Leaving Cert Area & Volume (Higher Level): Trapezoidal Rule & 3D Solids

Area and volume covers the area of 2D shapes (including the trapezoidal rule) and the surface area and volume of 3D solids. It is examined on Paper 2.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

The trapezoidal rule estimates the area under an irregular curve.
Volume and surface area formulas for common solids are in the formula and tables booklet.
Compound shapes are split into simpler shapes and added or subtracted.
Area and volume is examined on Paper 2.

Area and Volume explained

Basic Area Formulas

Essential area formulas you must know:

Rectangle: $A = l \times w$ (length × width)

Square: $A = s^2$ (side squared)

Triangle: $A = \frac{1}{2}bh$ (half base × height)

Parallelogram: $A = bh$ (base × height)

All areas are measured in square units (cm², m², etc.)

Basic Perimeter Formulas

Perimeter is the distance around a shape:

Rectangle: $P = 2l + 2w$ or $P = 2(l + w)$

Square: $P = 4s$

Triangle: $P = a + b + c$ (sum of all sides)

Any polygon: Add all side lengths

Perimeter is measured in linear units (cm, m, etc.)

Trapezium Area

A trapezium (or trapezoid) has one pair of parallel sides.

Area formula: $A = \frac{1}{2}(a + b)h$

where: - $a$ = length of first parallel side - $b$ = length of second parallel side - $h$ = perpendicular height between parallel sides

Memory aid: "Average of parallel sides times height"

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Key formulas

Name	Formula	Description
Area of Rectangle	$A = l \times w$	Area equals length times width
Area of Triangle	$A = \frac{1}{2}bh$	Area equals half base times height
Area of Circle	$A = \pi r^2$	Area of circle with radius r
Circumference	$C = 2\pi r$	Circumference of circle with radius r
Arc Length	$s = r\theta$	Arc length with angle θ in radians
Sector Area	$A = \frac{1}{2}r^2\theta$	Area of sector with angle θ in radians

Worked examples

Worked example 1

What is the area of a trapezium with parallel sides 6 m and 10 m, and height 4 m?

Use formula A = ½(a + b)h. A = ½(6 + 10) × 4 = ½(16) × 4 = 8 × 4 = 32 m²

Answer:

32 m²

Worked example 2

What is the area of a circle with radius 4 cm? (Use π ≈ 3.14)

Use formula A = πr². A = π(4)² = 16π ≈ 16 × 3.14 = 50.24 cm²

Answer:

50.24 cm²

Where students lose marks

Mixing up surface area and volume formulas.
Forgetting to convert units consistently (cm vs m).
Using the wrong number of strips in the trapezoidal rule.

Frequently asked questions

How does the trapezoidal rule work on the Leaving Cert?

Divide the region into equal-width strips, measure the height at each boundary, then apply the formula: h/2 × (first + last + 2 × middle values). The more strips, the more accurate the estimate.

Where do I find volume formulas?

Volume and surface area formulas for standard solids are provided in the exam formula and tables booklet.

Is area and volume on Paper 2?

Yes, area and volume is examined on Paper 2.

Authoritative sources

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