Learn about the surface area and volume of common 3D shapes.
Example 1: Surface Area vs. Volume
Explain the difference between surface area and volume.
Surface Area: The total area of all the faces (surfaces) of a three-dimensional object. Think of it as the amount of wrapping paper needed to cover the object. The unit is in square units (e.g., cm2).
Volume: The amount of space a three-dimensional object occupies. Think of it as the amount of liquid a container can hold. The unit is in cubic units (e.g., cm3).
Key Difference: Surface area is a measure of the exterior, while volume is a measure of the interior space.
Example 2: Surface Area of a Cuboid
Explain how to find the surface area of a cuboid and provide the formula.
Step 1: A **cuboid** is a 3D shape with 6 rectangular faces. Opposite faces are identical.
Step 2: To find the total surface area, you calculate the area of each of the six faces and add them together.
Step 3: Let the length be 'l', width be 'w', and height be 'h'. The pairs of identical faces have areas:
Top and Bottom: l × w
Front and Back: l × h
Left and Right Sides: w × h
Step 4: The formula for the total surface area of a cuboid is:
Surface Area = 2(lw + lh + wh)
Step 5: The unit of surface area is in square units (e.g., cm2).
Example 3: Volume of a Cuboid
Explain how to find the volume of a cuboid and provide the formula.
Step 1: The **volume** of a cuboid is the amount of space it occupies.
Step 2: To find the volume of a cuboid, you multiply its length, width, and height.
Step 3: Let the length be 'l', width be 'w', and height be 'h'. The formula for the volume of a cuboid is:
Volume = length × width × height or Volume = lwh
Step 4: The unit of volume is always in cubic units (e.g., cm3, m3).
Example 4: Surface Area and Volume of a Cube
Explain how to find the surface area and volume of a cube and provide the formulas.
Step 1: A **cube** is a special type of cuboid where all six faces are congruent squares. All edges have the same length.
Step 2: Let the length of an edge of the cube be 's'.
Step 3: **Surface Area of a Cube:** Since there are 6 identical square faces, and the area of one square face is side × side (s × s or s2), the total surface area is 6 times the area of one face.
Surface Area = 6 × side × side or Surface Area = 6s2
Step 4: **Volume of a Cube:** Using the cuboid volume formula (l × w × h) and knowing that for a cube l=w=h=s, the volume is side × side × side.
Volume = side × side × side or Volume = s3
Step 5: Remember to use square units for surface area and cubic units for volume.
Practice Mode
Enter a simple surface area or volume problem to solve.
Note: This basic solver can calculate the surface area or volume of cuboids and cubes. Specify the shape and its dimensions (e.g., "surface area of cuboid l=10 w=5 h=4", "volume of cube side=6", "SA of cube s=7", "Volume of cuboid 8x3x5").
Related Concepts
Explore these related mathematical concepts to deepen your understanding of surface area and volume.
3D Shapes
Objects that have three dimensions: length, width, and height.
Face
A flat surface of a 3D shape.
Edge
A line segment where two faces of a 3D shape meet.
Vertex (Vertices)
A point where three or more edges of a 3D shape meet.
Cuboid
A 3D shape with six rectangular faces.
Cube
A 3D shape with six square faces of equal size.
Units of Volume
Standard measures for volume, like cubic centimeters (cm3) or cubic meters (m3).
Units of Surface Area
Standard measures for area, like square centimeters (cm2) or square meters (m2).