Volume of 3D Shapes

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Volume of 3D Shapes

We can use formulae to calculate the volume of common shapes.

Prism

Volume = area of base × length.

Cuboid

Remember that a cuboid is a prism with rectangular base.
- Volume = width × height × length = area of base × length.

Cylinder

The area of a circle is πr² so:
- V = πr² × h
Where h is the height of the cylinder.

Volume of 3D Shapes

Sphere

The volume of a sphere is:
- V = ⁴⁄₃πr³

Hemisphere

A hemisphere is half a sphere, so its volume is half that of a sphere:
- V = ²⁄₃πr³

Cone

The volume of a cone is V = ¹⁄₃πr²h
Where h is the vertical height of the cone (not the slant height).

Pyramid

The volume of a pyramid is V = ¹⁄₃ × base area × vertical height.

Frustum

A frustum of a cone is a cone with its top removed.
The part that is removed is also a cone so the volume is:
- V = V_{(whole cone)} - V_{(removed cone)}

Number

1.1

Using Numbers

1.2

Fractions, Decimals & Percentages

1.2.1Adding Fractions 1.2.2Subtracting Fractions 1.2.3Multiplying Fractions 1.2.4Dividing Fractions 1.2.5Mixed Fractions 1.2.6Calculating Percentages of Numbers 1.2.7Calculating Fractions of Numbers 1.2.8Multiply Decimals 1.2.9Place Value 1.2.10Recurring Decimals 1.2.11Exam-Style Questions - Recurring Decimals 1.2.12Convert Fractions & Terminating Decimals 1.2.13Convert Recurring Decimals 1.2.14Convert Decimals & Percentages 1.2.15Compare Decimals 1.2.16Compare Fractions 1.2.17Exam-Style Questions - Percentages 1.2.18End of Topic Test - Fractions & Decimals 1.2.19Diagnostic Misconceptions - Adding Fractions

1.3

Powers & Roots

1.3.1Powers 1.3.2Roots 1.3.3Indices Advanced 1.3.4Estimating Powers 1.3.5Estimating Roots 1.3.6Multiplying Powers 1.3.7Dividing Powers 1.3.8Power Raised to Power 1.3.9Fractions Raised to a Power 1.3.10Numbers Raised to Negative Powers 1.3.11Roots of Numbers 1.3.12Numbers to the Power of 1 & 0 1.3.13Surds 1.3.14Simplify Surds 1.3.15Rationalise Denominator 1.3.16Exam-Style Questions - Indices 1.3.17End of Topic Test - Powers & Roots 1.3.18Diagnostic Misconceptions - Squaring 1.3.19Diagnostic Misconceptions - Simplifying Surds

1.4

Accuracy

1.4.1Decimal Rounding 1.4.2Significant Figure Rounding 1.4.3Estimation 1.4.4Truncation 1.4.5Limits of Accuracy 1.4.6Bounds Addition 1.4.7Bounds Subtraction 1.4.8Bounds Multiplication 1.4.9Bounds Division 1.4.10End of Topic Test - Rounding & Estimation 1.4.11Diagnostic Misconceptions - Upper Bounds

Algebra

2.1

Introduction to Algebra

2.1.1Algebra Basics 2.1.2Substituting Values 2.1.3Algebraic Vocabulary 2.1.4Collecting Terms 2.1.5End of Topic Test - Algebra Basics & Power Rules 2.1.6Expanding Brackets I 2.1.7Common Factors 2.1.8Expanding Brackets II 2.1.9Rearrange For x 2.1.10Rearrange with Powers 2.1.11Exam-Style Questions - Basics of Algebra 2.1.12End of Topic Test - Expanding & Rearranging

2.2

Manipulating Algebra

2.2.1How to Factorise 2.2.2Factorising Quadratics 2.2.3Algebraic Fractions - Addition & Subtraction 2.2.4Algebraic Fractions - Multiplication & Division 2.2.5Diagnostic Misconceptions - Common Denominators 2.2.6Factorising Quadratics (tt)2.2.7Simplify algebraic fractions - cancelling factors 2.2.8Simplify algebraic fractions - factorisation

2.3

Proofs & Functions

2.3.1Identities & Proofs 2.3.2Exam-Style Questions - Algebraic Proof 2.3.3End of Topic Test - Algebra & Proofs 2.3.4Functions 2.3.5Combine Functions 2.3.6Composite Functions 2.3.7Inverse Functions 2.3.8Functions (tt)

2.4

Straight Line Graphs

2.4.1Straight Line Graphs 2.4.2y = mx + c 2.4.3Grad + YIntercept 2.4.4Exam-Style Questions - Straight Line Graphs 2.4.5End of Topic Test - Functions & Straight Lines 2.4.6Diagnostic Misconceptions - The x Coefficient

2.5

Common Graphs

2.5.1Shape of Quadratic Graphs 2.5.2Roots of a Quadratic Graph 2.5.3Completing Squares 2.5.4Turning Points 2.5.5Common Functions 1 2.5.6Common Functions 2 2.5.7Trigonometric functions 2.5.8End of Topic Test - Common Graphs

2.6

Transformations & Tangents

2.6.1Transformations 2.6.2Tangent on a Circle

2.7

Properties of Graphs

2.7.1Estimate Gradient 2.7.2Estimate Area 2.7.3Represent Gradient & Area 2.7.4Distance-Time Graph 2.7.5Velocity-Time Graph 2.7.6End of Topic Test - Transformations & Graphs

2.8

Solving Equations

2.8.1Solve Linear Equations 2.8.2Linear Equations with Fractions 2.8.3Solve Linear Equations Graphically 2.8.4Quadratic ±2.8.5Factorise Quadratics 2.8.6Quadratic Formula 2.8.7Complete the Square 2.8.8Solve Quadratics Graphically 2.8.9End of Topic Test - Solving Equations 2.8.10Simultaneous Equations 2.8.11Quadratic Simultaneous Equations 2.8.12Simultaneous Equations - Graphs 2.8.13Simultaneous Equations - Word Problems 2.8.14Approximate Solutions - Number of Iterations 2.8.15Approximate Solutions - Iterations to 1 DP 2.8.16Approximate Solutions - Iteration Formula 2.8.17Exam-Style Questions - Simultaneous Equations 2.8.18End of Topic Test - Simultaneous Equations 2.8.19Diagnostic Misconceptions - Solving Quadratics

2.9

Inequalities

2.9.1Solving Inequalities 2.9.2Solving Double Inequalities 2.9.3Solving Quadratic Inequalities 2.9.4Representing Inequalities 2.9.5Exam-Style Questions - Inequalities 2.9.6End of Topic Test - Inequalities

2.10

Sequences

2.10.1Nth Term Rules 2.10.2Recurrence Relationships 2.10.3Find Nth Term in Linear Sequence 2.10.4Check If a Number is In a Sequence 2.10.5Exam-Style Questions - Sequences 2.10.6Arithmetic & Geometric Progressions 2.10.7Key Sequences 2.10.8Quadratic Sequences 2.10.9Find Nth Term in Quadratic Sequence 2.10.10Exam-Style Questions - Geometric Series 2.10.11End of Topic Test - Sequences 2.10.12Nth Term Rules (tt)

Ratio

3.1

Ratios in Practice

3.1.1Converting Between Units 3.1.2Formula Triangles 3.1.3Scale Factors 3.1.4Ratios in Shapes 3.1.5Diagnostic Misconceptions - Writing Ratios

3.2

Manipulating Ratios

3.2.1Simplifying Ratios 3.2.2Turn Ratios into Equations 3.2.3Ratio Notation & Reduction 3.2.4End of Topic Test - Units & Ratios 3.2.5Part : Part Ratio 3.2.6Part : Whole Ratio 3.2.7Two Ratios 3.2.8Differences in Ratios 3.2.9Fractions in Ratios 3.2.10Numbers as Fractions 3.2.11Ratios to Fractions 3.2.12Exam-Style Questions - Ratios 3.2.13End of Topic Test - Fractions & Ratios 3.2.14Diagnostic Misconceptions - Combining Ratios

3.3

Percentage & Interest

3.3.1Convert Percentages to Decimals 3.3.2Convert Percentages to Fractions 3.3.3Find the Percentage 3.3.4More/Less Percentage 3.3.5Price from Percentage Change 3.3.6Calculate Percentage Change from Price 3.3.7End of Topic Test - Percentages 3.3.8Profit & Interest & Discount 3.3.9Original Price after Discounts 3.3.10Simple Interest 3.3.11Compound Growth & Decay 3.3.12Compound Interest 3.3.13Compound Interest - After How Many Days?3.3.14End of Topic Test - Percentages in Finance

3.4

Proportion

3.4.1Proportion 3.4.2Direct Proportion 3.4.3Inverse Proportion 3.4.4Exam-Style Questions - Proportionality

3.5

Gradient

3.5.1Gradient on a Straight Line 3.5.2Direct & Inverse Proportions as Graphs 3.5.3Gradient of a Curve 3.5.4Positive & Negative Gradients 3.5.5End of Topic Test - Proportion & Gradient

Geometry

4.1

Introduction to Geometry

4.1.1Geometry Basics 4.1.2Geometry Basics 2 4.1.3Regular Polygons 4.1.4Triangles & Angles 4.1.5Midpoint 4.1.6Bisectors & Perpendiculars 4.1.7Locus 4.1.8Angle rules 4.1.9Parallel Angle Rules 4.1.10Angles in a Polygon 4.1.11Angles in a Triangle 4.1.12Exam-Style Questions - Angles in Shapes 4.1.13End of Topic Test - Geometry

4.2

Triangles & Quadrilaterals

4.2.1Types of Triangle 4.2.2Angles & Triangles 4.2.3Congruence 4.2.4Similarity 4.2.5Quadrilaterals

4.3

Transformations

4.3.1Types of Transformations 4.3.2Invariant Points 4.3.3Effects of Transformations 4.3.4Exam-Style Questions - Reflection

4.4

Circle Basics

4.4.1Properties of a Circle 1 4.4.2Chords 4.4.3Properties of a Circle 2 4.4.4Area of a Circle 4.4.5Circumference of a Circle 4.4.6Fractions of Circles 4.4.7Exam-Style Questions - Circle Formulae 4.4.8End of Topic Test - Triangles & Circles

4.5

Circle Theorems

4.5.1Perpendicular Bisector of a Chord 4.5.2Angle at the Centre 4.5.3Angle of a Semicircle 4.5.4Opposite Angles in Cyclic Quadrilaterals 4.5.5Tangent Lengths 4.5.6Angle in the Same Segment 4.5.7Alternate Segment Theorem 4.5.8Exam-Style Questions - Circle Theorems 4.5.9Diagnostic Misconceptions - Circumference & Radius

4.6

Measurements & Units

4.6.1Length 4.6.2Area 4.6.3Volume 4.6.4Capacity 4.6.5Time 4.6.6Money 4.6.7Mass 4.6.8Bearings 4.6.9Scale Factor & Enlargement 4.6.10Exam-Style Questions - Enlargements 4.6.11Diagnostic Misconceptions - Hours

4.7

Calculating Area

4.7.1Area of 2D Shapes 4.7.2Exam-Style Questions - Area 4.7.3End of Topic Test - Circle Theorems, Units & Area

4.8

Triangle Formulae

4.8.1Pythagoras' Theorem 4.8.2Pythagoras in 3D 4.8.3Trigonometry 4.8.4Exact Values 4.8.5Sine Rule 4.8.6Cosine Rule 4.8.7Area of a Triangle 4.8.8Exam-Style Questions - Trigonometry & Area 4.8.9Diagnostic Misconceptions - Bearings from Angles 4.8.10Diagnostic Misconceptions - Pythagoras' Theorem

4.9

3D Shapes

4.9.1Types of 3D Shape 4.9.2Plans & Elevations 4.9.3Nets 4.9.4Volume of 3D Shapes 4.9.5Volume of a Cuboid 4.9.6Volume of a Cylinder 4.9.7Volume of a Sphere 4.9.8Volume of a Hemisphere 4.9.9Volume of a Cone 4.9.10Volume of a Pyramid 4.9.11Surface Area & Composite Shapes 4.9.12Surface Area of a Cuboid 4.9.13Surface Area of a Cylinder 4.9.14Surface Area of a Sphere/Hemisphere 4.9.15Surface Area of a Cone 4.9.16Exam-Style Questions - Volume

4.10

Vectors

4.10.1Vectors Introduction 4.10.2Vectors 4.10.3Translation 4.10.4Scalar 4.10.5Exam-Style Questions - Vectors 4.10.6End of Topic Test - Trigonometry, Shapes & Vectors 4.10.7Diagnostic Misconceptions - Column Vectors

Probability

5.1

Introduction to Probability

5.1.1Randomness & Fairness 5.1.2Sum of Probabilities 5.1.3Relative & Expected Frequency 5.1.4Frequency Tables 5.1.5Using Frequency Trees 5.1.6Complete Frequency Trees

5.2

Representing Probabilities

5.2.1Probability 5.2.2Probability Scale 5.2.3Sample Space Diagrams 5.2.4Sets 5.2.5Venn Diagrams 5.2.6Probabilities in Venn Diagrams 5.2.7Exam-Style Questions - Venn Diagrams

5.3

Probability Calculations

5.3.1Theoretical Probabilities 5.3.2Multiple Events 5.3.3Probability Calculations 5.3.4Dependent Events 5.3.5Independent Events 5.3.6Probability Tree Diagram 5.3.7Complex Tree Diagrams 5.3.8Conditional Probability 5.3.9Conditional Probability From a Two Way Table 5.3.10Conditional Probability Venn Diagrams 5.3.11Conditional Probability Tree Diagrams 5.3.12Exam-Style Questions - Probability 5.3.13End of Topic Test - Probability 5.3.14Diagnostic Misconceptions - Denominator

Statistics

Jump to other topics

Number

1.1

Using Numbers

1.2

Fractions, Decimals & Percentages

1.3

Powers & Roots

1.4

Accuracy

Algebra

2.1

Introduction to Algebra

2.2

Manipulating Algebra

2.3

Proofs & Functions

2.4

Straight Line Graphs

2.5

Common Graphs

2.6

Transformations & Tangents

2.6.1Transformations 2.6.2Tangent on a Circle

2.7

Properties of Graphs

2.7.1Estimate Gradient 2.7.2Estimate Area 2.7.3Represent Gradient & Area 2.7.4Distance-Time Graph 2.7.5Velocity-Time Graph 2.7.6End of Topic Test - Transformations & Graphs

2.8

Solving Equations

2.9

Inequalities

2.10

Sequences

Ratio

3.1

Ratios in Practice

3.1.1Converting Between Units 3.1.2Formula Triangles 3.1.3Scale Factors 3.1.4Ratios in Shapes 3.1.5Diagnostic Misconceptions - Writing Ratios

3.2

Manipulating Ratios

3.3

Percentage & Interest

3.4

Proportion

3.4.1Proportion 3.4.2Direct Proportion 3.4.3Inverse Proportion 3.4.4Exam-Style Questions - Proportionality

3.5

Gradient

3.5.1Gradient on a Straight Line 3.5.2Direct & Inverse Proportions as Graphs 3.5.3Gradient of a Curve 3.5.4Positive & Negative Gradients 3.5.5End of Topic Test - Proportion & Gradient

Geometry

4.1

Introduction to Geometry

4.2

Triangles & Quadrilaterals

4.2.1Types of Triangle 4.2.2Angles & Triangles 4.2.3Congruence 4.2.4Similarity 4.2.5Quadrilaterals

4.3

Transformations

4.3.1Types of Transformations 4.3.2Invariant Points 4.3.3Effects of Transformations 4.3.4Exam-Style Questions - Reflection

4.4

Circle Basics

4.5

Circle Theorems

4.6

Measurements & Units

4.7

Calculating Area

4.7.1Area of 2D Shapes 4.7.2Exam-Style Questions - Area 4.7.3End of Topic Test - Circle Theorems, Units & Area

4.8

Triangle Formulae

4.9

3D Shapes

4.10

Vectors

Probability

5.1

Introduction to Probability

5.1.1Randomness & Fairness 5.1.2Sum of Probabilities 5.1.3Relative & Expected Frequency 5.1.4Frequency Tables 5.1.5Using Frequency Trees 5.1.6Complete Frequency Trees

5.2

Representing Probabilities

5.2.1Probability 5.2.2Probability Scale 5.2.3Sample Space Diagrams 5.2.4Sets 5.2.5Venn Diagrams 5.2.6Probabilities in Venn Diagrams 5.2.7Exam-Style Questions - Venn Diagrams

5.3

Probability Calculations

Statistics

Practice questions on Volume of 3D Shapes

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

1
V = πr<sup>2</sup> × height</br>What is this the formula for?Multiple choice
2
What is the formula for the volume of a prism?Multiple choice
3
What is the formula for the volume of a cuboid?Multiple choice
4
What is the formula for the volume of a cylinder?Multiple choice
5
What is the formula for the volume of a sphere?Multiple choice

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Nets

Volume of a Cuboid

Volume of 3D Shapes

Number

Using Numbers

Fractions, Decimals & Percentages

Powers & Roots

Accuracy

Algebra

Introduction to Algebra

Manipulating Algebra

Proofs & Functions

Straight Line Graphs

Common Graphs

Transformations & Tangents

Properties of Graphs

Solving Equations

Inequalities

Sequences

Ratio

Ratios in Practice

Manipulating Ratios

Percentage & Interest

Proportion

Gradient

Geometry

Introduction to Geometry

Triangles & Quadrilaterals

Transformations

Circle Basics

Circle Theorems

Measurements & Units

Calculating Area

Triangle Formulae

3D Shapes

Vectors

Probability

Introduction to Probability

Representing Probabilities

Probability Calculations

Statistics

Introduction to Statistics

Charts & Diagrams

Comparing Datasets

Number

Using Numbers

Fractions, Decimals & Percentages

Powers & Roots

Accuracy

Algebra

Introduction to Algebra

Manipulating Algebra

Proofs & Functions

Straight Line Graphs

Common Graphs

Transformations & Tangents

Properties of Graphs

Solving Equations

Inequalities

Sequences

Ratio

Ratios in Practice

Manipulating Ratios

Percentage & Interest

Proportion

Gradient

Geometry

Introduction to Geometry

Triangles & Quadrilaterals

Transformations

Circle Basics

Circle Theorems

Measurements & Units

Calculating Area

Triangle Formulae

3D Shapes

Vectors

Probability

Introduction to Probability

Representing Probabilities

Probability Calculations

Statistics