Geometry Geometric Constructions Worksheet (PDF)

Geometry math worksheet: Geometric Constructions – Maths PDF

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This constructions practice worksheet for Geometry covers geometric constructions (Common Core G-CO.D.12, G-CO.D.13), with a free PDF to print.

These 20 problems practice constructions with a compass and straightedge (Common Core G-CO.D.12, G-CO.D.13). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with benches, pavers, rugs and coordinates.

Geometry · Common Core State Standards: G-CO.D.12, G-CO.D.13

Constructions practice worksheet: exercises 1 to 10

Exercise 1 – Compass and straightedge

State what each tool does in a construction: the compass and the straightedge.

Exercise 2 – Equilateral triangle sides

Triangle ABC with AB = 6.

An equilateral triangle is constructed on segment AB of length 6 cm. Find the other two sides and the perimeter.

Exercise 3 – Choose the radius

To construct the perpendicular bisector of a segment 14 cm long, Jamal considers compass radii of 5 cm, 7 cm and 8 cm. Which radius works? Explain.

Exercise 4 – Point on the bisector

Point P is on the perpendicular bisector: arcs of the same radius (more than half of AB) from A and from B.

P lies on the perpendicular bisector of AB. \(PA = 2x + 1\) and \(PB = 3x – 4\). Find x and PA.

Exercise 5 – Bisect an angle

Angle bisector: arc across both rays, then two equal arcs meet at R.

The angle at V measures \(84^{\circ}\). After bisecting it, find the measure of each angle.

Exercise 6 – Bisector with x

An angle measures \((8x + 4)^{\circ}\). After constructing its bisector, each half measures \((3x + 12)^{\circ}\). Find x and the original angle.

Exercise 7 – Angles from 60 degrees

Keisha constructs a 60° angle using an equilateral triangle. She then bisects it twice. Find the angle after each bisection.

Exercise 8 – Add constructed angles

Explain how to build a 75° angle from a 60° angle using only constructions. Give the angles used.

Exercise 9 – Inscribed equilateral triangle

A circle has radius 6 cm. Joining every other vertex of an inscribed regular hexagon gives an inscribed equilateral triangle. A chord between vertices two steps apart has length \(r\sqrt{3}\). Find the side of the triangle in simplest radical form and as a decimal to the nearest hundredth.

Exercise 10 – Hexagon in a circle

A regular hexagon: the side equals the radius, so step the radius around the circle.

A regular hexagon is inscribed in a circle of radius 8 cm. Find the side length and the perimeter.

Exercises 11 and up

Exercise 11 – Square in a circle

A square: draw two perpendicular diameters and join their endpoints.

A square is inscribed in a circle of radius 5 inches. Find its side length and area.

Exercise 12 – Order the steps

Put these steps in order to construct a perpendicular bisector of AB: (a) draw line PQ, (b) draw arcs from B with the same radius, (c) draw arcs above and below AB from A with a radius greater than half of AB.

Exercise 13 – Copy an angle for parallels

A transversal crosses line l, forming a 58° angle. Keisha copies this angle at another point on the transversal so that the two angles are corresponding. Why are the two lines parallel, and what is the corresponding same-side interior angle?

Exercise 14 – Why the bisector works

Arcs of radius 9 on AB = 14.

A perpendicular bisector is built with arcs of radius 9 on a segment AB of length 14. Find PM, the distance from the crossing point P to the midpoint M.

Exercise 15 – Bisector proof by SSS

In an angle bisector construction, VP = VQ = 4 and PR = QR = 6. Name the congruence test that proves that triangle VPR is congruent to triangle VQR and state which angles are congruent.

Exercise 16 – Perpendicular bisector equation

The bisector passes through M(2, 4).

Find an equation of the perpendicular bisector of the segment with endpoints A(-1, 2) and B(5, 6).

Exercise 17 – Paver pattern

A garden path uses regular hexagonal pavers. Each paver is inscribed in a circle of radius 8 inches. Find the distance across the paver from one vertex to the opposite vertex, and the perimeter.

Exercise 18 – Equal distance from two gates

Two park gates A and B are 100 yards apart. A bench is on the perpendicular bisector of AB and is 130 yards from gate A. How far is the bench from gate B, and how far from the midpoint M of the gates?

Exercise 19 – Rug design

A designer inscribes a square in a circular rug with diameter 12 feet. Find the side and the area of the square, and the fraction of the rug’s area that the square covers (use \(\pi = 3.14\)).

Exercise 20 – Copy a segment

Mei sets her compass to the length of segment AB, which is 7.5 cm, and marks three equal lengths in a row along a ray starting at A. How far is the third mark from A?

Key facts behind this constructions practice worksheet

Key facts

Constructions use only a compass and a straightedge.
Perpendicular bisector: equal arcs from both endpoints with radius more than half the segment.
Angle bisector: arc at the vertex, then equal arcs from the two crossing points.
Equilateral triangle: arcs of radius AB from A and from B.
A regular hexagon has side length equal to the radius of its circumscribed circle.

How to use this constructions practice worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the constructions practice worksheet as a PDF with the button below and keep it with your notes.

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