Lesson 3

Creating Cross Sections by Dilating

  • Let’s create cross sections by doing dilations.

3.1: Dilating, Again

Dilate triangle \(BCD\) using center \(P\) and a scale factor of 2.

Triangle BCD dilated using center P and a scale factor of 2.

Look at your drawing. What do you notice? What do you wonder?

3.2: Pyramid Mobile

Your teacher will give you sheets of paper. Each student in the group should take one sheet of paper and complete these steps:

  1. Locate and mark the center of your sheet of paper by drawing diagonals or another method.
  2. Each student should choose one scale factor from the table. On your paper, draw a dilation of the entire sheet of paper, using the center you marked as the center of dilation.
  3. Measure the length and width of your dilated rectangle and calculate its area. Record the data in the table.
  4. Cut out your dilated rectangle and make a small hole in the center.
scale factor, \(k\) length of scaled rectangle width of scaled rectangle area of scaled rectangle

Now the group as a whole should complete the remaining steps:

  1. Cut 1 long piece of string (more than 30 centimeters) and 4 shorter pieces of string. Make 4 marks on the long piece of string an equal distance apart.
  2. Thread the long piece of string through the hole in the largest rectangle. Tie a shorter piece of string beneath it where you made the first mark on the string. This will hold up the rectangle.
  3. Thread the remaining pieces of paper onto the string from largest to smallest, tying a short piece of string beneath each one at the marks you made.
  4. Hold up the end of the string to make your cross sections resemble a pyramid. As a group, you may have to steady the cross sections for the pyramid to clearly appear.

Is dilating a square using a factor of 0.9, then dilating the image using scale factor 0.9 the same as dilating the original square using a factor of 0.8? Explain or show your reasoning.


Imagine a triangle lying flat on your desk, and a point \(P\) directly above the triangle. If we dilate the triangle using center \(P\) and scale factor \(k=\frac12\) or 0.5, together the triangles resemble cross sections of a pyramid.

Image of pyramid. Apex, point p. Halfway from apex to base, small green triangular cross section. Base, larger blue triangle.

We can add in more cross sections. This image includes two more cross sections, one with scale factor \(k=0.25\) and one with scale factor \(k=0.75\). The triangle with scale factor \(k=1\) is the base of the pyramid, and if we dilate with scale factor \(k=0\) we get a single point at the very top of the pyramid.

A triangle laying flat with point P above it.

Each triangle’s side lengths are a factor of \(k\) times the corresponding side length in the base. For example, for the cross section with \(k=\frac12\), each side length is half the length of the base’s side lengths.

Glossary Entries

  • axis of rotation

    A line about which a two-dimensional figure is rotated to produce a three-dimensional figure, called a solid of rotation. The dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.

    Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.
  • cone

    A cone is a three-dimensional figure with a circular base and a point not in the plane of the base called the apex. Each point on the base is connected to the apex by a line segment.

  • cross section

    The figure formed by intersecting a solid with a plane.



  • cylinder

    A cylinder is a three-dimensional figure with two parallel, congruent, circular bases, formed by translating one base to the other. Each pair of corresponding points on the bases is connected by a line segment.

  • face

    Any flat surface on a three-dimensional figure is a face.

    A cube has 6 faces.

  • prism

    A prism is a solid figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a “pentagonal prism.”

    rectangular prism

    triangular prism

    pentagonal prism

  • pyramid

    A pyramid is a solid figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a “hexagonal pyramid.”

    square pyramid

    pentagonal pyramid

  • solid of rotation

    A three-dimensional figure formed by rotating a two-dimensional figure using a line called the axis of rotation.

    The axis of rotation is the dashed line. The green triangle is rotated about the axis of rotation line to form a solid of rotation.

    Green triangles shown as if they are spinning on the diagonal axis. They resemble a top that is spinning.
  • sphere

    A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.