Review the key concepts, formulae, and examples before starting your quiz.
🔑Concepts
Definition of 3D Shapes: Three-dimensional (3D) shapes are solid objects that have three dimensions: length, width, and height. Unlike 2D shapes which are flat, 3D shapes occupy space, which is measured as volume. Visually, these shapes look like real-world objects such as boxes, balls, or cans.
Faces, Edges, and Vertices: These are the primary properties used to identify 3D shapes. A 'Face' is a flat or curved surface (e.g., a cube has 6 flat faces). An 'Edge' is a line segment where two faces meet. A 'Vertex' (plural: vertices) is a point or corner where three or more edges meet.
Prisms: A prism is a 3D shape with two identical ends (bases) and flat sides. The shape of the base gives the prism its name. For example, a Rectangular Prism looks like a cereal box and has 6 faces, 12 edges, and 8 vertices. A Triangular Prism looks like a tent and has 5 faces, 9 edges, and 6 vertices.
Pyramids: A pyramid has one base and several triangular sides that meet at a single point called the apex. A Square-based Pyramid looks like the ancient Egyptian pyramids; it has 1 square base and 4 triangular faces, totaling 5 faces, 8 edges, and 5 vertices.
Curved 3D Shapes: Not all 3D shapes have flat faces and straight edges. A 'Sphere' is perfectly round like a ball and has 1 curved surface with no edges or vertices. A 'Cylinder' has 2 flat circular faces and 1 curved surface, looking like a soup can. A 'Cone' has 1 circular base, 1 curved surface, and 1 vertex at the top, looking like a party hat.
Nets of 3D Shapes: A net is a 2D pattern that can be folded to create a 3D shape. Imagine unfolding a cardboard box until it lies flat on the floor; that flat pattern is the net. For instance, the net of a cube consists of 6 squares arranged in a specific way (often looking like a cross) so that when folded, they meet to form a closed solid.
Euler's Formula: For many solid shapes (polyhedra), there is a mathematical relationship between the number of faces (), vertices (), and edges (). This rule helps verify if the count of properties for a shape like a prism or pyramid is correct.
📐Formulae
Euler's Formula for Polyhedra:
Volume of a Rectangular Prism:
Volume of a Cube: (where is the side length)
Surface Area of a Cube:
💡Examples
Problem 1:
Identify the number of faces, vertices, and edges for a Pentagonal Prism.
Solution:
- A pentagonal prism has two pentagonal bases. Each pentagon has 5 sides.
- Faces: 2 (bases) + 5 (rectangular sides) = faces.
- Vertices: Each pentagonal base has 5 corners. vertices.
- Edges: 5 edges on the top base, 5 on the bottom base, and 5 connecting the corners. edges.
- Check with Euler's Formula: . The calculation is correct.
Explanation:
To solve for properties of prisms, identify the base shape first. The number of side faces and edges is always relative to the number of sides on that base shape.
Problem 2:
Calculate the volume of a rectangular prism (box) that has a length of cm, a width of cm, and a height of cm.
Solution:
- Use the volume formula for a rectangular prism:
- Substitute the given values:
- Multiply length and width:
- Multiply the result by height:
- The final volume is .
Explanation:
Volume measures the space inside a 3D shape. For prisms, you multiply the area of the base () by the height (). Always remember to use cubic units (e.g., ).