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Here we will learn about the sphere shape, including how to identify a sphere shape based on its properties, how to identify a sphere shape within a composite shape, and how to find the volume and surface area of a sphere shape.

Students will first learn about spheres as part of geometry in 1 st grade. They will expand their learning in middle school and high school when they learn how to find the volume and surface area of a sphere shape.

The **sphere shape** is a three-dimensional shape with a curved surface. Unlike other 3D shapes, such as a cube, cuboid, cone, or cylinder, a sphere has no faces (no flat surfaces), no edges, and no vertex. Every point of its surface is an equal distance away from the center of the sphere.

A **hemisphere** is half of a sphere. It has one circular base, which is a flat face, and one edge. Like a sphere, it has no vertices.

The sphere is a three-dimensional shape with a curved surface. Unlike other 3D shapes, such as a cube, cuboid, cone, or cylinder, a sphere has no faces (no flat surfaces), no edges, and no vertex.

There are many real life examples of spherical objects.

For example,

Note: The following content does not apply until middle school or high school.

The **volume** of a sphere is the amount of space there is inside a sphere. The volume of a sphere is measured in cubic units.

The formula for the volume of a sphere is:

\text{Volume}=\cfrac{4}{3} \, \pi{r}^3

For example, find the volume of the sphere.

This radius of the sphere is 7 {~cm} .

\begin{aligned} \text{Volume}&=\cfrac{4}{3} \, \pi r^3 \\\\ &= \cfrac{4}{3} \times \pi \times 7^3\\\\ &=\cfrac{1372}{3} \, \pi\\\\ &=1436.755… \\\\ &=1440 \ cm^3 \ \text{(3sf)}\ \end{aligned}

**Step by step guide:** Volume of a sphere

The **surface area** of a sphere is the area which covers the outer surface of a sphere.

The surface area of a sphere is measured in square units.

The formula for the surface area of a sphere is:

\text{Surface Area}=4 \, \pi r^2

For example, find the surface area of the sphere.

This radius of the sphere is 7 \, cm.

\begin{aligned} \text{Surface area}&=4 \, \pi r^2\\\\ &=4 \times \pi \times 7^2\\\\ &=196 \, \pi\\\\ &=615.752…\\\\ &=616 \ cm^2 \ \text{(to 3 sf)}\ \end{aligned}

**Step by step guide:** Surface area of a sphere

How does this relate to 1 st grade math?

**Grade 1 – Geometry (1.G.2)**Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.

In order to determine if a shape is a sphere:

**Look at the shape and examine its properties.****Determine if it has the properties of a sphere.**

3a **If it does, name the shape as a sphere.**

3b **If it does not, describe the difference in properties.**

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEAssess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEIs this shape a sphere?

**Look at the shape and examine its properties.**

The shape has a curved surface, one flat face, and one edge. It has no vertices.

2**Determine if it has the properties of a sphere.**

A sphere has a curved surface, no faces, no edges, and no vertices. This shape has some properties in common with a sphere, but there are some differences.

3**If it does not, describe the difference in properties.**

The shape shown has one flat face and one edge, while a sphere has no faces or edges. The shape shown is a hemisphere which is half of a sphere.

Is this object a sphere?

**Look at the shape and examine its properties.**

The object has a curved surface, no flat faces, no edges, and no vertices.

**Determine if it has the properties of a sphere.**

A sphere has a curved surface, no flat faces, no edges, and no vertices. It has all of the properties of a sphere.

**If it does, name the shape as a sphere.**

This object is a sphere.

In order to identify spheres within composite shapes:

**Look at all the shapes within the composite shape and examine their properties.****Determine if any of the shapes have the properties of a sphere. If they do, identify them.****Answer the question about the composite shape.**

There is one sphere in this composite shape. What color is it?

**Look at all the shapes within the composite shape and examine their properties.**

There are three shapes within the composite shape. The purple shape is rectangular and has 6 faces, 12 edges, and 8 vertices.

The yellow shape has 2 circular faces, no vertices, 2 edges, and a curved surface. The blue shape has a curved surface, no flat faces, no edges, and no vertices.

**Determine if any of the shapes have the properties of a sphere. If they do, identify them.**

The blue shape on top has the same properties of a sphere.

**Answer the question about the composite shape.**

The sphere is blue.

How many spheres are in this composite shape?

**Look at all the shapes within the composite shape and examine their properties.**

There are 8 shapes in this composite shape.

- Red and purple shapes (bottom): These shapes are rectangular; they each have 6 faces, 12 edges, and 8 vertices.

- Blue and green shapes (middle): These shapes each have 2 circular faces, no vertices, 2 edges, and a curved surface.

- Red, orange and blue shapes (on top): These shapes have a curved surface, no flat faces, no edges, and no vertices.

**Determine if any of the shapes have the properties of a sphere. If they do, identify them.**

The red, orange, and blue shapes on top of the composite shape have the properties of a sphere.

**Answer the question about the composite shape.**

There are 3 spheres in this composite shape.

In order to calculate the volume of a sphere or the surface area of a sphere:

**Write down the formula.****Substitute the given values.****Work out the calculation.****Write the final answer.**

Calculate the volume of a sphere of radius 12 {~}cm . Give your answer to the nearest whole number.

**Write down the formula for the volume of a sphere.**

\text{Volume} =\cfrac{4}{3} \, \pi r^3

**Substitute the given values into the formula.**

V=\cfrac{4}{3} \times \pi \times 12^3

**Complete the calculation.**

\begin{aligned}&=2304 \, \pi \\\\ &=7238.229…\\\end{aligned}

**Write the final answer, including the units.**

Volume is measured in cubic units. The radius is given in centimeters, so the volume will be in cubic centimeters.

The volume of the sphere is 7240 {~cm^3} (to the nearest whole number).

Find the surface area of a sphere with the diameter of 20 {~m}. Leave your answer in terms of \pi.

**Write down the formula for the surface area of a sphere.**

\text{Surface Area }=4 \, \pi{r}^{2}

**Substitute the given values into the formula.**

The diameter of the sphere is 20 {~m} and so we need to use this to calculate the radius and the surface area of the sphere.

20\div2=10

The radius of the sphere is 10 {~m} .

We now know the radius of the sphere is 10 {~m}, so we substitute r=10 into the formula for the surface area.

SA=4\times\pi\times{10}^{2}

**Complete the calculation.**

SA=400 \, \pi

**Write the final answer, including the units.**

Surface area is measured in square units. The radius is measured in meters so the surface area is measured in square meters (m^2) , with the solution in terms of \pi .

The surface area of the sphere is 400 \, \pi {m^2} .

- Begin by introducing the concept of a sphere as a three-dimensional shape that is round and ball-shaped. Use simple language and emphasize that spheres are completely symmetrical.

- Utilize visual aids such as pictures, diagrams, or real-life objects to help students visualize what a sphere looks like. Show them examples of spheres such as balls, balloons, fruits like oranges, or even Earth globes to reinforce the concept.

- Compare spheres with other shapes like circles and cubes. Point out the differences and similarities between the shape of a sphere and other 2D and 3D shapes. Highlight that circles are flat two-dimensional shapes, while spheres are three-dimensional shapes with a curved surface.

**Thinking that all circles are spheres**

Children may confuse spheres with circles. They might think that any object with a circular shape, such as a coin or a drawing of a circle, is a sphere. It is important to explain that a circle is a two-dimensional shape (2D shape), while a sphere is a three-dimensional object (3D shape).

**Thinking that spheres are flat on the bottom**

Children may have difficulty understanding that a sphere is a completely symmetrical shape, meaning it doesn’t have a flat surface on the bottom. They might imagine a sphere as having a flat base like other solid shapes or objects they are familiar with, such as a cup or a plate.

**Labeling volume or surface area of a sphere in the incorrect units**

Volume is measured in cubic units such as {cm^3}, {m^3}, {ft^3}.

Surface area is measured in square units such as {cm^2}, {m^2}, {ft^2}.

1. Which shape is a sphere?

A sphere has a curved surface, no flat faces, no edges, and no vertices. Every point of its surface is an equal distance away from the center of the sphere.

The last shape has all of these properties and the others do not.

2. Which shape is NOT a sphere?

A sphere has a curved surface, no flat faces, no edges, and no vertices. Every point of its surface is an equal distance away from the center of the sphere.

All shapes except the second shape, which is a hemisphere, have these properties.

3. How many spheres are in the composite shape?

2

1

3

4

A sphere has a curved surface, no flat faces, no edges, and no vertices. Every point of its surface is an equal distance away from the center of the sphere.

The red shape on the top and the orange shape have all of these properties.

4. What color is the sphere in the composite shape?

red

green

yellow

blue

The green shape has all of these properties.

5. Find the volume of a sphere with radius 4.3 {~cm} . Round your answer to the nearest hundredth.

333.04 \mathrm{~cm}^3

330 \mathrm{~m}^3

58.09 \mathrm{~cm}^3

232.35 \mathrm{~cm}^3

We are finding the volume of a sphere, so we substitute the value of the radius r=4.3 into the formula V=\cfrac{4}{3} \, \pi{r}^{3}.

\begin{aligned}\text{Volume}&=\cfrac{4}{3} \, \pi r^3 \\\\ &= \cfrac{4}{3} \times \pi \times 4.3^3\\\\ &=333.0381428… \\\\ &=333.04\text{~cm}^{3} \ \text{(to the nearest hundredth)}\end{aligned}

6. Find the surface area of a sphere with radius 4.6 {~cm} . Round your answer to the nearest hundredth.

270\text{ cm}^2

407.72\text{ cm}^2

66.48\text{ cm}^2

265.90\text{ cm}^2

We are finding the surface area of a sphere, so we substitute the value of the radius r=4.6 into the formula SA=4 \, \pi {r^2} .

\begin{aligned}\text{Surface area}&=4 \, \pi r^2\\\\ &=4 \times \pi \times 4.6^2\\\\ &=265.9044022…\\\\ &=265.90 \text{~cm}^2 \ \text{(2dp)}\\\end{aligned}

A sphere is a three-dimensional shape that is perfectly round and ball-shaped. It has a curved surface that is the same distance from its center at all points.

Some examples of spheres are balls (like a basketball or soccer ball), oranges, marbles, globes, and soap bubbles.

A spheroid is a three-dimensional shape that is similar to a sphere but not perfectly round.

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