Overview
Three-dimensional solid figures can be represented by the two-dimensional pattern of polygons that create them. The pattern, called a net, is a visual representation that illustrates the formula for the surface area of the three-dimensional figure. If the net were folded, it would produce that figure.
Representation of Cylinders
The net for a cylinder consists of two circles adjoining a rectangle. This relates to the formula for the surface area of a cylinder; 2 πr2+ 2πrh. The 2 πr2 is the formula for the area of the two circles. In order for the rectangle to fit around the circumference of the circle, the width of the rectangle is 2πr. The height of the rectangle is the height of the cylinder. Since the formula for the area of a rectangle is length multiplied by width, the area of the rectangle will have the measurement 2πrh.
Representation of Rectangular Prisms
The net for a cube consists of the six square faces that make up the cube. A square has the same length, width, and height. It is a special type of a rectangular prism, also known as a rectangular cuboid. A rectangular prism also has 6 faces, in three parallel pairs that meet at right angles. The top and bottom faces are congruent, as are the two other pairs of opposite sides. The formula for the surface area of a rectangular prism is 2(lw +wh +lh), which the net illustrates perfectly.
Representations of Pyramids
Pyramids are solid figures with triangular faces that meet at a single point called an apex, and a polygon base. A tetrahedron is a special type of pyramid with 4 triangular faces, and a regular tetrahedron, with all triangles equilateral and congruent, is a Platonic solid. Another type of pyramid has a square or rectangular base and three triangular sides. The net that illustrates the pyramids has the base bounded by triangles on each side. A regular tetrahedron has a net with all four triangles inside a larger triangle.
Representations of Other Figures
Many other solid figures can be represented by their nets. For example, a cone with a circular base is represented by the circular base adjoined by a quarter circle. Solid figures have been extrapolated into more than three dimensions. A tesseract is a four-dimensional figure with three-dimensional faces. It has been used in surreal art, science fiction, music, and popular culture.
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