How to Construct a Square Using Compass and Ruler with Steps and Properties
What is Square?
A square is a quadrilateral and a polygon with 4 vertices and 4 sides enclosing four angles, to note. 360 degrees is the sum of the interior angles. In general, a quadrilateral has sides of varying lengths and angles of different measurements. However, with some of their sides and angles being equal, triangles, rectangles, etc. are special kinds of quadrilaterals.
A square is a regular quadrilateral in geometry, which implies it has :
Four equal sides.
Four equal angles (90-degree angles, or 100-gradian angles or right angles).
It can also be represented as a rectangle in which the length of two adjacent sides is equal.
A square with ABCD vertices will be denoted as □ABCD.
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Constructing a Square With a Compass
Square in construction is quite easy to draw as it requires only a few steps which are very easy to implement. While we focus on the square for construction, we have to take care of the following properties:
All the sides must be equal.
All the angles made by the sides of the square should be 90 degrees.
Therefore, the following are the steps for the construction of square:
Draw a reference line AB of 6cm using a ruler.
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Draw the 90 degree angle at A with the aid of the compass.
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Draw the 90 degree angle at B with the aid of the compass.
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Using an open compass 6 cm wide. Draw an arc with A as the centre; that cuts arms at a 90 degree angle. And mark the intersection point as C.
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Using the ruler find length measure of AC. The AC length should be 6 cm.
Using the compass again and take 6 cm width of the compass. With D as the middle, draw an arc that cuts arms at an angle of 90 degrees. And mark the junction point as D.
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Using the ruler and the BD length measurement. The BD length should be 6 cm and we'll get:
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Join CD.
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Use the CD ruler and calculate its length. CD should be 6 cm in length and we get the corresponding square ABCD.
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Constructing a Square in a Circle
The steps to inscribe a square in a circle are:
Draw a circle using your compass and mark the middle O.
Draw a circular diameter using your ruler, marking the endpoints A and B.
Build the perpendicular diameter bisector, AB.
Label the points where the circle is intersected by the bisector as C and D.
For the square to form, link points A to B to C to D.
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Solved Examples
1. Construct a square with a side of length 7cm.
Solution:
Draw a reference line AB of 7 cm using a ruler.
Draw the 90 degree angle at A with the aid of the compass.
Draw the 90 degree angle at B with the aid of the compass.
Using an open compass 6 cm wide. Draw an arc with A as the centre; that cuts arms at a 90 degree angle. And mark the intersection point as C.
Using the ruler find length measure of AC. The AC length should be 6 cm.
Using the compass again and take 7 cm width of the compass. With D as the middle, draw an arc that cuts arms at an angle of 90 degrees. And mark the junction point as D.
Using the ruler and the BD length measurement. The BD length should be 6 cm and
Join CD.
Use the CD ruler and calculate its length. The CD should be 7 cm in length and we get the corresponding square ABCD.
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2. Construct a circle with radius 5 cm and draw a square inside it.
Solution:
Draw a circle with radius 5 cm using your compass and mark the middle O.
Draw a circular diameter using your ruler, marking the endpoints A and B.
Build the perpendicular diameter bisector, AB.
Label the points where the circle is intersected by the bisector as C and D.
For the square to form, link points A to B to C to D.
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Did You Know?
A square is a regular quadrilateral since it has 4 sides of equal length.
Often, a square is a rectangle with equal sides and a right-angled rhombus.
A square's area equals the length of one side to the power of two (length squared).
FAQs on Construction of Square in Geometry
1. What is the construction of a square in geometry?
The construction of a square is the geometric process of drawing a quadrilateral with four equal sides and four right angles using a ruler and compass. A square has:
- All sides equal
- Each interior angle equal to 90°
- Diagonals that are equal and bisect each other at right angles
2. How do you construct a square when one side is given?
To construct a square when one side is given, draw perpendicular lines at both ends of the segment and mark equal lengths. Steps:
- Draw a line segment AB of given length.
- At point A, construct a perpendicular to AB.
- Mark point D on the perpendicular such that AD = AB.
- At point B, construct a perpendicular to AB.
- Mark point C such that BC = AB.
- Join C and D to complete square ABCD.
3. What are the properties of a square?
A square is a quadrilateral with equal sides and right angles. Its key properties are:
- All four sides are equal.
- Each interior angle is 90°.
- Diagonals are equal.
- Diagonals bisect each other at right angles.
- Diagonals bisect the vertex angles.
4. What is the formula for the area of a square?
The area of a square is calculated using the formula Area = side × side = s². For example:
- If side s = 5 cm
- Area = 5 × 5 = 25 cm²
5. What is the formula for the perimeter of a square?
The perimeter of a square is given by P = 4s, where s is the side length. For example:
- If s = 7 cm
- Perimeter = 4 × 7 = 28 cm
6. How do you construct a square using its diagonal?
To construct a square using its diagonal, use the property that diagonals bisect each other at right angles. Steps:
- Draw the given diagonal AC.
- Find its midpoint O.
- Draw a perpendicular line through O.
- With O as center and radius OA, mark points B and D on the perpendicular.
- Join A, B, C, and D to form square ABCD.
7. Why are the diagonals of a square perpendicular?
The diagonals of a square are perpendicular because a square is both a rectangle and a rhombus. Since it has:
- Equal sides (property of a rhombus)
- Right angles (property of a rectangle)
8. What is the length of the diagonal of a square?
The diagonal of a square is given by the formula d = s√2, where s is the side length. This comes from the Pythagoras theorem:
- d² = s² + s²
- d² = 2s²
- d = s√2
9. What instruments are required for the construction of a square?
The construction of a square requires basic geometric instruments. These include:
- Ruler (to draw line segments)
- Compass (to mark equal lengths)
- Protractor (optional, for checking 90° angles)
- Pencil
10. What is the difference between a square and a rectangle?
The main difference between a square and a rectangle is that all sides of a square are equal, while only opposite sides of a rectangle are equal. Key differences:
- Square: All sides equal; all angles 90°.
- Rectangle: Opposite sides equal; all angles 90°.
- Diagonals of both are equal, but only in a square they are also perpendicular.
