Construction

In the general sense, construction means to build something. But in geometry it has a special meaning. Here, construction is the act of drawing geometric shapes using only a compass and straightedge. No measuring of lengths or angles is allowed.

 Compass Straight-edge
For more see Introduction to constructions.

## Pages referring to 'construction'

How to construct the complementary angle to a given angle with compass and straightedge or ruler. A Euclidean construction.
Given an apex angle and leg length this page shows how to construct an isosceles triangle using a compass and straightedge or ruler.
Constructing 75, 105, 120, 135, 150 degree angles and more. Euclidean constructions with compass and straight edge (ruler). The table shows angles that can be obtained by combining simpler ones in various ways
Constructing a circle through 3 given points. Printable step-by-step instructions.
How to construct (draw) a regular pentagon inscribed in a circle. The largest pentagon that will fit in the circle, with each vertex touching the circle.
A brief introduction to constructions - creating various geometric objects with only compasses and straightedge or ruler. History and origins
How to copy a line segment with compass and straightedge or ruler. Given a line segment, this shows how to make another segemnt of the same length. A Euclidean construction.
How to copy a line segment with compass and straightedge or ruler. Given a line segment, this shows how to make another segemnt of the same length. A Euclidean construction.
How to copy a triangle with compass and straightedge or ruler. Given a triangle, this page shws how to make another triangle that is congruent to it. A Euclidean construction.
How to subtract one angle from another with compass and straightedge or ruler. A Euclidean construction.
How to subtract the lengths of two line segments with compass and straightedge or ruler. A Euclidean construction.
How to divide a line segment into equal parts with compass and straightedge or ruler. We start with a given line segment and divide it into any number of equal parts. In the applet we divide it into five parts but it can be any number. Using a compass and straightedge, we do this without measuring the line. A Euclidean construction
How to find the two foci of an ellipse given its width and height (major and minor axes). This can be used to find the two focus points when you are planning to draw an ellipse using the string and pins method. Uses a compass, no measuring is used. A Euclidean construction.
How to find the center of a circle with compass and straightedge or ruler. This method relies on the fact that, for any chord of a circle, the perpendicular bisector of the chord always passes through the center of the circle. By applying this twice to two different chords, the center is established where the two bisectors intersect. A Euclidean construction
How find the center of a circle using any right-angled object. This method works as a result of Thales Theorem. The diameter of a circle subtends a right angle to any point on the circle.
Shows how to construct add one given angle to another using a compass and straightedge or ruler. A proof is shown. A Euclidean construction
This construction shows how to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler. This both bisects the segment (divides it into two equal parts), and is perpendicular to it. Finds the midpoint of a line segmrnt. The proof shown below shows that it works by creating 4 congruent triangles. A Euclideamn construction.
How to bisect an angle with compass and straightedge or ruler. To bisect an angle means that we divide the angle into two equal (congruent) parts without actually measuring the angle. This Euclidean construction works by creating two congruent triangles. See the proof below for more on this.
How to construct (draw) a triangle given two angles and a non-included side with compass and straightedge or ruler. (AAS)
How to construct (draw) a triangle given one side and the angle at each end of it with compass and straightedge or ruler. It works by first copying the line segment to form one side of the triangle, then copy the two angles on to each end of it to complete the triangle. As noted below, there are four possible triangles that be drawn - they are all correct. A Euclidean construction.
On this page we show how to construct (draw) a 90 degree angle with compass and straightedge or ruler. There are various ways to do this, but in this construction we use a property of Thales Theorem. We create a circle where the vertex of the desired right angle is a point on a circle. Thales Theorem says that any diameter of a circle subtends a right angle to any point on the circle. A Euclidean construction.
This page shows how to construct (draw) the circumcircle of a triangle with compass and straightedge or ruler. The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. It's center is called the circumcenter, which is the point where the three perpedicular bisectors of the sides intersect. A Euclidean construction.
How to construct (draw) the centroid of a triangle with compass and straightedge or ruler. The centroid of a triangle is the point where its medians intersect. It is also the center of gravity of the triangle and one of the triangle's points of concurrency. It works by constructing two medians, which intersect at the centroid. A Euclidean construction.
This is not a true Euclidean construction but a practical way to draw an ellipse given its width and height and when mathematical precision is not so important. It is done using string and two pins, but it is sometimes called the 'Gardener's Ellipse', because it works well on a large scale, using rope and stakes, to lay out elliptical flower beds in formal gardens.
This page shows how to construct (draw) a 45 degree angle with compass and straightedge or ruler. It works by constructing an isosceles right triangle, which has interior angles of 45, 45 and 90 degrees. We use one of those 45 degree angles to get the result we need. See the proof below for more details. A Euclidean construction.
This page shows how to draw a perpendicular at a point on a line with compass and straightedge or ruler. It works by effectively creating two congruent triangles and then drawing a line between their vertices. A Euclidean construction.
This page shows how to cosntruct a perpendicular at the end of a ray with compass and straightedge or ruler. This construction works as a result of Thales Theorem. From this theorem, we know that a diameter of a circle always subtends a right angle to any point on the circle, so by using it in reverse we produce a right angle.
This page shows how to construct a perpendicular to a line through an external point, using only a compass and straightedge or ruler. It works by creating a line segment on the given line, then bisecting it. A Euclidean construction.
This page shows how to construct a right triangle given one leg and the hypotenuse with compass and straightedge or ruler.
This page shows how to construct a right triangle given one angle and the hypotenuse with compass and straightedge or ruler.
This page shows how to construct a right triangle given one angle and one leg (LA) with compass and straightedge or ruler.
This page shows how to construct a right triangle with two given leg lengths with compass and straightedge or ruler.
This page shows how to construct (draw) a square with a given side length with compass and straightedge or ruler. It works by first erecting a perpendicular and then drawing the three remaining sides all the same length. A Euclidean construction.
This page shows how to construct a triangle given the length of all three sides, with compass and straightedge or ruler. It works by first copying one of the line segments to form one side of the triangle. Then it finds the third vertex from where two arcs intersect at the given distance from each end of it. A Euclidean construction.
This page shows how to construct an equilateral triangle with compass and straightedge or ruler. It begins with a given line segment which is the length of each side of the desired equilateral triangle. It works because the compass width is not changed between drawing each side, guaranteeing they are all congruent (same length). It is similar to the 60 degree angle construction, because the interior angles of an equilateral triangle are all 60 degrees. A Euclidean construction.
This page shows how to construct one of the three altitudes of a triangle, using only a compass and straightedge or ruler. A Euclidean construction.
This page shows how to construct one of the three altitudes of an obtuse triangle, using only a compass and straightedge or ruler. A Euclidean construction.
This page shows how to draw the two possible tangents to a given circle through an external point with compass and straightedge or ruler. This construction assumes you are already familiar with constructing the perpendicular bisector of a line segment.
How to construct (draw) the incircle of a triangle with compass and straightedge or ruler. The three angle bisectors of any triangle always pass through its incenter. In this construction, we only use two, as this is sufficient to define the point where they intersect. We bisect the two angles and then draw a circle that just touches the triangles's sides. A Euclidean construction.
Median of a triangle construction with compass and straightedge or ruler. A triangle has three medians. They are lines linking each vertex to the midpoint of the opposite side. We first find the midpoint, then draw the median. A Euclidean construction.
How to construct the orthocenter of a triangle with compass and straightedge or ruler. The orthocenter is the point where all three altitudes of the triangle intersect. An altitude is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. A Euclidean construction
This page shows how to construct (draw) the two external tangents to two given circles with compass and straightedge or ruler.
This page shows how to construct (draw) the two internal tangents to two given circles with compass and straightedge or ruler.
This page shows how to construct a line parallel to a given line through a given point with compass and straightedge or ruler. This construction works by creating a rhombus. Since we know that the opposite sides of a rhombus are parallel, then we have created the desired parallel line. This construction is easier than the traditional angle method since it is done with just a single compass setting. A Euclidean construction.
How to draw an isosceles triangle given the base and altitude with compass and straightedge or ruler. The base is the unequal side of the triangle and the altitude is the perpendicular height from the base to the apex. It works by first copying the base segment, then constructing its perpendicular bisector. The apex is then marked up from the base. A Euclidean construction.
This page shows how to construct (draw) a 30 degree angle with compass and straightedge or ruler. It works by first creating a rhombus and then a diagonal of that rhombus. Using the properties of a rhombus it can be shown that the angle created has a measure of 30 degrees. See the proof below for more on this. A Euclidean construction.
This page shows to construct (draw) a 30 60 90 degree triangle with compass and straightedge or ruler. We are given a line segment to start, which will become the hypotenuse of a 30-60-90 right triangle. It works by combining two other constructions: A 30 degree angle, and a 60 degree angle. Because the interior angles of a triangle always add to 180 degrees, the third angle must be 90 degrees. A Euclidean construction. Includes a cool math animation showing the step-by-step procedure, and printable worksheet handouts. An OER resource.
This page shows how to construct (draw) a 60 degree angle with compass and straightedge or ruler. This construction works by creating an equilateral triangle. Recall that an equilateral triangle has all three interior angles 60 degrees. We use one of those angles to get the desired 60 degree result. See the proof below for more details. A Euclidean construction.
Given three points, it is always possible to draw a circle that passes through all three. This page shows how to construct (draw) a circle through 3 given points with compass and straightedge or ruler. It works by joining two pairs of points to create two chords. The perpendicular bisectors of a chords always passes through the center of the circle. By this method we find the center and can then draw the circle. A euclidean construction.
How to construct a line parallel to a given line that passes through a given point with compass and straightedge or ruler. It is called the 'translated triangle method' because it works by translating a triangle along one of its sides. The third vertex traces out a line parallel to that side. A Euclidean construction.
This page shows how to construct a line parallel to a given line that passes through a given point with compass and straightedge or ruler. It is called the 'angle copy method' because it works by using the fact that a transverse line drawn across two parallel lines creates pairs of equal corresponding angles. It uses this in reverse - by creating two equal corresponding angles, it can create the parallel lines. A Euclidean construction.
How to construct a regular hexagon given one side. The construction starts by finding the center of the hexagon, then drawing its circumcircle, which is the circle that passes through each vertex. The compass then steps around the circle marking off each side. A Euclidean construction.
How to construct (draw) a regular hexagon inscribed in a circle with a compass and straightedge or ruler. This is the largest hexagon that will fit in the circle, with each vertex touching the circle. Ina regular hexagon, the side length is equal to the distance from the center to a vertex, so we use this fact to set the compass to the proper side length, then step around the circle marking off the vertices. A Euclidean construction.
How to construct a square inscribed in a circle. The construction starts by drawing a diameter of the circle, then erecting a perpendicular as another diameter. The resulting four points define a square. A Euclidean construction.
Constructing the tangent to a circle at a given point on the circle with compass and straightedge or ruler. It works by using the fact that a tangent to a circle is perpendicular to the radius at the point of contact. It first creates a radius of the circle, then constructs the perpendicular bisector of the radius at the given point.
This page shows how to construct (draw) a triangle given two sides and the included angle with compass and straightedge or ruler. It works by first copying the angle, then copying the two segments on to the angle. A third line completes the triangle. A Euclidean construction
How to construct (draw) an isosceles triangle with compass and straightedge or ruler, given the length of the base and one side. First we copy the base segment. Then we use the fact that both sides of an isosceles triangle have the same length to mark the topmost point of the triangle that same distance from each end of the base. A Euclidean construction.
How to construct (draw) an equilateral triangle inscribed in a given circle with a compass and straightedge or ruler. This is the largest equilateral that will fit in the circle, with each vertex touching the circle. This is very similar to the construction of an inscribed hexagon, except we use every other vertex instead of all six. A Euclidean construction.
This page shows how to construct (draw) the circumcenter of a triangle with compass and straightedge or ruler. The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is also the center of the circumcircle, the circle that passes through all three vertices of the triangle. A Euclidean construction.
This page shows how to construct (draw) the midsegment of a triangle with compass and straightedge or ruler. The midsegment of a triangle is a line which links the midpoints of two sides of the triangle. A Euclidean construction.
This page shows how to construct (draw) the incenter of a triangle with compass and straightedge or ruler. The incenter of a triangle is the point where all three angle bisectors always intersect, and is the center of the triangle's incircle. A Euclidean construction.
Given an angle formed by two lines with a common vertex, this page shows how to construct another angle from it that has the same angle measure using a compass and straightedge or ruler. It works by creating two congruent triangles. A proof is shown below. A Euclidean construction
A list of the compass and straightedge construction worksheets available on this site
Printable step-by-step instructions for bisecting a line with compass and straightedge or ruler
Printable step-by-step instructions for drawing a 45 degree angle with compass and straightedge or ruler
Printable step-by-step instructions for drawing a 60 degree angle with compass and straightedge or ruler
Printable step-by-step instructions for drawing a 90 degree angle with compass and straightedge or ruler.
Printable step-by-step instructions for drawing an equilateral triangle with compass and straightedge or ruler
Printable step-by-step instructions for finding the altitude of an acute triangle with compass and straightedge or ruler
Printable step-by-step instructions for finding the altitude of an obtuse triangle with compass and straightedge or ruler
Printable step-by-step instructions for finding the circumcenter of a triangle with compass and straightedge or ruler
Printable step-by-step instructions for finding the midsegment of a triangle with compass and straightedge or ruler
Printable step-by-step instructions for bisecting an angle with compass and straightedge or ruler
Printable step-by-step instructions for constructing a hexagon given one side.
Printable step-by-step instructions for constructing a hexagon inscribed in a given circle
Printable step-by-step instructions for constructing a pentagon inscribed in a given circle
Printable step-by-step instructions for constructing a right triangle with given angle and hypotenuse with compass and straightedge or ruler. (HA)
Printable step-by-step instructions for constructing a right triangle with given angle and One given leg with compass and straightedge or ruler. (LA)
Printable step-by-step instructions for constructing a right triangle with given leg and hypotenuse with compass and straightedge or ruler.
Printable step-by-step instructions for constructing a right triangle with given leg lengths with compass and straightedge or ruler.
Printable step-by-step instructions for constructing a square inscribed in a circle with compass and straightedge or ruler.
Printable step-by-step instructions for constructing a square of a given side length with compass and straightedge or ruler
Printable step-by-step instructions for constructing an equilateral triangle inscribed in a given circle
Printable step-by-step instructions for constructing an isosceles triangle given a side and apex angle with compass and straightedge or ruler
Printable step-by-step instructions for constructing one of the two internal tangents to two given circles.
Printable step-by-step instructions for constructing tangents to two given circles.
Printable step-by-step instructions for constructing the complementary angle to a given angle with compass and straightedge or ruler
Printable step-by-step instructions for constructing the supplementary angle to a given angle with compass and straightedge or ruler
Printable step-by-step instructions for copying a line segment with compass and straightedge or ruler
Printable step-by-step instructions for copying a triangle with compass and straightedge or ruler
Printable step-by-step instructions for copying an angle with compass and straightedge or ruler
Printable step-by-step instructions for dividing a line segment into equal parts with compass and straightedge or ruler
Printable step-by-step instructions for drawing a 30 degree angle with compass and straightedge or ruler.
Printable step-by-step instructions for drawing a 30-60-90 triangle with compass and straightedge or ruler.
Printable step-by-step instructions for drawing a perpendicular at a point on a line with compass and straightedge or ruler.
Printable step-by-step instructions for drawing a perpendicular at the end of a ray with compass and straightedge or rule
Printable step-by-step instructions for drawing a perpendicular to a line through a point with compass and straightedge or ruler.
Printable instructions for drawing a tangent at a point on a circle with compass and straightedge or ruler.
Printable instructions for drawing a triangle given two sides and included angle with compass and straightedge or ruler.
Printable step-by-step instructions for drawing an angle with a protractor
Printable step-by-step instructions for drawing an ellipse with string and pins
Printable step-by-step instructions for drawing parallel through a point (rhombus method)
Printable step-by-step instructions for drawing parallel through a point with compass and straightedge or ruler
Printable step-by-step instructions for drawing parallel through a point with compass and straightedge or ruler
Printable instructions for drawing tangents to a circle from a point with compass and straightedge or ruler.
Printable step-by-step instructions for drawing the circumcircle of a triangle with compass and straightedge or ruler
Printable step-by-step instructions for drawing the incircle of a triangle with compass and straightedge or ruler
Printable step-by-step instructions for drawing the medians of a triangle with compass and straightedge or ruler
Printable step-by-step instructions for finding the center of a circle with any right angled object
Printable step-by-step instructions for finding the center of a circle with compass and straightedge or rule
Printable step-by-step instructions for finding the centroid of a triangle with compass and straightedge or ruler
Printable step-by-step instructions for finding the foci of an ellipse with compass and straightedge or ruler
Printable step-by-step instructions for finding the incenter of a triangle with compass and straightedge or ruler
Printable step-by-step instructions for finding the orthocenter of a triangle with compass and straightedge or ruler
Printable step-by-step instructions for measuring an angle using a protractor
Printable step-by-step instructions for adding two angles with compass and straightedge or ruler.
Printable step-by-step instructions for constructing a segment whose length is the difference of two given segments with compass and straightedge or ruler.
Printable step-by-step instructions for constructing a segment whose length is the sum of given segments with compass and straightedge or ruler.
Printable step-by-step instructions for constructing a triangle given two angles and non-included side (AAS) with compass and straightedge or ruler.
Printable step-by-step instructions for constructing the difference of two angles with compass and straightedge or ruler.
Printable step-by-step instructions for drawing a triangle given two angles and included side with compass and straightedge or ruler.
Printable step-by-step instructions for drawing an isosceles triangle with compass and straightedge or ruler
Printable step-by-step instructions for drawing an isosceles triangle with compass and straightedge or ruler - II
Printable strp-by-step instructions for drawing a triangle given three sides with compass and straightedge or ruler.
How to add the lengths of given line segments with compass and straightedge or ruler. A Euclidean construction.
How to construct the supplementary angle to a given angle with compass and straightedge or ruler. A Euclidean construction.