Slicing solids
Tonight's idea
Slice a solid with one flat cut. The new flat face is a cross-section. Cut a cube level and you get a square. Tilt the knife and other shapes appear.
Why these shapes? On each flat face of the cube, a flat knife leaves one straight line. So every side of the cross-section lies on a different face.
Number of sides of the cross-section = number of faces the knife crosses.
A cube has 6 faces, so no slice has more than 6 sides. A shape needs at least 3 sides to close up. So the choices are 3, 4, 5 or 6, and each can be done.
One more clue. A cube's faces come in 3 opposite pairs, and opposite faces are parallel. A flat cut crosses parallel faces in parallel lines. So if the knife crosses both faces of a pair, the cross-section has a pair of parallel sides.
Watch one: slicing off a corner
A cube has edges of 4 cm. Pick a corner. Mark a point 2 cm from it on each of the three edges that meet there. Cut through the marks.
The knife crosses 3 faces, so the cross-section is a triangle. On each face the cut does the same thing: it joins two marks, each 2 cm from the corner. So the three sides are equal, and the triangle is equilateral.
Cut deeper at the same tilt and the triangle grows. Once the knife reaches other faces, the shape gains sides.
Tilt the knife and count the sides: mathdetours.com/demos/slices.html.
Your turn
- Name each cross-section. (a) A cube cut parallel to a face. (b) A tin can (a cylinder) cut parallel to its base. (c) The same can cut straight down through the centre of its top. (d) A square-based pyramid cut parallel to its base.
- A box is 3 cm by 4 cm by 6 cm. (a) A cut parallel to a face gives a rectangle. Which three rectangles are possible, and what are their areas? (b) Stand the box on its 3 by 4 base and cut straight down along a diagonal of the base. That diagonal is 5 cm long. What shape is the cross-section, and what is its area?
- (a) A slice of a cube has 5 sides. How many faces did the knife cross, and how many did it miss? (b) A wedge of cheese has 2 triangle faces and 3 rectangle faces. What is the most sides a slice of it could have? (c) Same question for a pyramid with 4 triangle faces only.
- (a) Can a flat slice of a cube have 7 sides? Explain. (b) Explain why each side of the 6-sided slice in the picture is parallel to the side opposite it. (c) A regular pentagon has 5 equal sides, no two of them parallel. A friend says she sliced a cube and got one. Why must she be wrong?
Stretch: slicing a ball
Every flat slice of a ball is a circle. A ball has radius 5 cm. A slice passes 3 cm from the centre. Find the radius of the circle. (Draw the triangle joining the ball's centre, the circle's centre and a point on the circle's rim.) Where must you cut to get the biggest circle?
Answers
- (a) A square, the same size as the face. (b) A circle. (c) A rectangle, as tall as the can and as wide as its top. (d) A square, smaller than the base.
- (a) 3 by 4, area \(12\) cm²; 3 by 6, area \(18\) cm²; 4 by 6, area \(24\) cm². (b) A rectangle 5 cm wide and 6 cm tall. Area \(5 \times 6 = 30\) cm². It is bigger than any face.
- (a) It crossed 5 faces and missed 1. (b) The wedge has 5 faces, so at most 5 sides. A tilted cut through all five faces does give a pentagon. (c) 4 faces, so at most 4 sides. Slices of it are triangles or four-sided shapes.
- (a) No. Each side needs its own face, and a cube has only 6. (b) The slice has 6 sides, so it crosses all 6 faces. Opposite sides of the hexagon lie on opposite faces of the cube. Those faces are parallel, and a flat cut crosses parallel faces in parallel lines. (c) A 5-sided slice crosses 5 faces and misses 1. The missed face spoils one opposite pair. The other two pairs are both crossed, so the slice has two pairs of parallel sides. A regular pentagon has none.
Stretch. The triangle has a right angle at the circle's centre. Its longest side is the ball's radius, 5 cm, and one short side is 3 cm. By Pythagoras the other short side has square \(5^2 - 3^2 = 25 - 9 = 16\), so the radius of the circle is 4 cm. The biggest circle comes from a cut through the centre of the ball. Its radius is 5 cm.