What are the Steps for Construction of a Square?
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
1. What is a square and what are its key properties for construction?
A square is a special type of quadrilateral defined by two primary properties essential for its construction. It has four equal sides and four interior right angles (90°). Additionally, its diagonals are equal in length and bisect each other at a right angle. These properties are the fundamental rules used when constructing a square with geometric tools.
2. What are the basic geometric tools required for the construction of a square?
To accurately construct a square according to geometric principles, you will need the following tools:
A ruler or straightedge to draw straight line segments.
A compass to draw arcs and circles, which helps in marking off equal lengths and constructing angles.
A protractor to measure and verify the 90° angles, although a right angle can also be constructed using just a compass and a ruler.
3. How do you construct a square when the length of one side is given?
To construct a square with a given side length, say 's', follow these steps:
1. Draw a line segment AB of length 's'.
2. At point A, construct a perpendicular line (a 90° angle) using a compass or protractor.
3. Set your compass to the length of AB ('s') and cut an arc on the perpendicular line from point A. Mark this point as D.
4. With the same compass width 's', place the compass point at B and draw an arc. Then, place the compass point at D and draw another arc to intersect the previous one. Mark this intersection as C.
5. Join points D to C and B to C. The resulting figure ABCD is the required square.
4. What is the method to construct a square if only the length of its diagonal is provided?
If you are given the length of the diagonal, say 'd', the construction relies on the property that a square's diagonals bisect each other at 90°.
1. Draw the diagonal AC of length 'd'.
2. Construct the perpendicular bisector of the line segment AC. Let the midpoint be O.
3. From the midpoint O, use a compass to mark arcs of length d/2 (half the diagonal) on the perpendicular bisector on both sides. Label these points B and D.
4. Join the points A, B, C, and D in order. The quadrilateral ABCD is the required square.
5. What is the crucial difference between 'drawing' a square and 'constructing' a square in geometry?
The difference lies in precision and method. 'Drawing' a square is a visual representation or sketch, often done freehand or with a ruler, where the primary goal is to create a shape that looks like a square. 'Constructing' a square is a formal geometric process that uses only a compass and a straightedge to create a mathematically perfect figure based on its defined properties, such as equal sides and perfect 90-degree angles.
6. How is the construction of a square different from the construction of a rhombus?
While both a square and a rhombus have four equal sides, the key difference in their construction lies in the angles. The construction of a square requires creating four 90-degree angles at the vertices. In contrast, a rhombus does not have this restriction; its angles can vary. The construction of a rhombus often relies on its diagonals, which, unlike a square's, are not equal but do bisect each other perpendicularly.
7. How does the Pythagorean theorem help in verifying the construction of a square?
The Pythagorean theorem provides a powerful way to verify the accuracy of a constructed square. A diagonal divides the square into two right-angled triangles. If the side length of the square is 's' and the diagonal is 'd', then according to the theorem, s² + s² = d². This simplifies to 2s² = d². After constructing a square, you can measure a side and the diagonal. If these measurements satisfy the theorem, it confirms that the angles are indeed 90°, and the construction is accurate.
8. What is the real-world importance of knowing how to construct a perfect square?
Knowing how to construct a perfect square is fundamental in many practical fields. In architecture and engineering, it is crucial for creating stable foundations, floor plans, and ensuring that walls meet at precise right angles. In design and art, constructing squares is essential for creating grids, patterns, and symmetrical layouts. This basic geometric skill forms the foundation for more complex designs and precise manufacturing processes.
