6th Grade geometry worksheets: Perimeters, surface area, and volume measurements

Perimeter, surface area, and volume measurements are three important concepts in 6th-grade geometry that help students measure different aspects of shapes. Students can develop their spatial reasoning and problem-solving skills by practicing these concepts.

They will also learn how to use formulas and tools to find the measurements of various shapes. These skills can help them in many areas of math and science, such as algebra, physics, engineering, and more. They can also help students to understand the world around them better and to solve real-life problems involving shapes and space.

The perimeter of a shape is the total distance around its boundary. To find the perimeter of a shape, we need to add up the lengths of all its sides. For example, the perimeter of a square with a side length of 5 cm is 5 + 5 + 5 + 5 = 20 cm. The perimeter of a rectangle with a length of 8 cm and a width of 6 cm is 8 + 6 + 8 + 6 = 28 cm.

Some shapes have unique formulas for their perimeters. For example, the perimeter of a circle with a radius of r is 2πr, where π is approximately 3.14. The perimeter of a regular polygon with n sides and a side length of s is ns. A regular polygon is a shape that has equal sides and angles.

To measure the perimeter of a shape, we use tools such as rulers, tape measures, or string. We can also use reasoning and estimation to find the perimeter of a shape if we know some information about it. For example, if you know that a square has an area of 25 cm2, you can find its side length by taking the square root of 25, which is 5 cm. Then, you can find its perimeter by multiplying 5 by 4, which is 20 cm.

The surface area of a 3D shape is the total area of all its faces. To find the surface area of a 3D shape, we need to add up the areas of all its faces. For example, the surface area of a cube with an edge length of 4 cm is 6 × (4 × 4) = 96 cm2. This is because a cube has six faces that are all squares with an area of 4 × 4 cm2.

Some 3D shapes have unique formulas for their surface areas. For example, the surface area of a sphere with a radius of r is 4πr2. The surface area of a cylinder with a radius of r and a height of h is 2πr2 + 2πrh. The surface area of a cone with a radius of r and a slant height of l is πr2 + πrl.

To measure the surface area of a 3D shape, we can use tools such as rulers, protractors, or nets. A net is a flat pattern that can be folded to make a 3D shape. We can also use reasoning and estimation to find the surface area of a 3D shape if we know some information about it.

For example, if you know that a cube has a volume of 64 cm3, you can find its edge length by taking the cube root of 64, which is 4 cm. Then you can find its surface area by multiplying 6 by (4 × 4), which is 96 cm2.

Another important concept in geometry is volume. Volume is the amount of space that a 3D shape occupies. We can think of volume as how much water a shape can hold or how many cubes can fit inside a shape.

To find the volume of a 3D shape, we need to know its base area and height. The base area is the area of the face that the shape stands on. The height is the distance from the base to the opposite face. The formula for volume is:

Volume = Base area x Height

For example, to find the volume of a rectangular prism, multiply the area of the rectangular base by the height of the prism. The area of a rectangle is length x width, so the formula for volume is:

Volume = Length x Width x Height

Some 3D shapes have different formulas for volume, depending on their shape. For example, to find the volume of a cylinder, multiply the circular base's area by the cylinder's height. The area of a circle is pi x radius squared, so the formula for volume is:

Volume = Pi x Radius squared x Height

Here are some other common 3D shapes and their formulas for volume:

• Cube: Volume = Side cubed
• Sphere: Volume = 4/3 x Pi x Radius cubed
• Cone: Volume = 1/3 x Pi x Radius squared x Height
• Pyramid: Volume = 1/3 x Base area x Height

To help your 6^th graders understand these concepts better, let's look at some examples of common shapes and their measurements, i.e., perimeters, surface areas, and volumes. Students can use these examples as a reference when practicing their calculations.

• Rectangle: A rectangle is a 2D shape with four sides and four right angles. The opposite sides are equal and parallel.

  Perimeter = 2 x Length + 2 x Width

  Surface area = Length x Width

  Volume = Length x Width x Height

  For example, if a rectangle has a length of 10 cm, a width of 5 cm, and a height of 3 cm, then:

  Perimeter = 2 x 10 + 2 x 5 = 30 cm

  Surface area = 10 x 5 = 50 cm squared

  Volume = 10 x 5 x 3 = 150 cm cubed

• Circle: A circle is a 2D shape with no sides or angles. It has a center point and a radius, the distance from the center to any point on the circle.

  Perimeter (also called circumference) = 2 x Pi x Radius

  Surface area (also called area) = Pi x Radius squared

  Volume (of a sphere) = 4/3 x Pi x Radius cubed

  For example, if a circle has a radius of 4 cm, then:

  Perimeter = 2 x Pi x 4 = 25.13 cm (approx.)

  Surface area = Pi x 4 squared = 50.27 cm squared (approx.)

  Volume (of a sphere) = 4/3 x Pi x 4 cubed = 268.08 cm cubed (approx.)

• Triangle: A triangle is a 2D shape with three sides and three angles. The sum of the angles in any triangle is always 180 degrees.

  Perimeter = Side A + Side B + Side C

  Surface area (also called area) = Base x Height / 2

  Volume (of a pyramid) = Base area x Height / 3

  For example, if a triangle has a base of 6 cm, a height of 4 cm, and sides of length 5 cm, then:

  Perimeter = 5 + 5 + 6 = 16 cm

  Surface area = 6 x 4 / 2 = 12 cm squared

  Volume (of a pyramid) = Base area x Height /3

  = (6x4/2) x (4/3)

  =12 x (4/3)

  =16 cm cubed

Now that your 6^th graders have learned about perimeters, surface areas, and volumes, it's time to practice their skills. Here are some exercises from Mathskills4kids’ 6th-grade geometry worksheets that they can try to solve independently. They can use rulers, protractors, or calculators to measure shapes. They can also use reasoning and estimation to check their answers.

Find the perimeter, surface area, and volume of each shape below. Round your answers to two decimal places if necessary.

1. A square with a side length of 8 cm.
2. A circle with a radius of 5 cm.
3. A cube with an edge length of 6 cm.
4. A cylinder with a radius of 3 cm and height of 10 cm.
5. A cone with a radius of 4 cm and height of 12 cm.

Find the missing measurements of each shape below. Round your answers to two decimal places if necessary.

1. A rectangle with a length of 15 cm and a perimeter of 46 cm. What is the width and the surface area?
2. A circle with an area of 78.54 cm squared. What is the radius and the perimeter?
3. A sphere with a volume of 523.6 cm cubed. What is the radius and the surface area?
4. A pyramid with a base area of 36 cm squared and a volume of 48 cm cubed. What is the height and the surface area?
5. A triangular prism with a base area of 18 cm squared, height of 9 cm, and volume of 162 cm cubed. What is the length and the surface area?

Geometry is useful in the math class and also in everyday life. We can use geometric measurement concepts to solve real-life situations such as:

• How much paint do you need to cover a wall or a room?
• How much wrapping paper do you need to wrap a gift box?
• How much water can a tank or a bottle hold?
• How much space do you need to store or pack some items?
• How much land do you own or need to buy?
• How much fence do you need to enclose a garden or a yard?

To solve these problems, we need to identify the shapes involved and their dimensions. Then, use the appropriate formulas to find the measurements that are needed. We also need to pay attention to the units of measurement and convert them if necessary.

Here are some more examples of how to apply geometry concepts to real-life situations:

• If you want to paint a wall, you need to know its surface area to buy enough paint. You can use the formula for the surface area of a rectangle to find the answer.
• If you want to buy a new backpack, you need to know its volume to see how much stuff you can fit inside. You can use the formula for the volume of a prism to find the answer.
• If you want to make a fence around your garden, you need to know its perimeter to buy enough wood. You can use the formula for the perimeter of a polygon to find the answer.
• If you want to design a logo, you need to know how to use angles and shapes to create a symmetrical and appealing image. You can use protractors and rulers to measure angles and shapes.
• To plan a trip, know how to use maps and scales to estimate distances and time. You can use ratios and proportions to convert measurements.

As you can see, geometry is everywhere and can help us with many things. The more your students practice geometric measurement skills, the more confident and creative they will become.