📦 Mensuration, Coordinates & Vectors
This chapter covers area and perimeter of 2D shapes, volume and surface area of 3D solids, arc length and sector area, radian measure, coordinate geometry, and vectors. These are popular Paper 2 topics with questions often worth 8-12 marks each. Many formulas are given in the formula sheet, but you must know when and how to apply each one.
Mensuration and vectors are often combined in complex, multi-step problems. For example, you might need to find the volume of a combined solid (cone + hemisphere), then calculate coordinates using vectors. The key to these questions is breaking them down into manageable steps and being systematic about which formula to use for each part.
Areas of 2D Shapes
You should know the area formulas for all standard shapes. While some are given in the formula sheet, being familiar with them saves time during the exam. Always draw a diagram and label all dimensions clearly before calculating.
📏Areas of 2D Shapes
Area formulas for common 2D shapes. All must be memorised.
Arc Length & Sector Area
A sector is a "pie-slice" of a circle, bounded by two radii and an arc. An arc is a portion of the circumference. Both are calculated as a fraction of the full circle, using the angle at the centre. A segment is the region between a chord and its arc — its area is found by subtracting the area of the triangle from the area of the sector.
🥧Arc Length & Sector Area
Formulas for parts of circles, using the angle at the centre.
📝Worked Example 1: Arc Length and Sector Area
A sector has radius 10 cm and angle 72 degrees. Find (a) the arc length, (b) the sector area, (c) the perimeter of the sector.
Circle sector with radius 10 cm and angle 72°
📝Worked Example 2: Area of Segment
A chord subtends an angle of 120 degrees at the centre of a circle with radius 8 cm. Find the area of the minor segment.
Minor segment: radius 8 cm, angle 120°
📝Worked Example 2b: Radian Measure
In A-Math, angles are measured in radians instead of degrees. For E-Math, you should know the basic conversion. One complete revolution = 360 degrees = 2pi radians.
📝Worked Example 2c: Arc Length in Radians
A sector with radius 9 cm has arc length 6pi cm. Find the angle at the centre in degrees.
Sector with radius 9 cm and arc length 6\u03C0 cm
📝Worked Example 2d: Perimeter of Shaded Region
A circle with centre O and radius 10 cm has two radii OA and OB with angle AOB = 60 degrees. Find the perimeter of the minor segment (the shaded region between chord AB and arc AB).
Shaded region: radius 10 cm, angle 60°
A sector has radius 14 cm and angle 90°. What is the arc length?
Volumes & Surface Areas of 3D Solids
Volume and surface area formulas for common 3D shapes are given in the formula sheet, but you must understand when to use each formula and how to handle combined solids (e.g., a cone attached to a cylinder, or a hemisphere on top of a cuboid). For combined solids, calculate each part separately and add them up. Be careful about internal faces — surfaces where two solids meet are not part of the total external surface area.
📦Volumes & Surface Areas of 3D Solids
Volume and total surface area formulas for standard 3D shapes.
📝Worked Example 3: Volume of a Cone
A cone has base radius 6 cm and height 8 cm. Find (a) the volume, (b) the slant height, (c) the curved surface area.
Cone with base radius 6 cm, height 8 cm, and slant height 10 cm
📝Worked Example 4: Volume of Combined Solid
A solid consists of a hemisphere of radius 6 cm attached to a cylinder of the same radius and height 10 cm. Find the total volume and total surface area.
Combined solid: hemisphere (radius 6 cm) on a cylinder (height 10 cm)
📝Worked Example 5: Sphere Problem
A sphere has volume 288pi cm cubed. Find (a) the radius and (b) the surface area.
📝Worked Example 5b: Frustum (Truncated Cone)
A cone has base radius 12 cm and height 15 cm. A smaller cone of height 5 cm is cut from the top. Find the volume of the frustum (remaining solid).
Frustum formed by cutting a cone: R = 12 cm, r = 4 cm, frustum height = 10 cm
📝Worked Example 5c: Water in a Container
A cylindrical tank has base radius 20 cm and height 50 cm. It is filled to a depth of 30 cm with water. A solid sphere of radius 10 cm is dropped into the tank. By how much does the water level rise? Will the tank overflow?
📝Worked Example 5d: Prism Volume
A solid has a cross-section that is a trapezium with parallel sides 6 cm and 10 cm, and height 4 cm. The length of the prism is 15 cm. Find the volume.
3D Solids Exam Strategy
For combined solids, always identify the individual shapes first. Draw them separately and label all dimensions. Calculate the volume of each part, then add. For surface area of combined solids, be careful to subtract the areas where the solids are joined (internal faces are not part of the external surface area). For example, a hemisphere on a cylinder: the circular face where they meet is NOT part of the total surface area.
A hemisphere has radius 6 cm. What is its volume?
Coordinate Geometry
Coordinate geometry uses algebra to solve geometric problems in the Cartesian plane. The three fundamental tools are: the distance formula, the midpoint formula, and the gradient formula. These are used to find lengths of line segments, midpoints, equations of lines, and to prove geometric properties algebraically.
📍Coordinate Geometry
Formulas for distance, midpoint, and equations of lines in the Cartesian plane.
📝Worked Example 6: Distance, Midpoint, and Gradient
Points A(1, 2) and B(7, 10). Find (a) distance AB, (b) midpoint M, (c) equation of line AB.
y = 4x/3 + 2/3 showing gradient, y-intercept, and x-intercept
📝Worked Example 7: Perpendicular Bisector
Find the equation of the perpendicular bisector of the line segment joining P(2, 5) and Q(8, 1).
y = 3x/2 - 9/2 showing gradient, y-intercept, and x-intercept
📝Worked Example 7b: Finding Where Two Lines Intersect
Line L1 has equation y = 2x - 1. Line L2 has equation y = -x + 8. Find their point of intersection.
L1: y = 2x - 1 showing gradient, y-intercept, and x-intercept
📝Worked Example 7c: Equation of Line from Two Points
Find the equation of the line passing through A(-2, 3) and B(4, -1).
y = -2x/3 + 5/3 showing gradient, y-intercept, and x-intercept
📝Worked Example 7d: Collinear Points
Show that the points A(1, 2), B(3, 6), and C(5, 10) are collinear (lie on the same straight line).
y = 2x showing gradient, y-intercept, and x-intercept
📝Worked Example 7e: Parallel and Perpendicular Lines
Line L has equation 3x + 2y = 12. Find (a) the equation of the line parallel to L passing through (1, 4), and (b) the equation of the line perpendicular to L passing through (1, 4).
L: y = -3x/2 + 6 showing gradient, y-intercept, and x-intercept
Coordinate Geometry Key Facts
Parallel lines
Have the SAME gradient. .
Perpendicular lines
Have gradients that multiply to . . (Negative reciprocal.)
Horizontal lines
Have gradient 0. Equation: (constant).
Vertical lines
Have undefined gradient. Equation: (constant). NOT in the form .
x-intercept
Set in the equation and solve for .
y-intercept
Set in the equation and solve for . In , the y-intercept is .
Line L has gradient 3/4. What is the gradient of a line perpendicular to L?
Vectors in Two Dimensions
A vector has both magnitude (size) and direction. In O-Level E-Math, we work with column vectors and position vectors. Key operations include addition, subtraction, scalar multiplication, and finding magnitude. Two vectors are parallel if one is a scalar multiple of the other.
The position vector of a point A is the vector from the origin O to A, written as . The displacement vector from A to B is (destination minus start). This formula is fundamental and used in virtually every vector question.
➡️Vectors in Two Dimensions
Column vectors, vector addition, scalar multiplication, and position vectors.
📝Worked Example 8: Vector Operations
Given and . Find (a) vector AB, (b) midpoint M of AB, (c) magnitude of AB.
📝Worked Example 9: Proving Vectors Are Parallel
Given and . Show that a and b are parallel.
📝Worked Example 10: Vector Ratio Problem
P divides AB in the ratio 2:3. Given and . Find .
📝Worked Example 10b: Vector Triangle Problem
In triangle OAB, OA = a and OB = b. M is the midpoint of AB. Express OM in terms of a and b.
📝Worked Example 10c: Proving Parallelism with Vectors
OABC is a parallelogram. OA = a and OC = c. M is the midpoint of AB. N is a point on OB such that ON = (2/3)OB. Prove that M, N and C are collinear.
Common Vector Mistakes
The most common errors in vector questions:
- Wrong direction: AB = OB - OA, NOT OA - OB. Remember: destination minus start.
- Forgetting negative: AO = -OA. Reversing the direction negates the vector.
- Scalar vs vector: |a| is a scalar (number). a (bold) is a vector. Do not confuse them.
- Ratio errors: If P divides AB in ratio m:n, then AP = (m/(m+n)) AB, not (m/n) AB.
- Parallelism proof: To prove two vectors are parallel, show one is a SCALAR MULTIPLE of the other. Same direction is not enough to state without showing the scalar.
In triangle OAB, OA = a and OB = b. What is the position vector of the midpoint of AB?
⚠️Common Mistakes — Mensuration & Vectors
Why: Arc length uses circumference (2 pi r), sector area uses area (pi r^2). Do not mix them up.
Why: Mid-value = (lower bound + upper bound) / 2. Using boundaries instead of mid-values gives an incorrect mean.
Why: n/2 is on the cumulative frequency axis (vertical), not the data axis (horizontal). Read across, then down.
Why: Mode refers to the most frequent value/class. Class width is irrelevant when identifying the modal class.