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Lesson 5: Surface area

# 3D figures FAQ

## What are geometric solids?

Geometric solids are shapes that have three dimensions: length, width, and height. They are also called 3D shapes, because they look like they pop out of the paper. Some examples of geometric solids are cubes, rectangular prisms, pyramids, cones, cylinders, and spheres.
We can describe geometric solids by their faces, edges, and vertices. Faces are the flat surfaces. Edges are the lines where two faces meet. Vertices are the points where three or more edges meet.

## How do we find the volume of a rectangular prism with fractions?

Sometimes, the length, width, or height of a rectangular prism might not be a whole number, but a fraction. For example, a rectangular prism might have a length of start fraction, 3, divided by, 2, end fraction units, a width of 2 units, and a height of start fraction, 5, divided by, 4, end fraction units. To find the volume of a rectangular prism with fractions, we use the same formula as before, but we multiply fractions instead of whole numbers. We can use a calculator, or we can use these rules to multiply fractions:
• To multiply two fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers) separately. For example, start fraction, 3, divided by, 2, end fraction, times, start fraction, 4, divided by, 5, end fraction, equals, start fraction, 3, times, 4, divided by, 2, times, 5, end fraction, equals, start fraction, 12, divided by, 10, end fraction.
• To simplify a fraction, we divide the numerator and the denominator by the same number, if possible. For example, start fraction, 12, divided by, 10, end fraction, equals, start fraction, 12, divided by, 2, divided by, 10, divided by, 2, end fraction, equals, start fraction, 6, divided by, 5, end fraction.
So, we can find the volume of the rectangular prism with fractions this way:
\begin{aligned} \dfrac{3}{2} \times 2 \times \dfrac{5}{4} &= \dfrac{3 \times 2 \times 5}{2 \times 1 \times 4} \\\\ &= \dfrac{30}{8} \\\\ &= \dfrac{30 \div 2}{8 \div 2} \\\\ &= \dfrac{15}{4} \end{aligned}
So the volume of the rectangular prism above is start fraction, 15, divided by, 4, end fraction cubic units.

## What are nets of 3D figures?

Nets of 3D figures are 2D shapes that can be folded or cut out to make the faces of 3D figures. For example, a net of a cube is a group of 6 squares (one for each face) attached along certain edges. If we fold the smaller squares along the edges, we can make a cube. Nets can help us visualize and understand 3D figures better, and also help us find their surface area.
Here is a diagram of a rectangular prism.
A rectangular prism. The front face has a base of 2 units and a height of 3 units. The bottom face has a base of 2 units and a height of 4 units.
Here is one possible net for the rectangular prism, showing all 6 rectangular faces. Unlike with the cube, the faces are not all the same size.
A surface net of a rectangular prism. The net consists of 4 rectangles in a row, where the second rectangle from the left is also connected to a rectangle above it and a rectangle below it. The bottom of the left most rectangle is labeled as 3 units. The bottom of the second rectangle from the left is labeled as 2 units. The right side of the second rectangle from the left is labeled as 4 units. The left side of the rectangle located beneath the second rectangle is labeled as 3 units.

## How do we find the surface area of a 3D figure?

The surface area of a 3D figure is the total area of all its faces. To find the surface area of a 3D figure, we can use a net to see all its faces, and then add up the areas of each face. For example, to find the surface area of a rectangular prism with side lengths of 2, 3, and 4 units, we can use a net like this:
A surface net of a rectangular prism. The net consists of 4 rectangles in a row, where the second rectangle from the left is also connected to a rectangle above it and a rectangle below it. The bottom of the left most rectangle is labeled as 3 units. The bottom of the second rectangle from the left is labeled as 2 units. The right side of the second rectangle from the left is labeled as 4 units. The left side of the rectangle located beneath the second rectangle is labeled as 3 units.
The area of each of the top and bottom faces of the net is 2, times, 3, equals, 6 square units. The area of each of the first and third rectangle in the row is 3, times, 4, equals, 12 square units. The area of each of the second and fourth rectangle in the row is 2, times, 4, equals, 8 square units. We add all the areas to get the total surface area.
6, plus, 6, plus, 12, plus, 12, plus, 8, plus, 8, equals, 52
The rectangular prism has a surface area of 52 square units. Notice that we use square units, because we are measuring area, even though it is part of a 3D figure.

## Why do we need to know about geometric solids, volume, and surface area?

We can use geometric solids, volume, and surface area to measure, compare, design, and create different objects and spaces. For example, we can use volume to find out how much water a bottle can hold, or how much sand we need to fill a sandbox. We can use surface area to find out how much paint we need to cover a wall, or how much wrapping paper we need to wrap a gift. We can use nets to make models of buildings, sculptures, or origami. There are many more examples of how geometric solids, volume, and surface area are useful and fun in everyday life!