Geometry and trigonometry practice questions (IB Maths AI)

Sample questions

All 3 free IB Maths AI questions on this topic, from the free IB Maths AI SL Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 17 marks · Trigonometry
   A triangular field ABC has AB = 85 m, AC = 62 m and angle BAC = 48 degrees. A vertical flagpole of height 12 m stands at C.
   (a)

   Find the length of BC.

   [3]
   (b)

   Find the area of the field.

   [2]
   (c)

   Find the angle of elevation of the top of the flagpole from B.

   [2]
   Show answerFrom the free IB Maths AI SL Practice Exam →
2. Question 26 marks · Voronoi diagrams
   A town has three mobile-phone masts at A(0, 0), B(8, 0) and C(4, 6), where the coordinates are in kilometres. Every phone connects to the nearest mast, so the town is divided into three regions by a Voronoi diagram with sites A, B and C.
   (a)

   Write down the equation of the perpendicular bisector of AB.

   [1]
   (b)

   Find the equation of the perpendicular bisector of AC. Give your answer in the form $ax + by = c$, where $a$, $b$ and $c$ are integers.

   [3]
   (c)

   Hence find the coordinates of the vertex of the Voronoi diagram, the point that is equidistant from all three masts.

   [1]
   (d)

   A new house is built at (5, 2). Determine which mast the phones in this house will connect to.

   [1]
   Show answerFrom the free IB Maths AI SL Practice Exam →
3. Question 36 marks · Volume & surface area
   A grain silo consists of a cylinder of radius 3 m and height 8 m with a cone of height 4 m on top. The cone has the same radius as the cylinder. The silo stands on flat ground.
   (a)

   Calculate the volume of the silo.

   [3]
   (b)

   Show that the slant height of the cone is 5 m.

   [1]
   (c)

   The outside of the silo, excluding the base on the ground, is to be painted. Calculate the area to be painted.

   [2]
   Show answerFrom the free IB Maths AI SL Practice Exam →

Key terms

The definitions and formulas IB Maths AI expects for geometry and trigonometry, typeset as they appear on the paper.

Distance and midpoint in three dimensions

Pythagoras extended to three coordinates. The midpoint averages each coordinate; both are needed for questions about pyramids and cuboids.

$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$

Sine rule, cosine rule and triangle area

The tools for any triangle that is not right-angled: the sine rule for a side-angle pair, the cosine rule when two sides and the included angle (or three sides) are known.

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, \qquad c^2 = a^2 + b^2 - 2ab\cos C, \qquad A = \frac{1}{2}ab\sin C$$

Volume and surface area of solids

Cuboids, prisms, cylinders, pyramids, cones, spheres and hemispheres, often combined. Curved surface area of a cone uses the slant height $l$, not the vertical height.

$$V_{\text{cone}} = \frac{1}{3}\pi r^2 h, \quad A_{\text{curved cone}} = \pi r l, \quad V_{\text{sphere}} = \frac{4}{3}\pi r^3, \quad A_{\text{sphere}} = 4\pi r^2$$

Angles of elevation and depression, bearings

The angle above or below the horizontal to an object, and the direction measured clockwise from north as a three-figure angle. Draw the diagram with the horizontal marked before writing any trig ratio.

Arc length and sector area in degrees

The fraction $\frac{\theta}{360}$ of the circumference and of the circle's area. AI works in degrees unless the question says radians.

$$l = \frac{\theta}{360} \times 2\pi r, \qquad A = \frac{\theta}{360} \times \pi r^2$$

Perpendicular bisector

The line through the midpoint of a segment at right angles to it, every point of which is equidistant from the two endpoints. Its gradient is the negative reciprocal of the segment's.

Voronoi diagram

A division of the plane into cells, one per site, where each cell contains the points nearest to its site. Edges are perpendicular bisectors between neighbouring sites, vertices are equidistant from three sites, and the largest empty circle (the "toxic waste dump" problem) is centred on a vertex.

Frequently asked questions

Other IB Maths AI topics

• Number and algebra
• Functions
• Statistics and probability
• Calculus

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