Begin with Euclidean geometry and polygons. The sum of the interior angles of an n-sided convex polygon is (n − 2) × 180°, so a pentagon's interior angles total 540°. The exterior angles of any convex polygon, taken one at each vertex, always add up to 360°, regardless of the number of sides. For circles, master the standard theorems: the angle an arc subtends at the centre is twice the angle it subtends at any point on the remaining circumference; the angle in a semicircle is 90°; and opposite angles of a cyclic quadrilateral are supplementary — they add up to 180°. In mensuration, for a circle of radius r, an arc subtending angle θ at the centre has length (θ/360) × 2πr, and the corresponding sector has area (θ/360) × πr². Loci and construction questions draw on these same angle and distance properties, so revise them together.
In coordinate geometry of straight lines, the distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²] and the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). The gradient is m = (y₂ − y₁)/(x₂ − x₁); two lines are parallel when their gradients are equal and perpendicular when m₁ × m₂ = −1.
For trigonometry, the syllabus requires you to apply the special angles 30°, 45°, 60°, 75°, 90°, 105° and 135°. Memorise the exact values: sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1 and sin 60° = √3/2. These feed directly into angles of elevation and depression, and into bearings, which are measured clockwise from north and written as three-digit figures from 000° to 360° (a bearing of 045° is north-east). To solve any triangle, use the sine rule, a/sin A = b/sin B = c/sin C, or the cosine rule, c² = a² + b² − 2ab cos C; the area of a triangle with sides a and b enclosing angle C is ½ab sin C.
Introductory calculus rests on a small set of rules:
Drill timed past questions on each formula group until recall is automatic — the UTME rewards speed as much as accuracy.
1. What is the sum of the interior angles of a regular pentagon?
The sum of the interior angles of an n-sided polygon is (n − 2) × 180°; for a pentagon (n = 5), this gives (5 − 2) × 180° = 540°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
2. For any convex polygon, regardless of the number of sides, the sum of the exterior angles (one taken at each vertex) is always ___.
However many sides a convex polygon has, its exterior angles, one at each vertex, always add up to 360°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
3. What is the size of each interior angle of a regular hexagon?
The sum of interior angles of a hexagon is (6 − 2) × 180° = 720°; dividing by the 6 equal angles gives 720° ÷ 6 = 120°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
4. Each exterior angle of a regular polygon measures 40°. How many sides does the polygon have?
Since the exterior angles of a regular polygon are equal and always sum to 360°, the number of sides is 360° ÷ 40° = 9. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
5. Each interior angle of a regular polygon is 156°. Find the number of sides of the polygon.
The corresponding exterior angle is 180° − 156° = 24°, and since exterior angles sum to 360°, the number of sides is 360° ÷ 24° = 15. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
6. Angles on a straight line always add up to ___.
Angles that lie on a straight line are supplementary, summing to 180°, a basic Euclidean geometry theorem. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry theorems); New General Mathematics for Senior Secondary Schools (Channon et al.))
7. The sum of all the angles at a point (a full turn) is ___.
Angles that meet at a single point and go all the way round add up to a full turn of 360°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry theorems); New General Mathematics for Senior Secondary Schools (Channon et al.))
8. When two parallel lines are cut by a transversal, the alternate angles formed are ___.
A key Euclidean theorem on parallel lines states that alternate angles formed with a transversal are always equal. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry theorems); New General Mathematics for Senior Secondary Schools (Channon et al.))
9. The three interior angles of any triangle always add up to ___.
It is a fundamental Euclidean theorem that the interior angles of any triangle sum to 180°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry theorems); New General Mathematics for Senior Secondary Schools (Channon et al.))
10. The exterior angle of a triangle is always equal to ___.
By the exterior angle theorem, an exterior angle of a triangle equals the sum of the two interior angles that are not adjacent to it. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry theorems); New General Mathematics for Senior Secondary Schools (Channon et al.))
11. An arc of a circle subtends an angle of 70° at the centre. What angle does the same arc subtend at a point on the remaining part of the circumference?
The angle an arc subtends at the centre is twice the angle it subtends at the circumference, so the angle at the circumference is 70° ÷ 2 = 35°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (circles and angle properties); New General Mathematics for Senior Secondary Schools (Channon et al.))
12. An angle subtended at the circumference of a circle by a diameter (an angle in a semicircle) is always ___.
It is a standard circle theorem that the angle in a semicircle is a right angle, 90°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (circle theorems); New General Mathematics for Senior Secondary Schools (Channon et al.))
13. In a cyclic quadrilateral, one of a pair of opposite angles is 75°. What is the size of the angle opposite to it?
Opposite angles of a cyclic quadrilateral are supplementary, so the missing angle is 180° − 75° = 105°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (circle theorems); New General Mathematics for Senior Secondary Schools (Channon et al.))
14. How many diagonals does a regular octagon (8 sides) have?
The number of diagonals of an n-sided polygon is n(n − 3)/2; for n = 8, this gives 8 × 5 ÷ 2 = 20. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
15. In trigonometry, the angle of elevation of an object from an observer is best described as ___.
The angle of elevation is measured upward from the horizontal line of sight to a point above the observer's eye level. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (bearings, angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
16. The angle of depression of an object viewed from a higher point is the angle between ___.
The angle of depression is measured downward from the horizontal line of sight to an object below the observer's eye level. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (bearings, angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
17. A student standing 50 m from the foot of a flagpole observes the angle of elevation of the top of the flagpole to be 30°. Taking tan 30° = 1/√3 (√3 ≈ 1.73), calculate the height of the flagpole to 1 decimal place.
Height = distance × tan 30° = 50 × (1/1.73) ≈ 28.9 m, using the right-angled triangle formed by the flagpole, the ground and the line of sight. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (trigonometrical ratios of special angles; angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
18. From the top of a vertical cliff 60 m high, the angle of depression of a boat at sea is 60°. Taking tan 60° = √3 ≈ 1.732, find the distance of the boat from the base of the cliff to 1 decimal place.
The angle of depression equals the angle of elevation from the boat, so distance = height ÷ tan 60° = 60 ÷ 1.732 ≈ 34.6 m. (JAMB UTME e-Syllabus, Mathematics — angles of elevation and depression; New General Mathematics for SSS, Channon et al.) (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (trigonometrical ratios of special angles; angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
19. A ladder leaning against a vertical wall makes an angle of 60° with the horizontal ground. If the foot of the ladder is 4 m from the wall, find the length of the ladder (take cos 60° = 1/2).
Since cos 60° = adjacent/hypotenuse = 4/length, the length of the ladder = 4 ÷ (1/2) = 8 m. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (trigonometrical ratios of special angles; angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
20. From a point on level ground 20 m from the foot of a vertical tower, the angle of elevation of the top of the tower is 45°. Find the height of the tower.
Since tan 45° = 1, the height of the tower equals the horizontal distance, so the height is 20 m. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (trigonometrical ratios of special angles; angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
21. Points A and B lie on level ground in a straight line with the foot of a vertical tower, with B nearer to the tower. The angle of elevation of the top of the tower is 60° from B and 30° from A. If AB = 40 m, find the height of the tower, leaving your answer in surd form.
Setting up tan 60° = h/OB and tan 30° = h/(OB + 40) and solving the two equations simultaneously gives h = 20√3 m (≈34.6 m). (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (trigonometrical ratios of special angles; angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
22. In three-figure bearing notation, a bearing of 045° corresponds to which compass direction?
Bearings are measured clockwise from north, so 045° lies exactly halfway between north (000°) and east (090°), which is north-east. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (bearings, angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
23. From a point on level ground 15 m from the foot of a vertical pole, the angle of elevation of the top of the pole is 60°. Taking tan 60° = √3, find the height of the pole in surd form.
Height = distance × tan 60° = 15 × √3 = 15√3 m, which is approximately 26.0 m. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (trigonometrical ratios of special angles; angles of elevation and depression); New General Mathematics for Senior Secondary Schools (Channon et al.))
24. If the angle of elevation of the top of a tower from a point on the ground is 50°, what is the angle of depression of that same point as seen from the top of the tower?
The horizontal lines at the observer's eye and at the top of the tower are parallel, so the angle of elevation and the angle of depression are alternate angles and are therefore equal. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (bearings, angles of elevation and depression — Euclidean geometry theorems); New General Mathematics for Senior Secondary Schools (Channon et al.))
25. The sum of the interior angles of a convex pentagon is:
The sum of the interior angles of an n-sided polygon is (n − 2) × 180°; for a pentagon, n = 5, giving (5 − 2) × 180° = 540°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
26. The sum of the exterior angles of any convex polygon, taken one at each vertex, is:
Regardless of how many sides a convex polygon has, the sum of its exterior angles taken one at each vertex is always 360°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
27. Using π = 22/7, find the area of a circle of radius 7 cm.
Area of a circle = πr² = 22/7 × 7 × 7 = 154 cm². (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (mensuration of plane shapes and solids); New General Mathematics for Senior Secondary Schools (Channon et al.))
28. For a cuboid of length l, breadth b and height h, the volume is given by:
The volume of a cuboid is the product of its three dimensions: V = l × b × h. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (mensuration of plane shapes and solids); New General Mathematics for Senior Secondary Schools (Channon et al.))
29. The area of a trapezium with parallel sides a and b, and perpendicular height h between them, is given by:
The area of a trapezium equals half the sum of the parallel sides multiplied by the perpendicular distance between them. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (mensuration of plane shapes and solids); New General Mathematics for Senior Secondary Schools (Channon et al.))
30. Each interior angle of a regular hexagon measures:
The interior angle sum of a hexagon is (6 − 2) × 180° = 720°; sharing this equally among the 6 angles gives 720° ÷ 6 = 120°. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
31. A regular polygon has each exterior angle equal to 40°. How many sides does the polygon have?
Since the exterior angles of a convex polygon always sum to 360°, the number of sides is 360° ÷ 40° = 9. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (Euclidean geometry — polygons); New General Mathematics for Senior Secondary Schools (Channon et al.))
32. A sector of a circle of radius 21 cm subtends an angle of 60° at the centre. Using π = 22/7, calculate the length of the arc.
Arc length = (θ/360) × 2πr = (60/360) × 2 × 22/7 × 21 = 22 cm. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (lengths and areas of circles — arcs and sectors); New General Mathematics for Senior Secondary Schools (Channon et al.))
33. A sector of a circle of radius 14 cm has an angle of 90° at the centre. Using π = 22/7, calculate the area of the sector.
Sector area = (θ/360) × πr² = (90/360) × 22/7 × 14² = 154 cm². (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (lengths and areas of circles — arcs and sectors); New General Mathematics for Senior Secondary Schools (Channon et al.))
34. Using π = 22/7, calculate the volume of a cylinder of radius 7 cm and height 10 cm.
Volume of a cylinder = πr²h = 22/7 × 7² × 10 = 1540 cm³. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (mensuration of plane shapes and solids); New General Mathematics for Senior Secondary Schools (Channon et al.))
35. Using π = 22/7, calculate the volume of a cone with base radius 6 cm and height 14 cm.
Volume of a cone = ⅓πr²h = ⅓ × 22/7 × 6² × 14 = 528 cm³. (JAMB UTME e-Syllabus, Mathematics, Section III: Geometry and Trigonometry (mensuration of plane shapes and solids); New General Mathematics for Senior Secondary Schools (Channon et al.))