be very long, very short, or somewhere in between. The difference between a line and a line segment is that the line segment has two endpoints and a line goes on forever. A line segment is denoted by its two endpoints, as in \(\ \overline{C D}\).

A ray has one endpoint and goes on forever in one direction. Mathematicians name a ray with notation like \(\ \overrightarrow{E F}\), where point \(\ E\) is the endpoint and \(\ F\) is a point on the ray. When naming a ray, we always say the endpoint first. Note that \(\ \overrightarrow{F E}\) would have the endpoint at \(\ F\), and continue through \(\ E\), which is a different ray than \(\ \overrightarrow{E F}\), which would have an endpoint at \(\ E\), and continue through \(\ F\).

The term “ray” may be familiar because it is a common word in English. “Ray” is often used when talking about light. While a ray of light resembles the geometric term “ray,” it does not go on forever, and it has some width. A geometric ray has no width; only length.

Below is an image of ray \(\ E F\) or \(\ \overrightarrow{E F}\). Notice that the end point is \(\ E\).

Identify each line and line segment in the picture below.

Lines: \(\ \overleftrightarrow{C E}, \overleftrightarrow{B G}\)

Line segments: \(\ \overline{D F}, \overline{C E}, \overline{B G}\)

Identify each point and ray in the picture below.

Points: \(\ A, B, C, D\)

Rays: \(\ \overrightarrow{B C}, \overrightarrow{A D}, \overrightarrow{D A}\)

Which of the following is not represented in the image below?

• \(\ \overleftrightarrow{B G}\)
• \(\ \overrightarrow{B A}\)
• \(\ \overline{D F}\)
• \(\ \overrightarrow{A C}\)
• Incorrect. A line goes through points \(\ B\) and \(\ G\), so \(\ \overleftrightarrow{B G}\) is shown. \(\ \overleftrightarrow{B A}\), is not shown in this image.
• Correct. This image does not show any ray that begins at point \(\ B\) and goes through point \(\ A\).
• Incorrect. There is a line segment connecting points \(\ D\) and \(\ F\), so \(\ \overleftrightarrow{D F}\) is shown. \(\ \overleftrightarrow{B A}\), is not shown in this image.
• Incorrect. There is a ray beginning at point \(\ A\) and going through point \(\ C\), so \(\ \overrightarrow{A C}\) is shown. \(\ \overrightarrow{B A}\), is not shown in this image.

Lines, line segments, points, and rays are the building blocks of other figures. For example, two rays with a common endpoint make up an angle . The common endpoint of the angle is called the vertex .

The angle \(\ A B C\) is shown below. This angle can also be called \(\ \angle A B C\), \(\ \angle C B A\), or simply \(\ \angle B\). When you are naming angles, be careful to include the vertex (here, point \(\ B\) as the middle letter.

The image below shows a few angles on a plane. Notice that the label of each angle is written “point-vertex-point,” and the geometric notation is in the form \(\ \angle A B C\).

Sometimes angles are very narrow; sometimes they are very wide. When people talk about the “size” of an angle, they are referring to the arc between the two rays. The length of the rays has nothing to do with the size of the angle itself. Drawings of angles will often include an arc (as shown above) to help the reader identify the correct ‘side’ of the angle.

Think about an analog clock face. The minute and hour hands are both fixed at a point in the middle of the clock. As time passes, the hands rotate around the fixed point, making larger and smaller angles as they go. The length of the hands does not impact the angle that is made by the hands.

An angle is measured in degrees, represented by the degree symbol, which is a small circle at the upper right of a number. For example, a circle is defined as having 360 o . (In skateboarding and basketball, “doing a 360" refers to jumping and doing one complete body rotation.)

A right angle is any degree that measures exactly 90 o . This represents exactly one-quarter of the way around a circle. Rectangles contain exactly four right angles. A corner mark is often used to denote a right angle, as shown in right angle \(\ D C B\) below.

Angles that are between 0 o and 90 o (smaller than right angles) are called acute angles . Angles that are between 90 o and 180 o (larger than right angles and less than 180 o ) are called obtuse angles . And an angle that measures exactly 180 o is called a straight angle because it forms a straight line!

Label each angle below as acute, right, or obtuse.

\(\ \angle D A B\) and \(\ \angle M L N\) are acute angles.

\(\ \angle G F I\) is a right angle.

\(\ \angle T Q S\) is an obtuse angle.

Identify each point, ray, and angle in the figure below.

Identify the acute angles in the image below.

• \(\ \angle W A X, \angle X A Y, \text { and } \angle Y A Z\)
• \(\ \angle W A Y \text { and } \angle Y A Z\)
• \(\ \angle W A X \text { and } \angle Y A Z\)
• \(\ \angle W A Z \text { and } \angle X A Y\)
• Incorrect. \(\ \angle W A X\) and \(\ \angle Y A Z\) are both acute angles, but \(\ \angle X A Y\) is an obtuse angle. So only \(\ \angle W A X\) and \(\ \angle Y A Z\) are acute angles.
• Incorrect. \(\ \angle Y A Z\) is an acute angle, but \(\ \angle W A Y\) is an obtuse angle. Both \(\ \angle W A X\) and \(\ \angle Y A Z\) are acute angles.
• Correct. Both \(\ \angle W A X\) and \(\ \angle Y A Z\) are acute angles.
• Incorrect. \(\ \angle W A Z\) is a straight angle, and \(\ \angle X A Y\) is an obtuse angle. Both \(\ \angle W A X\) and \(\ \angle Y A Z\) are acute angles.

Measuring Angles with a Protractor

Learning how to measure angles can help you become more comfortable identifying the difference between angle measurements. For instance, how is a 135 o angle different from a 45 o angle?

Measuring angles requires a protractor , which is a semi-circular tool containing 180 individual hash marks. Each hash mark represents 1 o . (Think of it like this: a circle is 360 o , so a semi-circle is 180 o .) To use the protractor, do the following three steps:

• line up the vertex of the angle with the dot in the middle of the flat side (bottom) of the protractor,
• align one side of the angle with the line on the protractor that is at the zero degree mark, and
• look at the curved section of the protractor to read the measurement.

Supplemental Interactive Activity

For practice using a protractor, try out the activity below:

The example below shows you how to use a protractor to measure the size of an angle.

Use a protractor to measure the angle shown below.

Use a protractor to measure the angle.

Align the blue dot on the protractor with the vertex of the angle you want to measure.

Rotate the protractor around the vertex of the angle until the side of the angle is aligned with the 0 degree mark of the protractor.

Read the measurement, in degrees, of the angle. Begin with the side of the angle that is aligned with the 0 o mark of the protractor and count up from 0 o . This angle measures 38 o .

The angle measures 38 o .

What is the measurement of the angle shown below?

Geometric shapes and figures are all around us. A point is a zero-dimensional object that defines a specific location on a plane. A line is made up of an infinite number of points, all arranged next to each other in a straight pattern, and going on forever. A ray begins at one point and goes on towards infinity in one direction only. A plane can be described as a two-dimensional canvas that goes on forever.

When two rays share an endpoint, an angle is formed. Angles can be described as acute, right, obtuse, or straight, and are measured in degrees. You can use a protractor (a special math tool) to closely measure the size of any angle.

Curriculum  /  Math  /  7th Grade  /  Unit 6: Geometry  /  Lesson 16

Lesson 16 of 21

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Identify and describe two-dimensional figures that result from slicing three-dimensional figures.

Common Core Standards

Core standards.

The core standards covered in this lesson

7.G.A.3 — Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

Foundational Standards

The foundational standards covered in this lesson

5.G.B.3 — Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

Measurement and Data

5.MD.C.3 — Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Understand that a three-dimensional figure can be sliced in various ways that result in a two-dimensional cross-section . 
• Understand that a slice through a right pyramid or prism, parallel to the base, will have the same shape as the base.
• Identify the two-dimensional figures that result from various slices through right pyramids and prisms. 

Suggestions for teachers to help them teach this lesson

Visualizing the two-dimensional figures that result from slicing a three-dimensional figure can be challenging for some students. The following are two visual supports:

• This GeoGebra applet  Sections of Prisms and Cylinders  is a great interactive tool. The slicing plane can be moved around to show the different two-dimensional figures that are created.  
• This Math Shorts episode  Slicing Three Dimensional Figures is a good, quick introduction as well.

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Justine has a block of cheese in the shape of a square prism, as shown below.

She cuts the block in half. What are two different two-dimensional faces that could result from Justine slicing the cheese in half?

Guiding Questions

A right rectangular pyramid is shown below.

Describe how the following two-dimensional figures can be created by slicing the pyramid. 

a.   rectangle

b.   triangle

c.   trapezoid

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

A cube is sliced with a single straight cut, creating a two-dimensional cross-section. Name 2 different two-dimensional shapes that could result from the slice, and explain or draw how they are created.

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• EngageNY Mathematics Grade 7 Mathematics > Module 6 > Topic C > Lesson 16 — Examples, Exercises, Problem Set
• Illustrative Mathematics Cube Ninjas!
• MARS Formative Assessment Lessons for High School Representing 3D Objects in 2D
• EngageNY Mathematics Grade 7 Mathematics > Module 6 > Topic C > Lesson 17 — Examples, Exercises, Problem Set

Topic A: Angle Relationships

Identify and determine values of angles in complementary and supplementary relationships.

Use vertical, complementary, and supplementary angle relationships to find missing angles. 

Use equations to solve for unknown angles. (Part 1)

Use equations to solve for unknown angles. (Part 2)

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Topic B: Circles

Define circle and identify the measurements radius, diameter, and circumference. 

Determine the relationship between the circumference and diameter of a circle and use it to solve problems. 

Solve real-world and mathematical problems using the relationship between the circumference of a circle and its diameter. 

Determine the relationship between the area and radius of a circle and use it to solve problems.

Solve real-world and mathematical problems using the relationship between the area of a circle and its radius.

Solve problems involving area and circumference of two-dimensional figures (Part 1).

7.G.B.4 7.G.B.6

Solve problems involving area and circumference of two-dimensional figures (Part 2).

Topic C: Building Polygons and Triangles

Draw two-dimensional geometric shapes using rulers, protractors, and compasses. 

7.G.A.2 7.G.B.5

Determine if three side lengths will create a unique triangle or no triangle. 

Identify unique and identical triangles. 

Determine if conditions describe a unique triangle, no triangle, or more than one triangle.

Topic D: Solid Figures

Find the surface area of right prisms.

 Find the surface area of right pyramids.

Find the volume of right prisms and pyramids.

Solve real-world and mathematical problems involving volume.

Distinguish between and solve real-world problems involving volume and surface area.

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