Hemisphere shape

Here you will learn about the hemisphere shape, including what a hemisphere is and how to calculate the volume of a hemisphere and the surface area of a hemisphere.

Students will first learn about the hemisphere shape as part of geometry in elementary school, but they will learn calculations involving the hemisphere shape in high school.

What is a hemisphere shape?

A hemisphere shape is a three-dimensional shape that is half of a sphere.

To visualize this, let’s look at a sphere. A sphere is a 3D shape where every point of its surface is equidistant (the same distance) from the center of the sphere. It has a radius katex is not defined

A hemisphere is half of a sphere. The prefix “hemi” means half.

You may have come across the word hemisphere in the real life context of planet Earth. You can split the world into katex is not defined equal halves – the northern hemisphere and the southern hemisphere.

Properties of a hemisphere

A hemisphere is a geometric shape which has a curved surface area. The base of a hemisphere is a flat face, which is a circle.

A hemisphere has no vertices and one edge. It is not made up of polygons, so it is not a polyhedron.

Faces: katex is not defined

Edges: katex is not defined

Vertices: katex is not defined

Volume of a hemisphere

The volume of a hemisphere is based on the volume of a sphere.

To calculate the volume of a sphere, where katex is not defined is the radius of the sphere, you use the formula:

katex is not defined

To find the volume of a hemisphere you halve the volume of a sphere.

Here is the volume of a hemisphere formula, with katex is not defined as the radius of the hemisphere:

katex is not defined

or

katex is not defined

See also: Volume of a hemisphere

Surface area of a hemisphere

The surface area of a hemisphere is based on the surface area of a sphere.

To calculate the surface area of a sphere with radius katex is not defined you can use the formula

katex is not defined

So to find the curved surface area of the hemisphere, you need to halve the surface area of the sphere.

katex is not defined

However this would only give the curved surface area. If you want the total surface area, you need to add the area of the base of the hemisphere.

The area of the base, which is a circle, is given by katex is not defined

Here is the surface area of a hemisphere formula, for a hemisphere with radius katex is not defined

katex is not defined

See also: Surface area of a hemisphere

What is a hemisphere shape?

Common Core State Standards

How does this relate to high school math?

• High School – Geometry – Geometric Measurement & Dimension (HS.G.GMD.A.2)
  Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

How to work out the volume or surface area of a hemisphere

In order to find the volume or surface area of a hemisphere:

1. Write down the formula for the sphere.
2. Adapt the formula for the question.
3. Substitute in the value.
4. Write the final answer.

Hemisphere shape examples

Example 1: volume

The radius of a hemisphere is katex is not defined Find the volume of the hemisphere to the nearest whole number.

1. Write down the formula for the sphere.

You need to consider the formula for the volume of a sphere.

2Adapt the formula for the question.

The formula needs to be halved for the volume of a hemisphere.

3Substitute in the value.

Then you substitute in the value katex is not defined in place of katex is not defined

4Write the final answer.

The answer is katex is not defined

This rounds to give katex is not defined (to the nearest whole number) as the volume of the hemisphere.

Example 2: volume in terms of π

Find the volume of a hemisphere with radius katex is not defined Leave your answer in terms of katex is not defined

Example 3: curved surface area

Find the curved surface area of a hemisphere with diameter katex is not defined Give your answer to the nearest tenth.

Example 4: total surface area

Find the total surface area of a hemisphere with radius katex is not defined Give your answer to the nearest whole number.

Example 5: curved surface area in terms of π

Find the curved surface area of a hemisphere with radius katex is not defined Leave your answer in terms of katex is not defined

Example 6: total surface area in terms of π

Find the total surface area of a hemisphere with diameter katex is not defined Leave your answer in terms of katex is not defined

Teaching tips for hemisphere shape

• Use physical models of spheres and hemispheres. Show everyday items like hemispherical bowls, domes, and igloos.

• Incorporate problem-solving exercises and worksheets that require students to apply the formulas for volume of a hemisphere and surface area of a hemisphere in different contexts.

• Hands-on activity: Have students create both solid and hollow hemisphere models using materials like clay for solid hemispheres and papier-mâché or plastic for hollow hemispheres. Let students weigh the solid and hollow models to understand the difference in mass. Have them calculate the volume and surface area for each type, reinforcing the mathematical concepts.

Easy mistakes to make

• Using the incorrect formula
  There are several formulas that can be used in geometry, so you need to match the correct formula to the correct context.

• Mixing up total surface area and curved surface area
  Double check whether the question requires you to find the curved surface area of a hemisphere or the total surface area of a hemisphere.

• Rounding too soon
  It is important to not round the answer until the end of the calculation. This will mean your final answer is accurate. It is useful to keep your answer in terms of katex is not defined until you round the answer at the very end of the question.

• Using incorrect units
  For area you use square units such as katex is not defined
  For volume you use cube units such as katex is not defined

• Mixing up the radius and the diameter
  It is a common error to mix up radius and diameter. Remember, the radius is half of the diameter.

Related 3D shapes lessons

• 3D shapes
• 3D shape names
• Pyramid shape
• Square pyramid
• Triangular pyramid
• Cylinder
• Cone
• Angles of elevation and depression

The next lessons are

• Area of a quadrilateral
• Volume
• Surface area

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