Perimeter, Area and VolumeEdexcel GCSE Maths: Revision notes
Section 1
How do you calculate perimeter and area of 2D shapes?
Perimeter is the total distance around the outside of a shape. Area is the space inside a shape, measured in square units.
Triangles:
- Perimeter = sum of all three sides
- Area = ½ × base × height
- Height must be perpendicular to the base
Rectangles:
- Perimeter = 2(length + width)
- Area = length × width
Parallelograms:
- Perimeter = 2(side₁ + side₂)
- Area = base × perpendicular height
- The perpendicular height is not the slant side
Trapezoids (Trapeziums):
- Perimeter = sum of all four sides
- Area = ½(a + b) × h, where a and b are the parallel sides and h is the perpendicular height
Composite shapes:
- Break the shape into simpler rectangles, triangles, or other standard shapes
- Calculate the area of each component
- Add or subtract areas as appropriate
- For perimeter of composite shapes, only include the outer boundary
Students often confuse the slant height of a trapezium or parallelogram with the perpendicular height. Always measure height at 90° to the base. Using the wrong height will lose marks.
For composite shapes, sketch lines to divide the shape clearly into rectangles and triangles. Label all dimensions before calculating—examiners want to see your working method.
A trapezium has parallel sides of 5 cm and 9 cm, with perpendicular height 4 cm. Area = ½(5 + 9) × 4 = ½ × 14 × 4 = 28 cm².
Section 2
What are the circumference and area formulas for circles?
Circumference is the perimeter of a circle—the distance around its edge.
Circumference = πd or Circumference = 2πr, where d is diameter and r is radius
Area = πr², where r is the radius
Key points:
- The radius is half the diameter: r = d ÷ 2
- Leave answers in terms of π unless the question specifically asks for a decimal approximation
- When leaving answers in terms of π, write them as multiples like 12π cm² or 5π cm, not as decimals
Sectors and arcs:
- A sector is a 'slice' of a circle bounded by two radii and an arc
- An arc is a curved section of the circumference
- Arc length = (θ ÷ 360) × 2πr, where θ is the angle in degrees
- Area of sector = (θ ÷ 360) × πr²
Semi-circles and quarter-circles:
- Semi-circle arc length = πr (half the full circumference)
- Semi-circle area = ½πr²
- Quarter-circle arc length = ½πr
- Quarter-circle area = ¼πr²
The mark scheme expects answers in terms of π unless you're told otherwise. Writing '12π cm²' earns full marks; writing '37.7 cm²' will lose marks even if numerically close.
Using diameter instead of radius in area calculations is a very common error. Remember: Area = πr² uses the radius, not the diameter. If you're given diameter, divide by 2 first.
A sector has radius 6 cm and angle 120°. Arc length = (120 ÷ 360) × 2π × 6 = (1/3) × 12π = 4π cm. Area of sector = (120 ÷ 360) × π × 36 = 12π cm².
Section 3
How do you calculate surface area and volume of 3D solids?
Volume is the amount of space inside a 3D solid, measured in cubic units. Surface area is the total area of all outer faces.
Prisms (including cylinders):
- Volume = base area × length (or height)
- Surface area = sum of areas of all faces
Cylinders:
- Volume = πr²h, where r is radius and h is height
- Curved surface area = 2πrh
- Total surface area = 2πrh + 2πr² (curved surface plus two circular ends)
- Leave answers in terms of π
Pyramids:
- Volume = ⅓ × base area × height
- Surface area = base area + sum of the areas of all triangular faces
Cones:
- Volume = ⅓πr²h
- Curved surface area = πrl, where l is the slant height
- Total surface area = πrl + πr²
- Leave answers in terms of π
Spheres:
- Volume = ⁴⁄₃πr³
- Surface area = 4πr²
- Leave answers in terms of π
Key formula table:
| Solid | Volume | Surface Area |
|---|---|---|
| Prism | base area × height | Sum of all face areas |
| Cylinder | πr²h | 2πrh + 2πr² |
| Pyramid | ⅓ × base area × height | Base area + lateral faces |
| Cone | ⅓πr²h | πrl + πr² |
| Sphere | ⁴⁄₃πr³ | 4πr² |
For composite 3D solids, identify each component solid separately, calculate its volume or surface area, then combine them. Always show which solids you're using and your method clearly.
Using height instead of slant height (or vice versa) in cone formulas loses marks. Slant height l is used in πrl for curved surface area; perpendicular height h is used in ⅓πr²h for volume.
A cylinder has radius 3 cm and height 10 cm. Volume = π × 3² × 10 = 90π cm³. Total surface area = 2π(3)(10) + 2π(3)² = 60π + 18π = 78π cm².
Section 4
How do you solve problems involving composite 3D solids?
Composite 3D solids are made from combinations of standard solids such as cylinders, cones, pyramids, and spheres. These often appear in real-world contexts on the exam.
Method for solving composite solid problems:
- Identify each component solid carefully—sketch or describe what you see
- List the dimensions of each part, converting units if necessary
- Calculate volume or surface area of each component using the appropriate formula
- Combine the answers:
- For volume: add the volumes together (solids are joined, not overlapping)
- For surface area: remember that touching faces are hidden and not counted
- Show all working clearly—examiners award method marks
Important considerations:
- When two solids join, the area where they meet is not part of the total surface area
- If a cone sits on top of a cylinder, subtract the cone's base area and the cylinder's top area from your calculation
- Always leave answers in terms of π unless instructed otherwise
- Double-check that you're calculating what the question asks (volume, surface area, or both)
Common composite shapes:
- Cylinder with a cone on top (e.g., a sharpened pencil)
- Hemisphere on top of a cylinder (e.g., a capsule shape)
- Composite pyramids or prisms made from simpler rectangles
- Hemispheres (half spheres) attached to other solids
Draw and label the composite shape clearly, showing which faces are hidden when solids join. Write annotations showing where you're subtracting surface areas—this demonstrates understanding to the examiner.
Students often forget to subtract the area of touching faces. If a cone sits on a cylinder, you must subtract both the cone's base area AND the cylinder's top area from the total surface area calculation.
A shape consists of a cylinder (radius 2 cm, height 5 cm) with a hemisphere (radius 2 cm) on top. Volume = π(2)²(5) + ⅔π(2)³ = 20π + (16/3)π = (76/3)π cm³. For surface area, include the cylinder's curved side, hemisphere's curved surface, and cylinder's base—the cylinder's top is hidden.
Section 5
When and how do you use π in exact calculations?
The Edexcel GCSE specification requires you to leave answers in terms of π for circle and circular solid problems unless explicitly told otherwise.
When to leave answers in terms of π:
- All circle circumference problems (unless asked for a decimal approximation)
- All circle area problems
- All cylinder volume and surface area
- All cone volume and surface area
- All sphere volume and surface area
- Sectors and arcs
What "in terms of π" means:
- Write the answer as a number multiplied by π, e.g. 12π cm or 25π cm²
- Do not substitute π ≈ 3.14 or use a calculator to give a decimal
- Simplify fractions where possible, e.g. write 2π/3 m not 2.094... m
Examples of correct form:
- 15π cm (not 47.1 cm)
- 8π m² (not 25.1 m²)
- (16/3)π cm³ (simplified fraction form)
- 5π/2 cm (simplified fraction form)
When you may use decimal approximations:
- Only when the question explicitly says "correct to..." or "approximately"
- When working with composite shapes that include non-circular components, you may mix π expressions with decimal values, but π should remain as π
- For final answers to real-world problems where a decimal is contextually appropriate
Tip for calculations with fractions of π:
- Work with the numerical coefficient and π separately
- Example: (120 ÷ 360) × 2πr = (1/3) × 2π × 6 = (12π)/3 = 4π cm
Examiners use the phrase 'leave in terms of π' or 'give an exact answer' to signal this requirement. If you see these phrases, never substitute π with a decimal—you will lose marks.
Leaving answers in terms of π is like leaving √2 as √2 rather than converting it to 1.414... — both are exact forms that preserve mathematical precision better than approximations.
A semicircle has radius 5 cm. Arc length = πr = 5π cm (exact). Area = ½πr² = ½π(25) = 12.5π or (25/2)π cm² (exact). Do not write 15.7 cm or 39.3 cm² unless the question asks for approximation.
Must Know
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Perimeter and area formulas: Triangles (Area = ½bh), rectangles (Area = lw), parallelograms (Area = base × height), trapeziums (Area = ½(a+b)h). For composite shapes, break into simpler shapes and add or subtract areas.
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Circles and sectors: Circumference = 2πr or πd; Area = πr²; Arc length = (θ/360) × 2πr; Sector area = (θ/360) × πr². Always leave in terms of π unless told otherwise.
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Prisms and cylinders: Volume = base area × height (or πr²h for cylinders); Surface area = sum of all faces (or 2πrh + 2πr² for cylinders). Leave in terms of π.
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Pyramids, cones and spheres: Pyramid volume = ⅓ × base area × height; Cone volume = ⅓πr²h, surface area = πrl + πr²; Sphere volume = ⁴⁄₃πr³, surface area = 4πr². Leave in terms of π.
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Composite 3D solids: Identify each component solid, calculate its volume or surface area separately, then combine—but subtract the areas of hidden faces where solids touch. Always show working and leave in terms of π.
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Use of π: Leave all answers involving circles, cylinders, cones and spheres in the form (number)π unless the question explicitly asks for a decimal approximation or gives a context requiring rounding.
That's the notes covered.
Carry on to the next subtopic.