Last reviewed: 7 November 2026
Updated November 2026. These Grade 8 geometrical constructions notes explain how to construct parallel and perpendicular lines, divide a line proportionally, construct regular and irregular polygons up to a hexagon, and construct circles through the vertices of a triangle (circumcircle) and touching its sides (inscribed circle). Most constructions need only a ruler and a pair of compasses; a protractor is used when an angle cannot be constructed.
The steps and questions are original practice for Kenyan junior school learners, not an official paper. For the other Grade 8 maths topics, see our Grade 8 mathematics revision guide and the Grade 8 notes hub.
On this page
- What the Grade 8 Curriculum Covers
- Tools and Good Habits
- Perpendicular Bisector of a Line
- Perpendicular From a Point to a Line
- Parallel Lines
- Dividing a Line Proportionally
- Angle Properties of Polygons
- Constructing Regular Polygons up to a Hexagon
- Irregular Polygons
- Circles and Triangles
- Original Practice Questions With Worked Answers
- Frequently Asked Questions
What the Grade 8 Curriculum Covers
Geometrical constructions is a sub-strand under Geometry in the Kenya Institute of Curriculum Development’s Grade 8 Mathematics design. Learners construct parallel and perpendicular lines, divide a line proportionally, identify angle properties of polygons, construct regular and irregular polygons up to a hexagon, and construct circles passing through the vertices of a triangle and touching its sides. The design is listed on the KICD Grade Eight curriculum designs page.
Tools and Good Habits
Use a sharp pencil, a ruler, a pair of compasses that holds its radius, and a protractor. Leave all construction arcs visible; teachers mark the method, not just the final shape.
Perpendicular Bisector of a Line
- Draw line AB.
- With the compass set to more than half of AB, draw arcs above and below the line from A.
- Keep the same radius and repeat from B.
- Join the two points where the arcs cross. This line cuts AB in half at right angles.
Perpendicular From a Point to a Line
From point P above a line, draw an arc that cuts the line at two points. From each of those points, draw arcs below the line with the same radius. Join P to where these arcs meet. That line is perpendicular to the first line.
Parallel Lines
To draw a line through P parallel to AB: draw any line from P to meet AB at Q, copy the angle PQB at P using compass arcs (corresponding angles), and draw the line. Corresponding angles are equal when lines are parallel, which is the property this construction uses.
Dividing a Line Proportionally
To divide AB in the ratio 2 : 3:
- Draw a line AC at any acute angle to AB.
- Mark 5 equal steps (2 + 3) along AC with the compass.
- Join the 5th mark to B.
- Through the 2nd mark, draw a line parallel to that joining line. Where it meets AB divides it in the ratio 2 : 3.
Angle Properties of Polygons
The interior angles of an n-sided polygon add up to (n − 2) × 180°, and the exterior angles add up to 360°, as described in OpenStax Contemporary Mathematics’ section on polygons. Each interior angle of a regular pentagon is 108°; of a regular hexagon, 120°.
Constructing Regular Polygons up to a Hexagon
- Equilateral triangle: draw a side, then arcs of the same length from each end; join where they meet.
- Square: construct a perpendicular at one end of a side, mark the same length, and complete with arcs.
- Regular hexagon: draw a circle; keeping the same radius, step around the circle marking 6 points; join them. This works because a hexagon is made of 6 equilateral triangles.
- Regular pentagon: draw one side, use a protractor to measure 108° at each end, mark equal sides, and complete.
Irregular Polygons
Use the given sides and angles in order. Example: construct quadrilateral ABCD with AB = 6 cm, angle A = 90°, AD = 4 cm, angle B = 60°, BC = 5 cm. Construct the right angle at A, mark D, construct 60° at B, mark C, then join C to D.
Circles and Triangles
Circumscribed circle (through the vertices)
Construct the perpendicular bisectors of two sides of the triangle. They meet at the circumcentre. With this centre and radius to any vertex, draw the circle through all three vertices.
Inscribed circle (touching the sides)
Construct the bisectors of two angles of the triangle. They meet at the incentre. Drop a perpendicular from the incentre to one side; its length is the radius. Draw the circle, which touches all three sides.
Original Practice Questions With Worked Answers
- Which construction lines meet at the centre of a circle through a triangle’s vertices? The perpendicular bisectors of the sides.
- Which lines meet at the centre of the inscribed circle? The angle bisectors.
- Why can a regular hexagon be drawn by stepping the radius round a circle? Each side equals the radius, because the hexagon is made of six equilateral triangles.
- How many equal steps do you mark to divide a line in the ratio 3 : 4? 7.
- What is each interior angle of a regular pentagon? 108°.
Cover the answers and attempt each question first. For timed practice, use the Grade 8 revision papers or the Grade 8 CBC Challenge. These notes are free; full revision access is available with the KSh 20 Day Pass or the KSh 500 Monthly Pass.
Frequently Asked Questions
How do you construct a perpendicular bisector?
Draw equal arcs from both ends of the line, above and below it, and join the two crossing points.
How do you construct a regular hexagon?
Draw a circle and, with the same radius, mark six points around it; join the points in order.
What is the difference between a circumscribed and an inscribed circle?
A circumscribed circle passes through the triangle’s vertices; an inscribed circle touches each of its sides from inside.
How do you divide a line in a given ratio?
Mark equal steps along a slanting line, join the last step to the end of the line, and draw a parallel line through the dividing step.