How to Construct a Triangle Inscribed in a Circle Step by Step with Theorem and Examples
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 Constructing a Triangle Inscribed in a Circle in Geometry
1. What does it mean to construct a triangle inscribed in a circle?
A triangle inscribed in a circle is a triangle whose three vertices lie on the circumference of the circle. In this construction, each vertex of the triangle touches the circle, and the circle is called the circumcircle of the triangle. Every triangle can be inscribed in a unique circle, whose center is the point where the perpendicular bisectors of the sides meet.
2. How do you construct a triangle inscribed in a given circle?
To construct a triangle inscribed in a circle, choose any three distinct points on the circumference and join them with straight lines.
- Step 1: Draw the given circle with center O.
- Step 2: Mark three distinct points A, B, and C on the circle.
- Step 3: Join A to B, B to C, and C to A using a ruler.
3. What is the formula for the area of a triangle inscribed in a circle?
The area of a triangle inscribed in a circle of radius R is given by Area = (abc) / (4R), where a, b, and c are the side lengths of the triangle. This formula relates the triangle’s sides to the circumradius (R). For example, if a = 5, b = 6, c = 7 and R = 4, then Area = (5 × 6 × 7) / (4 × 4) = 210 / 16 = 13.125 square units.
4. How do you construct an equilateral triangle inscribed in a circle?
An equilateral triangle inscribed in a circle can be constructed by stepping the radius around the circle six times and joining alternate points.
- Step 1: Draw a circle with center O and radius r.
- Step 2: With the same radius, mark six equal points on the circumference.
- Step 3: Join every second point (for example, A, C, E).
5. What is the relationship between an inscribed angle and the central angle?
An inscribed angle is half the measure of the central angle subtending the same arc. If a central angle is 100°, then the inscribed angle intercepting the same arc is 50°. This is known as the Inscribed Angle Theorem, which is important when solving problems involving triangles inscribed in circles.
6. How do you find the circumcenter of a triangle inscribed in a circle?
The circumcenter of a triangle is found by constructing the perpendicular bisectors of its sides and locating their intersection point.
- Step 1: Draw the perpendicular bisector of side AB.
- Step 2: Draw the perpendicular bisector of side BC.
- Step 3: Their intersection point is the circumcenter O.
7. Can any triangle be inscribed in a circle?
Yes, every triangle can be inscribed in a circle, and this circle is called its circumcircle. The perpendicular bisectors of the triangle’s sides always intersect at one point, ensuring a unique circumcenter and circumradius for any triangle—acute, right, or obtuse.
8. How do you construct a right triangle inscribed in a circle?
A right triangle inscribed in a circle is formed when one side of the triangle is the diameter of the circle.
- Step 1: Draw a circle and mark diameter AB.
- Step 2: Choose any other point C on the circumference.
- Step 3: Join AC and BC.
9. What is Thales’ Theorem in relation to an inscribed triangle?
Thales’ Theorem states that the angle subtended by a diameter at the circumference of a circle is 90°. This means if a triangle is inscribed in a circle and one side is the diameter, the triangle formed is always a right triangle.
10. What are common mistakes when constructing a triangle inscribed in a circle?
Common mistakes when constructing a triangle inscribed in a circle include placing vertices inside the circle instead of on the circumference and inaccurate angle or bisector constructions.
- Not marking points exactly on the circle boundary.
- Drawing incorrect perpendicular bisectors when finding the circumcenter.
- Confusing an inscribed triangle with a circumscribed triangle.
