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The flat face of a cut through a solid · Geometry · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Look at the cut
  2. Name the shape
  3. Find its area
  4. Check

The steps for Slice a cylinder radius 3 height 8 through the center, up and down. Every solution names its steps, so you always know what comes next.

Worked examples

Slice a cube parallel to a face Easy

The question

Question: what shape is the cut, if we slice a cube parallel to the base?

cube, cut parallel to the base

A cross section is the flat face we see when we slice a solid.

cross section = the face of a cut

1 Look at the cut

Here is the cube.

We slice it parallel to the base.

The red face is the cut.

2 Name the shape

Flat, the cut looks like this.

Every face of a cube is a square.

A cut parallel to a face is the same square.

So it is a square.

= cut: a square

Careful!

All the sides of the cut are the same length, like the face.

So it is a square.

Careful: not a rectangle ✗

3 Check

Let's check: a cut parallel to the base has the shape of the base.

So the answer is right!

like the base: square

Let's look back.

First: look at the cut.

Then: name the shape.

Last: check.

Answer
Careful: not a rectangle ✗

▶ Watch it solved

Slice a cone through the top, up and down Medium

The question

Question: what shape is the cut, if we slice a cone through the top, up and down?

cone, cut through the top, up and down

A cross section is the flat face we see when we slice a solid.

cross section = the face of a cut

1 Look at the cut

Here is the cone.

We slice it through the top, up and down.

The red face is the cut.

2 Name the shape

Flat, the cut looks like this.

Up and down through the top: the cut is a triangle, as wide as the base, as tall as the cone.

So it is a triangle.

= cut: a triangle

Careful!

Not a circle.

Only a cut straight across shows the round part.

Careful: not a circle ✗

3 Check

Let's check: look at the cone from the side.

You see a triangle, the same as the cut through the middle.

So the answer is right!

side view: triangle

Let's look back.

First: look at the cut.

Then: name the shape.

Last: check.

Answer
Careful: not a circle ✗

▶ Watch it solved

Slice a cylinder radius 3 height 8 through the center, up and down Harder

The question

Question: what shape is the cut, if we slice a cylinder radius 3 height 8 through the center, up and down, and what is its area?

cylinder radius 3 height 8, cut through the center, up and down

A cross section is the flat face we see when we slice a solid.

cross section = the face of a cut

1 Look at the cut

Here is the cylinder.

We slice it through the center, up and down.

The red face is the cut.

2 Name the shape

Flat, the cut looks like this.

Up and down through the center: across, the cut is as wide as the circle; up, as tall as the cylinder.

So it is a rectangle.

= cut: a rectangle 6 by 8

Careful!

Not a circle.

Only a cut straight across shows the round part.

Careful: not a circle ✗

3 Find its area

A rectangle's area is length times width: 48.

= area = 6 × 8 = 48

4 Check

Let's check: look at the cylinder from the side.

You see a rectangle, the same as the cut through the middle.

So the answer is right!

side view: rectangle

Let's look back.

First: look at the cut.

Then: name the shape.

Then: find its area.

Last: check.

Answer
= area = 6 × 8 = 48

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. slice a cube parallel to a face
    Answer
    Careful: not a rectangle ✗
  2. slice a cylinder parallel to the base
    Answer
    Careful: parallel cut = the base shape
  3. slice a cylinder through the center, up and down
    Answer
    Careful: not a circle ✗
  4. slice a cone parallel to the base
    Answer
    Careful: parallel cut = the base shape
  5. slice a sphere through the center
    Answer
    Careful: think of the flat face
  6. slice a square pyramid parallel to the base
    Answer
    Careful: parallel cut = the base shape
  7. slice a box parallel to the base
    Answer
    Careful: parallel cut = the base shape

Medium

  1. slice a cone through the top, up and down
    Answer
    Careful: not a circle ✗
  2. slice a square pyramid through the top, up and down
    Answer
    Careful: not a square ✗
  3. slice a box parallel to the front
    Answer
    Careful: think of the flat face
  4. slice a box 6 by 4 by 3 parallel to the base
    Answer
    = area = 6 × 4 = 24
  5. slice a cube side 5 parallel to a face
    Answer
    = area = 5 × 5 = 25
  6. slice a cylinder radius 3 parallel to the base
    Answer
    = area = π × 3² ≈ 28.27
  7. slice a box 8 by 5 by 2 parallel to the front
    Answer
    = area = 8 × 2 = 16

Harder

  1. slice a cylinder radius 3 height 8 through the center, up and down
    Answer
    = area = 6 × 8 = 48
  2. slice a cylinder radius 5 parallel to the base
    Answer
    = area = π × 5² ≈ 78.54
  3. slice a box 10 by 3 by 7 parallel to the base
    Answer
    = area = 10 × 3 = 30
  4. slice a cylinder radius 4 height 10 through the center, up and down
    Answer
    = area = 8 × 10 = 80
  5. slice a cube side 12 parallel to a face
    Answer
    = area = 12 × 12 = 144
  6. slice a box 9 by 6 by 4 parallel to the front
    Answer
    = area = 9 × 4 = 36