Exploring Some Geometric Themes · Lesson 4 of 13
Imagining Solids and Reading Their Profiles
“Use mental cutting, paper models, and different viewpoints to connect flat outlines with solid shapes.”
Predict shapes created by cutting corners from squares and triangles. Check mental constructions with drawings and paper models. Describe a solid’s profile from a specified viewpoint. Explain why one or two profiles may fit more than one solid.
Hold a small box in front of you. From one direction it may show a square; after a turn it may show a rectangle. The object has not changed, but the information reaching you has. Geometry helps us describe both the solid and the view.
Before drawing detailed solids, we will practise imagining changes to simpler shapes. Mental pictures become more reliable when we connect them to exact instructions, then check them with a sketch or model.
Build It in Your Imagination
Picture your written name, then try to read its letters in reverse order. You are operating on a mental image rather than changing a physical object. In geometry, the same skill helps us imagine a fold, a cut, or a turn before we carry it out.
A useful habit is to keep one feature fixed as a reference. For a square, you might keep the bottom side horizontal. As you imagine the cuts, track which vertices disappear and which new edges appear. If the picture becomes unclear, draw an intermediate stage instead of guessing.
Cutting the Corners of a Square
Draw a square and locate the midpoint of each side. Cut off a corner by joining the two midpoints next to it. Repeat at all four corners. The remaining central shape has its vertices at the four original side midpoints.
The four removed pieces are congruent isosceles right triangles. The inner shape has four equal sides and four right angles, so it is another square, tilted relative to the original. Turning a square does not change its name.
Problem
A square of side 8 cm has all four corners cut off through the side midpoints. Find the area left.
- 1.The original area is 8 × 8 = 64 cm².
- 2.Each corner triangle has perpendicular sides 4 cm and 4 cm, so its area is 1/2 × 4 × 4 = 8 cm².
- 3.Four triangles remove 4 × 8 = 32 cm². The remaining tilted square has area 64 − 32 = 32 cm².
- 4.The four removed triangles can also be rearranged into a square of area 32 cm², giving a second check of the equal-area split.
Now place each cut one-third of the way along the two sides meeting at a corner. A middle part of every original side remains, and each cut creates an additional side. Four old side portions plus four new cuts give eight sides: an octagon.
This octagon is not regular. For original side 3 units, a retained original side has length 1 unit, but a cut across two perpendicular 1-unit legs has length √2 units. Eight sides alone do not guarantee eight equal sides.
Cutting an Equilateral Triangle
Divide each side of an equilateral triangle into three equal parts. At each corner, join the nearest division points and cut away the small corner triangle. Three original middle-third segments remain, and three new cutting edges appear. The remaining shape is a hexagon.
Each corner triangle is equilateral: its two original side portions are equal, and the included angle is 60°. Therefore every cut edge has the same length as a middle-third segment. The remaining interior angles are 120°. In this particular construction the hexagon is regular.
Problem
A square and an equilateral triangle each have side 9 cm. Cut off their corners at the nearest one-third points. Describe the remaining shapes.
- 1.The square keeps four middle-third edges and gains four cutting edges, forming an octagon.
- 2.Its original-side pieces are 3 cm long, but the new cut edges are 3√2 cm long, so it is not regular.
- 3.The triangle keeps three middle-third edges and gains three cut edges. All six are 3 cm long, forming a regular hexagon.
Profiles of Solids
A profile is the outline seen from a specified direction. For a standard geometric view, imagine looking straight towards the object with parallel viewing lines. Features behind other features may be hidden. The outline records the extent of the object in the two visible directions.
A circular cylinder standing upright has a circular top profile and a rectangular front profile. An upright circular cone has a circular top profile and a triangular front profile. A sphere has a circular outline from every direction. These comparisons show why the viewing direction belongs in the description.
| Solid and orientation | One profile | Another profile |
|---|---|---|
| Upright cylinder | Circle from above | Rectangle from the front |
| Upright cone | Circle from above | Triangle from the front |
| Triangular prism viewed end-on and from a suitable side | Triangle | Rectangle |
| Upright truncated cone | Circle from above | Trapezium from the front |
| Pentagonal prism viewed end-on and from a suitable side | Pentagon | Rectangle |
A truncated cone, also called a frustum of a cone, is made by cutting the top from a cone with a plane parallel to its circular base. Its side outline is a trapezium because its two circular ends have different diameters. Naming this example helps connect an unfamiliar profile pair to a familiar starting solid.
Problem
An upright solid has a circular top view and a triangular front view. Suggest a familiar solid and explain why an upright cylinder does not fit.
- 1.The circle suggests a circular horizontal extent, which both a cylinder and a cone can have.
- 2.A cone tapers to a point, so its front outline is triangular.
- 3.A cylinder has parallel straight sides in this view, giving a rectangle. An upright cone is therefore a suitable answer. The views suggest a model; they do not prove that no more complicated solid could fit.
A triangular profile does not mean that the entire object is a flat triangle. A cone, a pyramid, and an end-on triangular prism can all give triangular outlines. A profile loses depth information.
One View Can Hide a Difference
A cube and a longer square-ended cuboid can both show a square when viewed directly at one end. Their depth is hidden in that view. Looking from the side reveals the difference: the cube gives a square while the longer cuboid gives a rectangle.
Even a pair of profiles may leave several possibilities. Use a model as a proposed answer, then test it against every stated view. If more than one model fits, report that ambiguity instead of inventing a measurement or a hidden feature.
Predict, Turn, and Check
Collect a box, a cylindrical container, and a paper cone. Use eye-level or top views to compare predictions with observations.
- Choose and label a front direction before moving any object.
- Sketch the front and top outlines you expect to see.
- Turn or reposition yourself to check each view while keeping track of the original front.
- Record one feature that was visible in one view but hidden in the other.
Your drawings need not show artistic shading. Their purpose is to communicate the outline and preserve the chosen directions. The habit of fixing a front direction will be essential when you draw three coordinated views.
Quiz
What remains when a square’s four corners are cut through adjacent side midpoints?
Why is the octagon made with one-third corner cuts not regular?
Which familiar upright solid has a circular top profile and a triangular front profile?
What does one profile generally fail to show?
What remains after one-third corner cuts on an equilateral triangle?
Practice Problems
- A square of side 12 cm is cut through its side midpoints. Find the remaining area.
- An equilateral triangle has side 15 cm. After one-third corner cuts, find the perimeter of the remaining hexagon.
- Suggest familiar solids with rectangle-and-circle and rectangle-and-triangle profile pairs.
- Explain how a cube and a cuboid measuring 4 cm by 4 cm by 9 cm can share one profile.
- A learner identifies a sphere after seeing one circular profile. Explain what additional evidence would help.
1. Each removed corner triangle has legs 6 cm and area 18 cm². 2. The four remove 72 cm² from 144 cm². 3. The remaining square has area 72 cm².
Key Takeaways
Mental pictures become reliable when cuts, turns, and viewpoints are specified. Midpoint corner cuts turn a square into a smaller tilted square. One-third cuts give a nonregular octagon from a square and a regular hexagon from an equilateral triangle. A profile shows an outline from one direction and loses depth information. Test a proposed solid against every given view.