Easy Construction of an Inscribed Triangle: Explained with Diagrams
In this chapter, we will help you learn how to construct an equilateral triangle inscribed in a circle with the help of a compass and a ruler or straightedge. You will also get familiar with the construction of the largest equilateral triangle that will fit in the circle, having each vertex touching the circle. This is quite the same as the construction of an inscribed hexagon, besides that we use every other vertex instead of all six.
How to Draw Equilateral Triangle in a Circle with a Compass and Ruler?
Here, we are not acquainted with the information about the circle. That is we don’t know the diameter of the circle (and thus the radius) of the circle. We are also not aware of where the centre is. However, we have a compass and a ruler or straightedge in hand. Now, follow the suggested steps below and you will get your triangle inscribed in a circle.
Step I: Use the ruler to construct two chords on the circle.
Step II: Draw a perpendicular bisector to the chords using the compass. This perpendicular bisector must meet at the centre of the circle.
Step III: Make a line from the centre to one of the farthest points of one of the chords.
Step IV: Take the help of the compass in order to draw two lines which form a 30 degree with the radius drawn by step 3 and on the opposite side.
Step V: Stretch out these two lines to make two chords on the circle. These two chords therefore make an angle of 60 degrees to each other.
Step VI: Connect the other extreme of the two chords made by step 4. This together with the two chords makes the needed equilateral triangle.
How to Construct Triangle Inscribe Given One Length
Using 3 methods, we will be performing constructions of an equilateral triangle given the length of one side, and the remaining two will be to draw an equilateral triangle inscribed in a circle.
Method 1:
Given: one side length measurement of the triangle.
Construct: an equilateral triangle.
Steps to Follow:
Set your compass point on A and calculate the distance to point B. Swing an arc of this size below or above the line segment.
Without disturbing or altering the span on the compass, place the compass point on B and swing the same arc, bisecting with the 1st arc.
Label the point of bisection as the 3rd vertex of the equilateral triangle.
Proof of Construction
Circle A is congruent to circle B, seeing that they were each created using the same length of radius, AB. Because AB and AC are length measurements of radii of circle A, they are equivalent to one another. In the same manner, AB and BC are radii of circle B, and are equivalent to one another. Thus, AB = AC = BC by method of substitution (or transitive property). Since congruent line segments consist of equal lengths, equal segments and ΔABC are thus equilateral (containing 3 congruent sides).
Method 2:
Modification of the construction of a regular hexagon inscribed in a circle.
Given: A piece of paper
Construct: An equilateral triangle inscribed in a circle.
Steps to Follow:
Place your drawing compass to construct a circle; make sure to keep the compass span.
Put a dot, marked as A anywhere on the circumference of the circle to play a part of an initial point.
Without altering the span on the compass, establish the compass point on A and swing a small arc going through the circumference of the circle.
Without altering the span on the compass, shift the compass point to the bisection of the previous arc and the circumference and draw another small arc on the circumference of the circle.
Repeat this process of "stepping" around the circle till the time you return to point A.
Beginning at A, join every other arc on the circle to create the equilateral triangle.
FAQs on How to Construct a Triangle Inscribed in a Circle
1. What are the basic steps to construct an equilateral triangle inscribed in a circle?
The standard method to construct an equilateral triangle inscribed in a circle using a compass and straightedge is as follows:
- Step 1: Draw the circle with centre O and a specific radius, r.
- Step 2: Mark any point on the circumference, let's call it A.
- Step 3: Keeping the compass width set to the radius (r), place the compass point on A and draw an arc that cuts the circle at a new point, B.
- Step 4: Move the compass point to B and repeat the process to get point C. Continue this around the circle to get six equally spaced points.
- Step 5: Using a straightedge, connect every alternate point. For example, join points A, C, and E.
2. What are the key geometric properties of a triangle inscribed in a circle?
When a triangle is inscribed in a circle, it has several important properties:
- All three vertices of the triangle lie on the circumference of the circle.
- The circle is called the circumcircle of the triangle.
- The centre of the circle is known as the circumcentre of the triangle.
- The circumcentre is the point where the perpendicular bisectors of all three sides of the triangle intersect.
- The radius of the circle is called the circumradius.
3. How do you find the side length of an equilateral triangle inscribed in a circle of a given radius?
There is a direct formula that connects the side length of an inscribed equilateral triangle to the radius of its circumcircle. The formula is: Side (a) = Radius (R) × √3. For example, to find the side length of an equilateral triangle inscribed in a circle with a radius of 10 cm, you would calculate: a = 10 × √3 ≈ 10 × 1.732 = 17.32 cm. This relationship is fundamental for solving problems involving such geometric figures.
4. What is the difference between constructing a triangle inscribed in a circle and constructing an incircle for a triangle?
These are two opposite geometric constructions.
- A triangle inscribed in a circle (or circumscribed circle) has all its vertices on the circle's circumference. To find the circle's center (the circumcentre), you construct the perpendicular bisectors of the triangle's sides.
- An incircle of a triangle is a circle drawn inside a triangle that touches all three sides. To find its center (the incentre), you construct the angle bisectors of the triangle's vertices.
5. What type of triangle is formed if one of its sides is the diameter of the circle it is inscribed in?
If one side of a triangle inscribed in a circle is also the diameter of that circle, the triangle is always a right-angled triangle. This is a direct application of Thales's theorem (or the angle in a semicircle theorem), which states that the angle subtended by a diameter at any point on the circumference is always 90 degrees. The diameter becomes the hypotenuse of the right-angled triangle.
6. Why are the circumcentre and centroid the same point in an inscribed equilateral triangle?
In an equilateral triangle, all sides are equal, which gives it unique symmetrical properties. The lines drawn as perpendicular bisectors (which meet at the circumcentre) are the exact same lines as the medians (which meet at the centroid) and the angle bisectors. Because these special lines are identical in an equilateral triangle, their points of concurrency—the circumcentre, centroid, and incentre—all coincide at the same single point.
7. How can you prove that the triangle constructed using the radius-arc method is equilateral?
You can prove the construction is equilateral using basic geometry. When you connect the vertices of the inscribed triangle (say, A, C, E) to the center of the circle (O), you form three isosceles triangles (△AOC, △COE, △EOA). Since the construction method creates six equal arcs on the circumference, the central angle for each of these three triangles is 360°/3 = 120°. In each isosceles triangle, the two base angles are (180° - 120°)/2 = 30°. Each angle of the main triangle △ACE is the sum of two such base angles (e.g., ∠CAE = ∠CAO + ∠EAO = 30° + 30° = 60°). Since all three angles are 60 degrees, the triangle is equilateral.
8. Is it possible to accurately construct an inscribed equilateral triangle without a compass?
No, for a geometrically accurate construction as per the CBSE/NCERT curriculum, a compass is essential. A compass performs two critical functions: drawing a perfect circle where all points are equidistant from the center, and transferring the radius length accurately to create equal arcs on the circumference. Without a compass, you could only sketch an approximation, which would not be a formal geometric construction and would lack the precision required.
