UPCAT Mathematics — Perimeter, Area, Volume & Equation of a LineExam Answer Templates

Examiner tip

For 1-mark questions, write the formula first, then substitute directly

Model answer

P = 2(l + w) = 2(12 + 8) = 2(20) = 40 cm

Answer structure

• Write formula, substitute values, and calculate in one line [1 mark]

Scoring breakdown

Common mark deductions

• Missing units
• Incorrect formula
• Calculation error

Key phrases to include

• P = 2(l + w)
• units (cm)

Examiner tip

Always state what's given first, especially when specific values like π are provided

Model answer

Given: r = 7 m, π = 22/7
Circumference = 2πr
C = 2 × (22/7) × 7 = 2 × 22 = 44 m

Answer structure

• State given values [0.5 mark]
• Write correct formula [0.5 mark]
• Substitute and calculate correctly [1 mark]

Scoring breakdown

Common mark deductions

• Not stating given values
• Using wrong value of π
• Missing final units

Key phrases to include

• Given:
• C = 2πr
• substitute
• units

Examiner tip

For Heron's formula questions, clearly show the semi-perimeter calculation first

Model answer

Given: a = 3 cm, b = 4 cm, c = 5 cm
Semi-perimeter s = (a + b + c)/2 = (3 + 4 + 5)/2 = 6 cm
Using Heron's formula: A = √[s(s-a)(s-b)(s-c)]
A = √[6(6-3)(6-4)(6-5)] = √[6 × 3 × 2 × 1] = √36 = 6 cm²

Answer structure

• State given values and calculate semi-perimeter [1 mark]
• Write Heron's formula correctly [1 mark]
• Substitute values and calculate final answer [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect semi-perimeter calculation
• Wrong formula
• Arithmetic errors in final calculation

Key phrases to include

• Semi-perimeter s =
• Heron's formula
• A = √[s(s-a)(s-b)(s-c)]
• cm²

Examiner tip

Multi-part questions require clear labeling (a), (b) and separate calculations for each part

Model answer

Given: Length = 4 m, Width = 3 m, Height = 2 m, Cost = ₱2 per litre

(a) Volume calculation:
V = l × w × h
V = 4 × 3 × 2 = 24 m³

(b) Cost calculation:
Volume in liters = 24 m³ × 1000 = 24,000 litres
(Since 1 m³ = 1000 liters)
Total cost = 24,000 × ₱2 = ₱48,000

Therefore, volume = 24 m³ and cost = ₱48,000

Answer structure

• State given values clearly [1 mark]
• Calculate volume using correct formula [2 marks]
• Convert volume to liters [1 mark]
• Calculate total cost [1 mark]

Scoring breakdown

Common mark deductions

• Forgetting unit conversion
• Missing currency symbol
• Not showing step-by-step calculation

Key phrases to include

• Given:
• V = l × w × h
• 1 m³ = 1000 liters
• Therefore

Examiner tip

Always show the slope calculation first, then choose one of the given points for substitution

Model answer

Given points: (2, 3) and (4, 7)
Slope m = (y₂ - y₁)/(x₂ - x₁) = (7 - 3)/(4 - 2) = 4/2 = 2
Using point-slope form: y - y₁ = m(x - x₁)
y - 3 = 2(x - 2)
y - 3 = 2x - 4
y = 2x - 1

Answer structure

• Calculate slope using slope formula [1 mark]
• Apply point-slope form correctly [1 mark]
• Simplify to get final equation [1 mark]

Scoring breakdown

Common mark deductions

• Incorrect slope calculation
• Wrong point substitution
• Algebraic errors in simplification

Key phrases to include

• Slope m =
• point-slope form
• y - y₁ = m(x - x₁)

Examiner tip

Remember that cone volume has the 1/3 factor - this is the most common error

Model answer

Given: r = 3 cm, h = 4 cm
Volume of cone = (1/3)πr²h
V = (1/3) × π × 3² × 4 = (1/3) × π × 9 × 4 = 12π cm³

Answer structure

• State given values and formula [1 mark]
• Substitute and calculate correctly [1 mark]

Scoring breakdown

Common mark deductions

• Forgetting the 1/3 factor
• Squaring error
• Missing units

Key phrases to include

• Given:
• V = (1/3)πr²h
• cm³

Examiner tip

Clearly identify which measurements are the parallel sides in trapezoid problems

Model answer

Given: b₁ = 8 cm, b₂ = 12 cm, h = 5 cm
Area of trapezoid = (1/2)(b₁ + b₂)h
A = (1/2)(8 + 12) × 5 = (1/2) × 20 × 5 = 50 cm²

Answer structure

• State given values and correct formula [1 mark]
• Calculate area with proper units [1 mark]

Scoring breakdown

Common mark deductions

• Using wrong values for parallel sides
• Formula error
• Missing square units

Key phrases to include

• Given:
• A = (1/2)(b₁ + b₂)h
• parallel sides
• cm²

Examiner tip

Always state the mathematical condition first, then verify it with the given values

Model answer

Given slopes: m₁ = 2/3 and m₂ = -3/2
For perpendicular lines: m₁ × m₂ = -1
Checking: (2/3) × (-3/2) = -6/6 = -1
Since m₁ × m₂ = -1, the lines are perpendicular.

Answer structure

• State the condition for perpendicular lines [1 mark]
• Verify the condition and conclude [1 mark]

Scoring breakdown

Common mark deductions

• Not stating the perpendicular condition
• Calculation error
• No clear conclusion

Key phrases to include

• m₁ × m₂ = -1
• perpendicular
• Since
• Therefore

Examiner tip

When working backwards from volume, show each algebraic step clearly

Model answer

Given: V = 36π cm³
Volume of sphere = (4/3)πr³
36π = (4/3)πr³
Dividing both sides by π: 36 = (4/3)r³
Multiplying both sides by 3/4: r³ = 36 × 3/4 = 27
Therefore: r = ∛27 = 3 cm

Answer structure

• Write volume formula and set up equation [1 mark]
• Solve for r³ correctly [1 mark]
• Find cube root and state final answer [1 mark]

Scoring breakdown

Common mark deductions

• Wrong sphere volume formula
• Algebraic errors
• Incorrect cube root

Key phrases to include

• V = (4/3)πr³
• Dividing both sides
• r³ =
• ∛

Examiner tip

Write the distance formula first, then substitute coordinates carefully

Model answer

Given points: A(1, 2) and B(4, 6)
Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
d = √[(4-1)² + (6-2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5 units

Answer structure

• Write distance formula [1 mark]
• Substitute values and calculate [1 mark]

Scoring breakdown

Common mark deductions

• Wrong formula
• Coordinate substitution errors
• Arithmetic mistakes

Key phrases to include

• Distance formula
• d = √[(x₂-x₁)² + (y₂-y₁)²]
• units

Examiner tip

When asked to explain reasonableness, relate your answer back to the properties of the shape

Model answer

Given: Regular hexagon with side length = 6 cm
Perimeter of regular polygon = n × s (where n = number of sides)
P = 6 × 6 = 36 cm
This is reasonable because a hexagon has 6 equal sides, so the total perimeter is 6 times one side length.

Answer structure

• Apply correct formula for regular polygon [1 mark]
• Provide reasonable explanation [1 mark]

Scoring breakdown

Common mark deductions

• Wrong formula application
• Missing explanation
• Unclear reasoning

Key phrases to include

• Regular polygon
• P = n × s
• 6 equal sides
• reasonable

Examiner tip

Always convert diameter to radius first - this is a very common oversight

Model answer

Given: Diameter = 14 m, so radius = 7 m, height = 10 m
Volume of cylinder = πr²h
V = π × 7² × 10 = π × 49 × 10 = 490π m³
Using π ≈ 22/7: V = 490 × (22/7) = 1540 m³
In liters: 1540 m³ × 1000 = 1,540,000 liters

Answer structure

• Convert diameter to radius [0.5 mark]
• Apply cylinder volume formula [1 mark]
• Calculate volume and convert to liters [1.5 marks]

Scoring breakdown

Common mark deductions

• Using diameter instead of radius
• Forgetting unit conversion
• Calculation errors

Key phrases to include

• radius = diameter/2
• V = πr²h
• 1 m³ = 1000 liters

Examiner tip

Start with slope-intercept form, then rearrange to get all variables on one side

Model answer

Given: y-intercept = -3, slope = 4
Slope-intercept form: y = mx + b
y = 4x + (-3) = 4x - 3
To convert to standard form (Ax + By = C):
4x - y = 3

Answer structure

• Write in slope-intercept form first [1 mark]
• Convert correctly to standard form [1 mark]

Scoring breakdown

Common mark deductions

• Sign errors
• Incorrect rearrangement
• Wrong final form

Key phrases to include

• y = mx + b
• standard form
• Ax + By = C

Examiner tip

For composite figures, calculate each part separately then combine

Model answer

Given: Rectangle 8 cm × 5 cm, semicircle on longer side (diameter = 8 cm)
Area of rectangle = l × w = 8 × 5 = 40 cm²
For semicircle: radius = 8/2 = 4 cm
Area of semicircle = (1/2)πr² = (1/2) × π × 4² = 8π cm²
Total area = 40 + 8π cm²

Answer structure

• Calculate rectangle area [1 mark]
• Calculate semicircle area correctly [1 mark]
• Add both areas for total [1 mark]

Scoring breakdown

Common mark deductions

• Using diameter instead of radius
• Forgetting the 1/2 factor for semicircle
• Not adding areas

Key phrases to include

• Area of rectangle =
• radius =
• Area of semicircle =
• Total area =

Examiner tip

Break down multi-step problems clearly with numbered steps - it helps you avoid errors and earns partial marks

Model answer

Given: Line 3x + 2y = 6 and point (1, -2)

Step 1: Find slope of given line
3x + 2y = 6
2y = -3x + 6
y = -3x/2 + 3
Slope m₁ = -3/2

Step 2: Find slope of parallel line
For parallel lines: m₁ = m₂
Therefore: m₂ = -3/2

Step 3: Use point-slope form
y - y₁ = m(x - x₁)
y - (-2) = -3/2(x - 1)
y + 2 = -3x/2 + 3/2
y = -3x/2 + 3/2 - 2
y = -3x/2 - 1/2

Therefore, the equation is y = -3x/2 - 1/2 or 3x + 2y = -1

Answer structure

• Convert given line to slope-intercept form [1 mark]
• Identify slope of given line [1 mark]
• State parallel line condition [1 mark]
• Apply point-slope form [1 mark]
• Simplify to final equation [1 mark]

Scoring breakdown

Common mark deductions

• Errors in slope extraction
• Wrong parallel condition
• Algebraic mistakes in simplification

Key phrases to include

• Step 1
• slope-intercept form
• parallel lines
• m₁ = m₂
• point-slope form
• Therefore

Do

• Use correct formula immediately
• Show final answer with units
• Keep calculation neat and direct

Don't

• Don't show excessive working for simple calculations
• Don't forget units
• Don't waste time on explanations

Do

• Write 'Given:' clearly
• Show formula before substituting
• Display calculation steps
• Include proper units

Don't

• Don't skip the formula statement
• Don't combine too many steps
• Don't forget intermediate steps

Do

• Break solution into clear steps
• Show all algebraic manipulations
• Verify your answer makes sense
• Use proper mathematical notation

Don't

• Don't skip algebraic steps
• Don't make careless calculation errors
• Don't present working in confusing order

Do

• Use clear headings for different parts
• Show all formula derivations
• Include diagrams where helpful
• Provide concluding statement
• Check reasonableness of answer

Don't

• Don't rush through any section
• Don't skip verification steps
• Don't present messy or unclear working
• Don't forget to label multi-part answers