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Introduction to solid geometry: mixed assessment

Read the idea, work independently, then explain what changed.

TOPIC 01

Introduction to solid geometry: mixed assessment

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

Connect the chapter skills without relying on the order of the exercises.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

How many vertices does an 10-gonal prism have?

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the defining faces, cross-sections and generating elements of the solid.

Hint 2

Each base contributes n vertices.

Worked solution

Identify the defining faces, cross-sections and generating elements of the solid.
Calculate or simplify this relation.
$V=2n=2(10)=20$
Corresponding base vertices are distinct and joined by lateral edges.

The requested value is 20.

Checks and common pitfalls: Corresponding base vertices are distinct and joined by lateral edges.

Think first. Reveal a hint when the class is ready.

02 / Foundation#Your turn

Recover the actual depth from its half-scale drawn length.

d'=8

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish the actual spatial object from the stated oblique drawing convention.

Hint 2

Undo the one-half scale factor.

Worked solution

Distinguish the actual spatial object from the stated oblique drawing convention.
Calculate or simplify this relation.
$d=2d'=16$
The same recovery is not applied to axes drawn at full scale.

The requested value is 16.

Checks and common pitfalls: The same recovery is not applied to axes drawn at full scale.

Think first. Reveal a hint when the class is ready.

03 / Foundation#Your turn

A sphere has radius r. Write its volume as cπ and find c.

r=7

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Choose the correct surface or volume formula and keep units and similarity powers consistent.

Hint 2

Use the cube of the radius.

Worked solution

Choose the correct surface or volume formula and keep units and similarity powers consistent.
Calculate or simplify this relation.
$V=\frac43\pi r^3=\frac{1372}3\pi$
Diameter must be halved before substitution if it is the given measurement.

The requested value is 457.33333333.

Checks and common pitfalls: Diameter must be halved before substitution if it is the given measurement.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Solve both parts and justify the conditions used. Part A: State the intersection of two distinct planes known to share two distinct points. Part B: Similar solids have linear scale factor 3 from small to large. Find the surface-area ratio.

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use precise incidence conditions; diagrams alone do not prove spatial relationships.

Hint 2

B: Choose the correct surface or volume formula and keep units and similarity powers consistent.

Worked solution

Part A reasoning
Use precise incidence conditions; diagrams alone do not prove spatial relationships.
Calculate or simplify this relation.
$A,B\in\alpha\cap\beta,\ A\ne B\Rightarrow AB\subseteq\alpha\cap\beta$
Distinct intersecting planes meet in a line, not a finite segment.
Part B reasoning
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Calculate or simplify this relation.
$S_{\text{large}}/S_{\text{small}}=3^2=9$
The volume ratio would instead be 27.

A: The entire line through those two points. B: The requested value is 9.

Checks and common pitfalls: Distinct intersecting planes meet in a line, not a finite segment. The volume ratio would instead be 27.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Solve both parts and justify the conditions used. Part A: State a sufficient criterion for two distinct planes to be parallel using lines in one plane. Part B: How many planes contain two distinct parallel lines? Explain.

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Apply a parallelism criterion with all its incidence and outside-plane conditions.

Hint 2

B: Use precise incidence conditions; diagrams alone do not prove spatial relationships.

Worked solution

Part A reasoning
Apply a parallelism criterion with all its incidence and outside-plane conditions.
Calculate or simplify this relation.
$l,m\subseteq\alpha,\ l\cap m\ne\varnothing,\ l\parallel\beta,\ m\parallel\beta\Rightarrow\alpha\parallel\beta$
Two independent directions constrain the plane orientation.
Part B reasoning
Use precise incidence conditions; diagrams alone do not prove spatial relationships.
Calculate or simplify this relation.
$l\parallel m,\ l\ne m\Rightarrow\exists!\alpha:l,m\subseteq\alpha$
The three selected points are noncollinear and determine a unique plane.

A: Two intersecting lines in one plane must each be parallel to the other plane. B: The requested value is 1.

Checks and common pitfalls: Two independent directions constrain the plane orientation. The three selected points are noncollinear and determine a unique plane.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Solve both parts and justify the conditions used. Part A: Find the dihedral angle between the coordinate planes x=0 and y=0. Part B: A line l is outside plane α and parallel to a line m in α. State and justify the conclusion.

\begin{gathered}\text{B: }l\not\subseteq\alpha,\quad m\subseteq\alpha,\quad l\parallel m\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

Hint 2

B: Apply a parallelism criterion with all its incidence and outside-plane conditions.

Worked solution

Part A reasoning
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Calculate or simplify this relation.
$\mathbf n_1=(1,0,0),\quad\mathbf n_2=(0,1,0),\quad\mathbf n_1\cdot\mathbf n_2=0$
The section gives two perpendicular coordinate axes.
Part B reasoning
Apply a parallelism criterion with all its incidence and outside-plane conditions.
Calculate or simplify this relation.
$l\parallel m\subseteq\alpha,\ l\not\subseteq\alpha\Rightarrow l\parallel\alpha$
Without the outside-plane condition, l could be contained in α.

A: The requested value is 90. B: l is parallel to α.

Checks and common pitfalls: The section gives two perpendicular coordinate axes. Without the outside-plane condition, l could be contained in α.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

07 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: A plane parallel to a pyramid base cuts halfway between apex and base. Find the ratio of the small cross-section area to the base area. Part B: Find the distance from P to the plane.

\begin{gathered}\text{B: }P=(8,2,11),\quad\alpha:z=0\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Identify the defining faces, cross-sections and generating elements of the solid.

Hint 2

B: Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

Worked solution

Part A reasoning
Identify the defining faces, cross-sections and generating elements of the solid.
Calculate or simplify this relation.
$\lambda=1/2;\quad S_{\text{section}}/S_{\text{base}}=\lambda^2=1/4$
Area ratios are squares of linear ratios.
Part B reasoning
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Calculate or simplify this relation.
$H=(8,2,0),\quad PH=11$
Distance to a plane is the perpendicular length, not distance to the origin.

A: The requested value is 0.25. B: The requested value is 11.

Checks and common pitfalls: Area ratios are squares of linear ratios. Distance to a plane is the perpendicular length, not distance to the origin.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

A vertical edge has actual length 10 and vertical scale one. Find its drawn length.

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish the actual spatial object from the stated oblique drawing convention.

Hint 2

Read the scale assigned to the vertical axis.

Worked solution

Distinguish the actual spatial object from the stated oblique drawing convention.
Calculate or simplify this relation.
$h'=1\cdot10=10$
The half-depth rule applies only to the specified receding direction.

The requested value is 10.

Checks and common pitfalls: The half-depth rule applies only to the specified receding direction.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: Similar solids have linear scale factor 3 from small to large. Find the surface-area ratio. Part B: In the 45° half-depth oblique convention, an original depth is d. Find its drawn length.

\begin{gathered}\text{B: }d=18\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Choose the correct surface or volume formula and keep units and similarity powers consistent.

Hint 2

B: Distinguish the actual spatial object from the stated oblique drawing convention.

Worked solution

Part A reasoning
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Calculate or simplify this relation.
$S_{\text{large}}/S_{\text{small}}=3^2=9$
The volume ratio would instead be 27.
Part B reasoning
Distinguish the actual spatial object from the stated oblique drawing convention.
Calculate or simplify this relation.
$d'=18/2=9$
This is a drawing convention, not a change in the solid.

A: The requested value is 9. B: The requested value is 9.

Checks and common pitfalls: The volume ratio would instead be 27. This is a drawing convention, not a change in the solid.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: How many planes contain two distinct parallel lines? Explain. Part B: A cone has radius r and height h. Write its volume as cπ and find c.

\begin{gathered}\text{B: }r=9,\quad h=3\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Use precise incidence conditions; diagrams alone do not prove spatial relationships.

Hint 2

B: Choose the correct surface or volume formula and keep units and similarity powers consistent.

Worked solution

Part A reasoning
Use precise incidence conditions; diagrams alone do not prove spatial relationships.
Calculate or simplify this relation.
$l\parallel m,\ l\ne m\Rightarrow\exists!\alpha:l,m\subseteq\alpha$
The three selected points are noncollinear and determine a unique plane.
Part B reasoning
Choose the correct surface or volume formula and keep units and similarity powers consistent.
Calculate or simplify this relation.
$V=\frac13\pi r^2h=81\pi$
Do not omit the factor one third.

A: The requested value is 1. B: The requested value is 81.

Checks and common pitfalls: The three selected points are noncollinear and determine a unique plane. Do not omit the factor one third.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Solve both parts and justify the conditions used. Part A: A line l is outside plane α and parallel to a line m in α. State and justify the conclusion. Part B: How many vertices does an 10-gonal prism have?

\begin{gathered}\text{A: }l\not\subseteq\alpha,\quad m\subseteq\alpha,\quad l\parallel m\end{gathered}

Use the domain, units and sampling assumptions stated in the question.
Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

A: Apply a parallelism criterion with all its incidence and outside-plane conditions.

Hint 2

B: Identify the defining faces, cross-sections and generating elements of the solid.

Worked solution

Part A reasoning
Apply a parallelism criterion with all its incidence and outside-plane conditions.
Calculate or simplify this relation.
$l\parallel m\subseteq\alpha,\ l\not\subseteq\alpha\Rightarrow l\parallel\alpha$
Without the outside-plane condition, l could be contained in α.
Part B reasoning
Identify the defining faces, cross-sections and generating elements of the solid.
Calculate or simplify this relation.
$V=2n=2(10)=20$
Corresponding base vertices are distinct and joined by lateral edges.

A: l is parallel to α. B: The requested value is 20.

Checks and common pitfalls: Without the outside-plane condition, l could be contained in α. Corresponding base vertices are distinct and joined by lateral edges.

Reasoning checklist · self / teacher assessment

Solve part A with its stated restrictions.
Solve part B using an appropriate representation.
Give the reasoning and check conditions in both parts.

Think first. Reveal a hint when the class is ready.

12 / Standard#Your turn

Find the distance from P to the plane.

P=(8,2,11),\quad\alpha:z=0

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

Hint 2

Drop a perpendicular parallel to the z-axis.

Worked solution

Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Calculate or simplify this relation.
$H=(8,2,0),\quad PH=11$
Distance to a plane is the perpendicular length, not distance to the origin.

The requested value is 11.

Checks and common pitfalls: Distance to a plane is the perpendicular length, not distance to the origin.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Verify Euler’s relation for an 12-gonal prism by giving V,E,F.

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Identify the defining faces, cross-sections and generating elements of the solid.

Hint 2

Count two bases and n lateral faces.

Worked solution

Identify the defining faces, cross-sections and generating elements of the solid.
Calculate or simplify this relation.
$2n-3n+(n+2)=2\quad(n=12)$
This verifies the relation for this convex polyhedron family; it is not a proof for every polyhedron.

V=24, E=36, F=14; V−E+F=2.

Checks and common pitfalls: This verifies the relation for this convex polyhedron family; it is not a proof for every polyhedron.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

14 / Transfer#Your turn

In the 45° half-depth oblique convention, an original depth is d. Find its drawn length.

d=18

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Distinguish the actual spatial object from the stated oblique drawing convention.

Hint 2

The receding-axis scale is one half.

Worked solution

Distinguish the actual spatial object from the stated oblique drawing convention.
Calculate or simplify this relation.
$d'=18/2=9$
This is a drawing convention, not a change in the solid.

The requested value is 9.

Checks and common pitfalls: This is a drawing convention, not a change in the solid.

Think first. Reveal a hint when the class is ready.

15 / Transfer#Your turn

A cone has radius r and height h. Write its volume as cπ and find c.

r=9,\quad h=3

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Choose the correct surface or volume formula and keep units and similarity powers consistent.

Hint 2

A cone occupies one third of the corresponding cylinder volume.

Worked solution

Choose the correct surface or volume formula and keep units and similarity powers consistent.
Calculate or simplify this relation.
$V=\frac13\pi r^2h=81\pi$
Do not omit the factor one third.

The requested value is 81.

Checks and common pitfalls: Do not omit the factor one third.

Think first. Reveal a hint when the class is ready.

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

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Teacher preparation and assessment

Question sequence

Ask students to name the relevant condition before calculating.

Board plan

Compare valid methods and annotate their conditions.

Anticipated thinking

A correct final value may still hide a missing assumption.

Assessment checklist

Check the method, conditions, reasoning and interpretation separately.