CONSTRUCTION
MEANING OF CONSTRUCTION
Definition:
Construction in Mathematics means drawing geometrical figures accurately using only a pair of compasses, a ruler (straight edge), and a pencil — without using a protractor.
It helps us to create accurate angles and shapes by following specific steps.
The main instruments used are:
- Ruler – for drawing straight lines.
- Compass – for drawing arcs and circles.
- Pencil – for marking points and drawing lines.
- Eraser – for cleaning unnecessary marks.
1. CONSTRUCTION OF AN ANGLE OF 90°
Before we can construct 45° or 30°, we must first know how to construct a 90° angle because both will depend on it.
STEPS TO CONSTRUCT A 90° ANGLE
- Draw a straight line and name it AB.
- With A as the centre, draw an arc of any radius to cut the line at a point C.
- With C as centre, draw another arc of the same radius to cut the first arc at a point D.
- Draw a straight line from A through D.
The line AD makes an angle of 90° with AB.
∠BAD = 90°
Steps (Text Diagram Explanation):
Step 1: Draw a straight line AB.
A________________________B
Step 2: With A as the centre, draw an arc to cut AB at C.
A____)____C______________B
Step 3: With C as centre, draw another arc to cut the first arc at D.
D
/ )
/ )
A____)__/___)___________B
C
Step 4: Draw a line from A through D.
D
/
/
A______/
\
\
B
Result: ∠BAD = 90°
2. CONSTRUCTION OF AN ANGLE OF 45°
Definition:
A 45° angle is half of a 90° angle.
So, to construct a 45° angle, we first construct a 90° angle, then bisect it.
STEPS TO CONSTRUCT A 45° ANGLE
- Draw a straight line and name it AB.
- With A as centre, draw an arc of any radius to cut the line at C.
- With C as centre, draw another arc of the same radius to cut the first arc at D.
- Join A to D. This gives you ∠DAB = 90°.
- To bisect the 90° angle, with C and D as centres, draw two arcs of equal radius to cut each other at point E.
- Draw a line from A through E.
∠EAB = 45°
EXPLANATION
The 45° line divides the 90° angle exactly into two equal halves.
Each half measures 45°.
Step 1: Draw a line AB and construct a 90° angle (∠BAD).
D
/
/
A______/
\
\
B
Step 2: With C and D as centres, draw two arcs that meet at E.
D
/\
/ \
A______/____\____B
E
Step 3: Draw line AE through E to bisect the 90°.
D
/|
/ |
/ |
A_____/___|____B
E
Result: ∠EAB = 45°
3. CONSTRUCTION OF AN ANGLE OF 60°
We must also know how to construct a 60° angle because it helps in constructing a 30° angle.
STEPS TO CONSTRUCT A 60° ANGLE
- Draw a straight line and name it AB.
- With A as centre, draw an arc of any radius cutting AB at C.
- Without changing the compass width, place the compass on C and draw another arc to cut the first arc at D.
- Draw a straight line from A through D.
∠DAB = 60°
Step 1: Draw line AB.
A________________________B
Step 2: With A as centre, draw an arc cutting AB at C.
A____)____C______________B
Step 3: Without changing compass width, draw another arc from C cutting the first arc at D.
D
/ )
/ )
A____)__/___)___________B
C
Step 4: Draw line AD.
D
/
/
A______/
\
\
B
Result: ∠DAB = 60°
4. CONSTRUCTION OF AN ANGLE OF 30°
Definition:
A 30° angle is half of a 60° angle.
So, we can construct a 60° angle first, then bisect it.
STEPS TO CONSTRUCT A 30° ANGLE
- Draw a straight line and name it AB.
- With A as centre, draw an arc to cut AB at C.
- With the same compass width, draw another arc with C as centre to cut the first arc at D.
- Draw a straight line from A through D. ∠DAB = 60°.
- To bisect the 60° angle: With C and D as centres, draw two arcs of equal radius to cut each other at E.
- Draw a line from A through E.
∠EAB = 30°
Step 1: Draw line AB and construct 60° (∠DAB).
D
/
/
A______/
\
\
B
Step 2: With C and D as centres, draw arcs that meet at E.
D
/\
/ \
A______/____\____B
E
Step 3: Draw line AE through E.
D
/|
/ |
/ |
A_____/___|____B
E
Result: ∠EAB = 30°
SUMMARY TABLE
| Angle | Construction Method | Key Idea |
|---|
| 90° | Use two equal arcs to form perpendicular | Basic right angle |
| 45° | Bisect 90° angle | Half of right angle |
| 60° | Use one compass width (equilateral triangle principle) | One-third of 180° |
| 30° | Bisect 60° angle | Half of 60° |
IMPORTANCE OF ANGLE CONSTRUCTION
It helps us to:
- Draw accurate angles in geometry.
- Make correct designs in technical drawing.
- Solve practical problems in engineering, carpentry, and architecture.
- Draw correct geometric shapes such as triangles, hexagons, and squares.
COPYING GIVEN ANGLES
MEANING
Definition:
To copy a given angle means to construct another angle that is exactly equal in size to the given one, without using a protractor.
We do this by using only a pair of compasses, a ruler (straight edge), and a pencil.
This construction helps us to make an identical angle in a different position.
INSTRUMENTS NEEDED
- Ruler (for drawing straight lines)
- Compass (for drawing arcs)
- Pencil (for marking)
- Eraser (for cleaning unwanted marks)
STEPS TO COPY A GIVEN ANGLE
Let the given angle be ∠ABC, and we want to copy it at another point P.
- Step 1: Draw a straight line PQ (this will be one arm of the new angle).
- Step 2: Place the compass point on B, the vertex of the given angle. Draw an arc that cuts both arms of the angle ∠ABC at points D and E.
- Step 3: Without changing the compass width, place the compass point on P (the vertex of the new angle). Draw a similar arc cutting PQ at point R.
- Step 4: Now, measure the distance between the two points D and E (where the arc cuts the arms of the original angle) using your compass.
- Step 5: With the compass still open at that same width, place the compass point on R and draw another small arc that crosses the first arc drawn from P. Let them meet at point S.
- Step 6: Draw a straight line from P through S.
The angle ∠QPS is equal to the given angle ∠ABC.
Step 1 — Given angle (∠ABC)
D
/
/
B __/______ C
\
\
A
Step 2 — Draw base line PQ (new vertex at P)
P________________________Q
Step 3 — Draw arc from B that cuts both arms at D and E
D
/)
/ )
B __/__)____ C
\
\
A
Step 4 — Draw the same arc from P cutting PQ at R
P___)_____________Q
Step 5 — Mark distance DE on the new arc from R to get point S
P___)___S_________Q
Step 6 — Join P to S
S
/
/
P__/______________Q
Result: ∠QPS is a copy of ∠ABC
EXPLANATION
The arcs ensure that both angles have the same opening (that is, they measure the same number of degrees).
So the new angle ∠QPS is a perfect copy of the original ∠ABC.
PRACTICAL EXAMPLE
Suppose you are given ∠ABC that measures 65°.
If you follow the construction steps correctly (without a protractor), the copied angle ∠QPS will also measure 65°.
This shows that compass and ruler constructions give us accurate and equal angles.
IMPORTANCE OF COPYING ANGLES
It helps us to:
- Transfer angles from one place to another accurately.
- Construct shapes like triangles and polygons with exact angles.
- Draw technical or architectural diagrams with correct angular measurements.
- Practice precision and accuracy in geometric work.
SUMMARY OF CONSTRUCTION STEPS
| Step | Action | Purpose |
|---|
| 1 | Draw a base line (PQ) | New line for the copied angle |
| 2 | Draw an arc from the vertex of the given angle | To mark equal distances on both arms |
| 3 | Draw a similar arc from the new point | To prepare the same shape for the new angle |
| 4 | Measure the distance between arc intersections | To get the exact opening of the angle |
| 5 | Transfer this distance to the new arc | To make both angles equal |
| 6 | Join the new vertex to the new intersection | To complete the copied angle |
CONSTRUCTION OF SIMPLE PLANE SHAPES
MEANING
Definition:
The construction of simple plane shapes means drawing accurate geometric figures (like triangles, squares, rectangles, etc.) using only a compass, ruler, and pencil, without using a protractor or set square.
It helps us to draw shapes correctly and understand their geometric properties.
INSTRUMENTS USED
- Ruler – for drawing straight lines and measuring lengths.
- Compass – for drawing circles and arcs.
- Pencil – for marking and drawing.
- Eraser – for cleaning unwanted marks.
SIMPLE PLANE SHAPES TO CONSTRUCT
- Triangle (Equilateral, Isosceles, and Right-angled)
- Square
- Rectangle
Let us learn how to construct each one properly, step by step.
1. CONSTRUCTION OF AN EQUILATERAL TRIANGLE
Definition:
An equilateral triangle is a triangle that has all three sides equal and all angles equal to 60°.
Equilateral triangle (AB = BC = CA)
C
/ \
/ \
/_____\
A B
STEPS TO CONSTRUCT AN EQUILATERAL TRIANGLE
- Draw a straight line AB of any given length (for example, 6 cm).
- With A as centre, and radius equal to the length of AB, draw an arc above the line.
- With B as centre, and the same radius, draw another arc to cut the first arc at C.
- Join A to C and B to C.
The triangle ABC is an equilateral triangle.
PROPERTIES
- All sides are equal.
- All angles are 60°.
2. CONSTRUCTION OF AN ISOSCELES TRIANGLE
Definition:
An isosceles triangle is a triangle that has two sides equal and two angles equal.
Isosceles triangle (AC = BC)
C
/ \
/ \
/_____\
A B
STEPS TO CONSTRUCT AN ISOSCELES TRIANGLE
- Draw a straight line AB of the required base length (for example, 7 cm).
- With A as centre, draw an arc with radius 6 cm.
- With B as centre, draw another arc with the same radius to cut the first arc at C.
- Join A to C and B to C.
The triangle ABC is an isosceles triangle.
PROPERTIES
- AC = BC (the two sides are equal).
- The base angles at A and B are equal.
3. CONSTRUCTION OF A RIGHT-ANGLED TRIANGLE
Definition:
A right-angled triangle is a triangle that has one angle equal to 90°.
Right-angled triangle (right angle at A)
C
|\
| \
|__\
A B
STEPS TO CONSTRUCT A RIGHT-ANGLED TRIANGLE
- Draw a straight line AB of length 6 cm (the base).
- At point A, construct a 90° angle using your compass (as learnt earlier).
- On the 90° line, mark point C such that AC = 4 cm (the height).
- Join B to C.
The triangle ABC is a right-angled triangle at A.
PROPERTIES
- One angle is 90°.
- The side opposite the right angle is the hypotenuse (longest side).
4. CONSTRUCTION OF A SQUARE
Definition:
A square is a four-sided figure (quadrilateral) with all sides equal and each angle equal to 90°.
Square (all sides equal, all angles 90°)
A________B
| |
| |
|________|
D C
STEPS TO CONSTRUCT A SQUARE
- Draw a straight line AB of 5 cm.
- At A, construct a 90° angle.
- On the 90° line, mark AD = 5 cm.
- At B, also construct a 90° angle on the same side and mark BC = 5 cm.
- Join C and D.
The figure ABCD is a square.
PROPERTIES
- All sides are equal (AB = BC = CD = DA).
- All angles are 90°.
5. CONSTRUCTION OF A RECTANGLE
Definition:
A rectangle is a quadrilateral with opposite sides equal and all angles equal to 90°.
Rectangle (opposite sides equal, all angles 90°)
A____________B
| |
| |
|____________|
D C
STEPS TO CONSTRUCT A RECTANGLE
- Draw a straight line AB = 8 cm (the length).
- At A, construct a 90° angle and mark AD = 5 cm (the breadth).
- At B, construct another 90° angle and mark BC = 5 cm.
- Join C and D.
The figure ABCD is a rectangle.
PROPERTIES
- Opposite sides are equal (AB = CD, and AD = BC).
- All angles are 90°.
6. CONSTRUCTION OF A REGULAR HEXAGON (OPTIONAL EXTENSION)
Definition:
A regular hexagon is a six-sided polygon with all sides and angles equal.
Regular hexagon (constructed from equal arcs on a circle)
__/‾\__
/ \
| |
\__ __/
\/
STEPS TO CONSTRUCT A REGULAR HEXAGON
- Draw a circle with centre O and any convenient radius.
- With the same radius, place the compass point anywhere on the circle and mark off equal arcs along the circumference to get six points.
- Join the six points together in order.
The figure is a regular hexagon.
SUMMARY TABLE
| Shape | Key Properties | How to Construct |
|---|
| Equilateral triangle | All sides and angles equal (60°) | Use equal arcs from both ends of base |
| Isosceles triangle | Two sides equal | Use same radius from both ends of base |
| Right-angled triangle | One angle 90° | Use perpendicular construction |
| Square | All sides equal, all angles 90° | Construct perpendicular lines and equal lengths |
| Rectangle | Opposite sides equal, all angles 90° | Construct 90° at each corner |
| Regular hexagon | Six equal sides | Use circle and equal arcs along circumference |
IMPORTANCE OF PLANE SHAPE CONSTRUCTION
- It helps us to Draw accurate shapes in geometry and technical drawing.
- It helps us to Develop good drawing and measuring skills.
- It helps us to Understand the properties of different plane figures.
- It helps us to Apply geometry in real-life work such as building, carpentry, and engineering.
JSSCE-STYLE CONSTRUCTION PRACTICE — QUESTIONS WITH SOLUTIONS
Instructions: Use only a ruler, a pair of compasses, and a pencil for all constructions unless the question states otherwise.
-
Question 1 — Construct a 90° angle at point A on a line AB.
Solution (steps):
- Draw line AB and mark point A.
- With A as centre, draw an arc that cuts AB at C.
- With C as centre and same radius, draw an arc to cut the first arc at D.
- Join A to D. Line AD is perpendicular to AB, so ∠BAD = 90°.
A________________B
(arc from A) C
\
) (arc from C)
D
Then join A to D to form the right angle.
-
Question 2 — Construct a 45° angle at A on AB.
Solution (steps):
- Construct 90° at A by the method in Question 1.
- With centres at the two arc intersection points used to form the 90°, draw two small arcs that intersect at E (bisector construction).
- Join A to E. Line AE bisects the 90°, so ∠ = 45°.
Start with 90°:
D
/
/
A/____B
Bisect with arcs meeting at E, then join A to E.
-
Question 3 — Construct a 60° angle at A on AB.
Solution (steps):
- Draw line AB and with A as centre draw an arc meeting AB at C.
- Without changing the compass width, draw an arc from C to meet the first arc at D.
- Join A to D. ∠DAB = 60°.
A_____)____C_____B
\
) arc from C meets first arc at D
D
Join A to D to make 60°.
-
Question 4 — Construct a 30° angle at A on AB.
Solution (steps):
- Construct 60° at A by Question 3.
- With centres at the arc intersection points used for 60°, draw arcs to bisect the 60° angle; the bisector gives 30°.
Construct 60° then bisect:
D
/ \
A \
B
Bisect the 60° with arcs; join A to intersection to get 30°.
-
Question 5 — Copy the given angle ∠XYZ at point P.
Given diagram:
X
\
\
Y
\
Z
Solution (steps):
- Place compass at Y and draw an arc cutting YX and YZ at A and B.
- At P draw a ray PQ and with the same radius draw an arc cutting PQ at R.
- Measure distance AB with compass. With centre R mark off an arc cutting the first arc at S.
- Join P to S. ∠QPS is equal to ∠XYZ.
Given ∠XYZ, arcs at A and B. Transfer arc and chord AB to PQ to get S. Join P-S to copy angle.
-
Question 6 — Construct an equilateral triangle with side 6 cm.
Solution (steps):
- Draw base AB of 6 cm.
- With A as centre and radius 6 cm draw arc; with B as centre and radius 6 cm draw arc; call intersection C.
- Join A to C and B to C. Triangle ABC is equilateral (all sides 6 cm).
C
/ \
/ \
A____B
All sides = 6 cm.
-
Question 7 — Construct an isosceles triangle with base AB = 8 cm and equal sides 6 cm.
Solution (steps):
- Draw base AB = 8 cm.
- With A as centre and radius 6 cm draw an arc; with B as centre and radius 6 cm draw arc; intersection is C.
- Join A to C and B to C. AC = BC = 6 cm; triangle is isosceles.
C
/ \
/ \
A___B (AB = 8 cm, AC = BC = 6 cm)
-
Question 8 — Construct a right-angled triangle with base AB = 7 cm and height from A equals 5 cm (right angle at A).
Solution (steps):
- Draw AB = 7 cm.
- At A construct a perpendicular line using the 90° method.
- Mark point C on the perpendicular such that AC = 5 cm.
- Join B to C. Triangle ABC is right-angled at A.
C
|
| (AC = 5)
A______B (AB = 7)
Join B to C.
-
Question 9 — Construct a square ABCD with side 4 cm.
Solution (steps):
- Draw AB = 4 cm.
- At A construct perpendicular and mark AD = 4 cm.
- At B construct perpendicular in same direction and mark BC = 4 cm.
- Join C to D. ABCD is a square.
A____B
| |
| |
D____C
All sides = 4 cm.
-
Question 10 — Construct rectangle ABCD with length AB = 9 cm and breadth AD = 5 cm.
Solution (steps):
- Draw AB = 9 cm.
- At A construct perpendicular and mark AD = 5 cm.
- At B construct perpendicular and mark BC = 5 cm.
- Join C and D to complete rectangle.
A________B (9 cm)
| |
| |
D________C (5 cm)
-
Question 11 — Construct a regular hexagon of side 3 cm using a circle.
Solution (steps):
- Draw circle with centre O and radius 3 cm.
- Place compass at a point on circle and stepping around the circumference mark off equal arcs six times to get points A, B, C, D, E, F.
- Join A-B-C-D-E-F-A. The polygon is a regular hexagon with side 3 cm.
A__B
/ \
F C
\ /
E--D
All sides = radius = 3 cm (on circle).
-
Question 12 — Construct angle of 45° at A, then from its ray mark a point X so that AX = 6 cm. Draw a perpendicular to AX through X.
Solution (steps):
- Construct 90° at A and bisect to get 45° ray.
- On that ray mark point X such that AX = 6 cm.
- At X construct a perpendicular using the 90° construction method applied with X as vertex; draw small arc and perpendicular line through X.
A
\
\ (45°)
X (AX = 6)
|
| (perpendicular at X)
-
Question 13 — Copy ∠M (given) onto point T and then construct a triangle with the copied angle at T and base of length 8 cm.
Solution (steps):
- Copy angle ∠M to point T by arc-transfer method (see Question 5).
- On the copied angle ray from T mark a point U so that TU = any convenient length.
- On the base place endpoints to make base 8 cm and join to T to form triangle.
Copy angle at T, then draw base AB of 8 cm and join A and B to T to form triangle TAB with angle at T copied from ∠M.
-
Question 14 — Construct a triangle ABC with AB = 7 cm, AC = 6 cm and ∠A = 60°.
Solution (steps):
- Draw ray from A and construct ∠A = 60°.
- On one side from A mark B such that AB = 7 cm.
- On the other side from A mark C such that AC = 6 cm.
- Join B to C to complete triangle ABC.
From A draw two rays with 60° between them. Mark B on one ray at 7 cm and C on the other at 6 cm. Join B-C.
-
Question 15 — Construct a trapezium ABCD with parallel sides AB = 10 cm and CD = 6 cm and height = 4 cm (AB is longer base).
Solution (steps):
- Draw base AB = 10 cm.
- At A draw a perpendicular and mark point A1 such that AA1 = 4 cm; at B draw perpendicular and mark B1 such that BB1 = 4 cm. Line A1B1 is the line of the other base.
- On line A1B1 mark CD = 6 cm somewhere between A1 and B1 so that CD is parallel to AB. Join A to C and B to D to form trapezium ABCD.
A____________B (10 cm)
| |
| |
A1____C__D___B1 (A1B1 line at height 4 cm; CD = 6 cm)
Join A-C and B-D to complete trapezium.
Notes and examination tips:
- Always show compass arcs and mark points clearly when presenting constructions in an exam.
- Label all points used for arcs and intersections; examiners award marks for correct construction steps and correct labelling.
- If the question requests a measure (for example AB = 6 cm), use a ruler to measure precisely and show the measurement on your diagram.
- Practice the arc-transfer method for copying angles; it is used often in construction problems.
