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Shape and Space - Properties of 3D Shapes (Prisms, Pyramids, Cylinders)

Grade 6IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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3D Shape Components: Every 3D shape is defined by its faces, edges, and vertices. Faces are the flat surfaces (like the square sides of a cube), edges are the straight lines where two faces meet, and vertices are the corner points where edges intersect. For example, a cube has 6 faces, 12 edges, and 8 vertices.

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Prisms: A prism is a 3D shape with two identical, parallel bases and a constant cross-section throughout its length. This means if you 'slice' it anywhere parallel to the base, the shape of the slice remains the same. A triangular prism looks like a tent, with two triangular ends and three rectangular sides.

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Pyramids: A pyramid has a polygon base and triangular faces that all meet at a single point called the apex. The shape is named after the shape of its base. For instance, a square-based pyramid looks like a classic Egyptian pyramid, featuring a square bottom and four triangular faces meeting at the top.

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Cylinders: A cylinder is a 3D object with two congruent circular bases and one curved side. While it is not a polyhedron (because it has curved surfaces), it behaves like a prism in terms of volume. Visually, it looks like a soup can or a tube, where the top and bottom circles are connected by a smooth, rounded surface.

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Euler’s Formula: This mathematical rule describes the relationship between the components of any polyhedron (3D shapes with flat faces). It states that the number of Vertices (VV) minus the number of Edges (EE) plus the number of Faces (FF) will always equal 22. This is expressed as V−E+F=2V - E + F = 2.

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Nets of 3D Shapes: A net is a 2D pattern that can be folded to create a 3D shape. Imagine unfolding a cardboard box and laying it flat on a table. The net of a cube consists of 6 squares arranged in a 'T' or cross shape, which, when folded, perfectly encloses the 3D volume.

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Properties of regular 3D shapes: In Grade 6, we focus on identifying shapes by their properties. For example, a rectangular prism (or cuboid) has 6 faces that are all rectangles, while a cube is a special type of rectangular prism where all 6 faces are identical squares.

📐Formulae

V−E+F=2V - E + F = 2 (Euler's Formula)

V=Area of Base×HeightV = \text{Area of Base} \times \text{Height} (General Volume of a Prism)

V=l×w×hV = l \times w \times h (Volume of a Rectangular Prism)

V=πr2hV = \pi r^2 h (Volume of a Cylinder)

SA=6s2SA = 6s^2 (Surface Area of a Cube, where ss is the side length)

💡Examples

Problem 1:

A rectangular prism has a length of 12 cm12\text{ cm}, a width of 4 cm4\text{ cm}, and a height of 5 cm5\text{ cm}. Calculate the volume of the prism.

Solution:

V=l×w×hV = l \times w \times h \ V=12×4×5V = 12 \times 4 \times 5 \ V=48×5V = 48 \times 5 \ V=240 cm3V = 240\text{ cm}^3

Explanation:

To find the volume of a rectangular prism, we use the formula V=l×w×hV = l \times w \times h. First, multiply the length (1212) by the width (44) to find the area of the base (48 cm248\text{ cm}^2), then multiply that by the height (55) to find the total space inside.

Problem 2:

Verify Euler's Formula for a square-based pyramid, which has 55 faces and 55 vertices.

Solution:

  1. Identify the number of Edges (EE): A square-based pyramid has 44 edges around the base and 44 edges leading to the apex, so E=8E = 8. \ 2. Plug values into Euler's Formula: V−E+F=2V - E + F = 2 \ 3. Substitute: 5−8+55 - 8 + 5 \ 4. Calculate: −3+5=2-3 + 5 = 2 \ 5. Conclusion: 2=22 = 2.

Explanation:

Euler's Formula states that V−E+FV - E + F must equal 22. By counting the edges of the pyramid (88) and using the given number of vertices (55) and faces (55), we show that the equation holds true.