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Key Facts: Madrid PAU Technical Drawing Applied to Arts and Design II Exam

90 Min

Official examination time limit

Madrid PAU Organising Commission UCM 2026

0–10

Official grading scale for PAU subject papers

Spanish University Access Regulations

Min 4.0

Minimum mark required in Access Phase to compute composite score

Community of Madrid University Admissions

EUR 93.02

Base registration fee for compulsory Access Phase in Madrid

Public University Fees Decree Madrid

100

High-quality practice questions in this OpenExamPrep bank

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Madrid PAU Technical Drawing Applied to Arts and Design II tests students on plane geometry, tangencies, dihedral system projections, axonometric and conic perspective, and ISO normalization over 90 minutes.

Sample Madrid PAU Technical Drawing Applied to Arts and Design II Practice Questions

Try these sample questions to review concepts for the Madrid PAU Technical Drawing Applied to Arts and Design II exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1In planar triangle geometry, which notable point is defined as the common intersection of the three perpendicular bisectors (mediatrices) of the triangle's sides?
A.Circumcenter (Circuncentro)
B.Incenter (Incentro)
C.Orthocenter (Ortocentro)
D.Centroid (Baricentro)
Explanation: The circumcenter is the point where the three perpendicular bisectors of a triangle intersect. It serves as the center of the circumscribed circle passing through all three vertices. The incenter is formed by angle bisectors, the orthocenter by altitudes, and the centroid by medians.
2Point P lies at a distance d = 13 cm from the center O of a circle with radius R = 5 cm. What is the power of point P with respect to this circle?
A.64 cm²
B.144 cm²
C.169 cm²
D.25 cm²
Explanation: The power of a point P with respect to a circle of radius R centered at O is calculated as P = d² - R². Substituting d = 13 cm and R = 5 cm gives P = 13² - 5² = 169 - 25 = 144 cm². This value also equals the square of the length of the tangent segment from P to the circle.
3What is the defining geometric locus property of the radical axis (eje radical) of two non-concentric planar circles?
A.The line connecting the centers of both circles
B.The set of points from which both circles subtend equal visual angles
C.The straight line containing all points that have equal power with respect to both circles
D.The line perpendicular to the segment connecting the two centers passing through the midpoint
Explanation: The radical axis of two non-concentric circles is defined as the locus of all points in the plane that have equal power with respect to both circles. It is always a straight line perpendicular to the line of centers.
4In a planar inversion transformation with center O and positive constant of inversion K = 36 cm², point A is located 4 cm from O on a ray. What is the distance from O to the inverse point A'?
A.32 cm
B.144 cm
C.4.5 cm
D.9.0 cm
Explanation: By definition of planar inversion, points A and A' lie on the same ray from O such that OA · OA' = K. Substituting OA = 4 cm and K = 36 cm² yields OA' = 36 / 4 = 9.0 cm.
5Which fundamental geometric condition determines the line tangent to a circle at a specified point T on its circumference?
A.The tangent line is perpendicular to the radius drawn to the point of contact T
B.The tangent line forms a 45-degree angle with the radius at T
C.The tangent line passes through the center of the circle
D.The tangent line is parallel to the diameter passing through T
Explanation: A line is tangent to a circle at point T if and only if it is perpendicular (forms a 90-degree angle) to the radius OT connecting the center to the contact point T.
6To construct the two tangent lines from an exterior point P to a circle centered at O, which classic geometric construction is used to locate the tangent points T1 and T2?
A.Draw an equilateral triangle on segment OP
B.Draw Thales' circle using segment OP as diameter; its intersections with the circle yield T1 and T2
C.Inscribe a square inside the original circle and project its vertices to P
D.Bisect the radius of the circle and draw a parallel line through P
Explanation: Thales' circle theorem states that an angle inscribed in a semicircle is a right angle. Drawing an auxiliary circle with diameter OP intersects the original circle at T1 and T2, where angle OPT1 and OPT2 are 90 degrees, defining the exact points of tangency.
7Three non-concentric circles have non-collinear centers. How is their unique radical center (centro radical) constructed?
A.By finding the centroid of the triangle formed by their three centers
B.By constructing the circumcenter of the triangle formed by their radii
C.By determining the mutual intersection point of the three pairwise radical axes of the circles
D.By drawing the common external tangents and finding their midpoint
Explanation: The radical center of three circles with non-collinear centers is the single point having equal power with respect to all three circles. It is found at the intersection of the three pairwise radical axes.
8What is the exact algebraic expression for the golden ratio φ (número de oro / razón áurea)?
A.(1 + √3) / 2 ≈ 1.366
B.(1 + √2) / 2 ≈ 1.207
C.√5 - 1 ≈ 1.236
D.(1 + √5) / 2 ≈ 1.618
Explanation: The golden ratio φ is defined by the positive solution to the quadratic equation φ² - φ - 1 = 0, which yields φ = (1 + √5) / 2 ≈ 1.618033988...
9In ruler-and-compass geometry, a regular pentagon is inscribed in a circle of radius R. What is the length of its side s in terms of R?
A.s = (R / 2) · √(10 - 2√5) ≈ 1.176 R
B.s = R · √3 ≈ 1.732 R
C.s = R · √2 ≈ 1.414 R
D.s = R · (√5 - 1) ≈ 1.236 R
Explanation: The side of an inscribed regular pentagon in a circle of radius R is s = 2R · sin(36°) = (R / 2) · √(10 - 2√5) ≈ 1.1756 R. R·√3 is the side of an inscribed equilateral triangle, and R·√2 is the side of an inscribed square.
10In a planar inversion transformation with center O, what is the inverse image of a straight line r that passes directly through the center of inversion O?
A.A circle passing through O
B.The line r itself (invariant as a set of points)
C.A parabola with vertex at O
D.A circle not passing through O
Explanation: Any straight line passing through the center of inversion O is transformed into itself (the line is invariant as a whole entity, though individual points exchange positions along rays from O).

About the Madrid PAU Technical Drawing Applied to Arts and Design II Exam

The Technical Drawing Applied to Arts and Design II examination for the PAU / Selectividad in the Community of Madrid evaluates 2º Bachillerato students on plane geometry, tangency, descriptive geometry (dihedral system), axonometric/conic perspective, ISO normalization, and geometric applications in product, graphic, and architectural design.

Exam sponsor: PAU Organising Commission of the Community of Madrid / Universidad Complutense de Madrid (UCM). The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

90-minute written and practical graphic examination divided into technical construction problems, descriptive geometry (dihedral orthographic projection), axonometric and perspective representation, and technical normalization applied to product and visual design.

Time Limit

90 minutes (1.5 hours)

Passing Score

Marked on a 0–10 scale. Minimum 4.0 required in Access Phase to combine with Bachillerato GPA (60% Bachillerato + 40% PAU >= 5.0 to pass).

Exam / Certification Fees

EUR 93.02 base registration fee for compulsory Access Phase in Community of Madrid (or ~11.63 € per optional subject in voluntary phase).

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

25%

Applied Plane Geometry & Tangency

Fundamental geometric constructions, triangle geometry, polygons, power of a point, radical axis/center, planar inversion, and tangency problems solved by power and transformations.

25%

Orthographic & Descriptive Systems for Design

Dihedral (multiview orthographic) system, representation of geometric primitives, auxiliary plane changes, rotations, rabatment (abatimiento), true shapes, intersections, and planar sections of polyhedra and curved surfaces.

20%

Perspective & Axonometric Systems in Applied Arts

Isometric, dimetric, and Cavalier axonometric projections, reduction factors, scale factors, and central conic perspective (one-point and two-point perspective) using horizon line, vanishing points, and distance points.

15%

Technical Normalization, CAD & Scale Drawing

UNE-EN-ISO drawing standards, line types, sectioning, dimensioning (acotación) principles, standard scales (reduction, enlargement), graphic scale construction, CAD layer management, and digital drafting principles.

15%

Geometry in Product & Graphic Design

Application of planar curves (ellipse, parabola, hyperbola, ovals, ogees, spirals, cycloids) in product design, modular grids, golden ratio (proportion), logo geometry, graphic packaging, and surface developments.

Preparing for the Madrid PAU Technical Drawing Applied to Arts and Design II Exam

What You Need to Know

  • Passing score: Marked on a 0–10 scale. Minimum 4.0 required in Access Phase to combine with Bachillerato GPA (60% Bachillerato + 40% PAU >= 5.0 to pass).
  • Assessment: 90-minute written and practical graphic examination divided into technical construction problems, descriptive geometry (dihedral orthographic projection), axonometric and perspective representation, and technical normalization applied to product and visual design.
  • Time limit: 90 minutes (1.5 hours)
  • Exam / certification fees: EUR 93.02 base registration fee for compulsory Access Phase in Community of Madrid (or ~11.63 € per optional subject in voluntary phase). Official sources

Using Our Practice Resources

  • Work through all 100 available questions
  • Review every answer and explanation
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Madrid PAU Technical Drawing Applied to Arts and Design II: Suggested Study Strategy

1Master fundamental tangency principles based on power of a point, radical axis, radical center, and inversion to solve complex tangential arcs quickly.
2Practice auxiliary plane changes, rotations, and abatimiento (rabatment) in the dihedral system to solve true shape and distance problems accurately.
3Memorize isometric (scale factor 0.816) and Cavalier oblique axonometric reduction ratios, as well as two-point perspective constructions using principal distance and measuring points.
4Review UNE-EN-ISO 128 and ISO 129 standards for line types, sectioning conventions, and dimensioning rules (acotación) applied to design objects.
5Study geometric applications in design including golden ratio proportions, surface developments of prisms/pyramids/cones, and conic sections (ellipse, parabola, hyperbola).

Frequently Asked Questions

What is the format and duration of the Madrid PAU Technical Drawing Applied to Arts II exam?

The examination lasts 90 minutes (1.5 hours). It consists of practical graphic drawing exercises and technical conceptual items covering plane geometry, dihedral orthographic projection, axonometric and conic perspective, and design normalization.

What drawing materials are required for the exam in Madrid?

Candidates must bring technical drawing instruments: compass, set squares (escuadra y cartabón), graduated ruler, protractor, drawing pencils (2H, HB, 2B), eraser, sharpener, and a non-programmable scientific calculator if quantitative calculations are required.

What passing score is needed for university admission in Madrid?

The exam is graded on a 0 to 10 scale. In the compulsory Access Phase, a minimum grade of 4.0 is required to calculate the admission mark (60% Bachillerato average + 40% PAU Access Phase, reaching at least 5.0 overall).

Which universities coordinate the PAU in the Community of Madrid?

The PAU is organized by the Joint PAU Commission of Madrid public universities: Universidad Complutense de Madrid (UCM), Universidad Autónoma de Madrid (UAM), Universidad Politécnica de Madrid (UPM), Universidad Carlos III de Madrid (UC3M), Universidad Rey Juan Carlos (URJC), and Universidad de Alcalá (UAH).

How does Technical Drawing Applied to Arts and Design II differ from standard Technical Drawing II?

While standard Technical Drawing II focuses heavily on complex engineering descriptive geometry, the Arts and Design variant emphasizes applied geometry in graphic design, packaging, product forms, surface developments, perspective rendering, modular composition, and visual proportion.

Are digital drawing tools or CAD software permitted during the PAU exam?

No, the official PAU examination is executed manually on paper using traditional drafting instruments. However, theoretical CAD concepts (layers, vector entities, SNAP modes, digital scaling) are assessed conceptually.