2.7.2

Estimate Area

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Estimating Area Under a Graph

Straight lines only - exact calculation

Straight lines only - exact calculation

  • To calculate the area under a graph, split the area into trapeziums and rectangles and work out the area of each section.
  • Use the formula:
    • Area (Trapezium) = 1⁄2 × base × (a + b)
    • Where a and b are the sides of the trapezium.
Curved line - estimate

Curved line - estimate

  • If the graph is not made up of straight lines you can estimate the area by splitting the graph approximately into trapeziums and doing the same calculation.
Overestimate or underestimate

Overestimate or underestimate

  • An area estimate is an overestimate when the trapeziums are above the curve and an underestimate when the trapeziums are below the curve.
  • In the example the first and second sections are underestimates and the third is an overestimate.
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1

Number

1.1

Using Numbers

1.1.1Equal Symbols1.1.2Negative Numbers1.1.3BIDMAS1.1.4Inverse Operations1.1.5End of Topic Test- Ordering1.1.6Factors1.1.7Multiples1.1.8Highest Common Factor1.1.9Lowest Common Multiple1.1.10Number of Ways to Do a Task1.1.11Number of Combinations1.1.12Number of Ways To Arrange Objects1.1.13End of Topic Test - Factors, Multiples & Combos1.1.14Standard Form1.1.15Ordinary Numbers to Standard Form1.1.16Addition in Standard Form1.1.17Subtraction in Standard Form1.1.18Multiplication in Standard Form1.1.19Division in Standard Form1.1.20Exam-Style Questions - Standard Form & Primes1.1.21Long Division1.1.22Prime Numbers1.1.23End of Topic Test - Standard Form & Long Division1.1.24Diagnostic Misconceptions - 1 NOT a Prime Number1.1.25Diagnostic Misconceptions - BIDMAS1.1.26Diagnostic Misconceptions - Adding Negatives
1.2

Fractions, Decimals & Percentages

1.2.1Adding Fractions1.2.2Subtracting Fractions1.2.3Multiplying Fractions1.2.4Dividing Fractions1.2.5Mixed Fractions1.2.6Calculating Percentages of Numbers1.2.7Calculating Fractions of Numbers1.2.8Multiply Decimals1.2.9Place Value1.2.10Recurring Decimals1.2.11Exam-Style Questions - Recurring Decimals1.2.12Convert Fractions & Terminating Decimals1.2.13Convert Recurring Decimals1.2.14Convert Decimals & Percentages1.2.15Compare Decimals1.2.16Compare Fractions1.2.17Exam-Style Questions - Percentages1.2.18End of Topic Test - Fractions & Decimals1.2.19Diagnostic Misconceptions - Adding Fractions
1.3

Powers & Roots

1.3.1Powers1.3.2Roots1.3.3Indices Advanced1.3.4Estimating Powers1.3.5Estimating Roots1.3.6Multiplying Powers1.3.7Dividing Powers1.3.8Power Raised to Power1.3.9Fractions Raised to a Power1.3.10Numbers Raised to Negative Powers1.3.11Roots of Numbers1.3.12Numbers to the Power of 1 & 01.3.13Surds1.3.14Simplify Surds1.3.15Rationalise Denominator1.3.16Exam-Style Questions - Indices1.3.17End of Topic Test - Powers & Roots1.3.18Diagnostic Misconceptions - Squaring1.3.19Diagnostic Misconceptions - Simplifying Surds
1.4

Accuracy

1.4.1Decimal Rounding1.4.2Significant Figure Rounding1.4.3Estimation1.4.4Truncation1.4.5Limits of Accuracy1.4.6Bounds Addition1.4.7Bounds Subtraction1.4.8Bounds Multiplication1.4.9Bounds Division1.4.10End of Topic Test - Rounding & Estimation1.4.11Diagnostic Misconceptions - Upper Bounds
2

Algebra

2.1

Introduction to Algebra

2.1.1Algebra Basics2.1.2Substituting Values2.1.3Algebraic Vocabulary2.1.4Collecting Terms2.1.5End of Topic Test - Algebra Basics & Power Rules2.1.6Expanding Brackets I2.1.7Common Factors2.1.8Expanding Brackets II2.1.9Rearrange For x2.1.10Rearrange with Powers2.1.11Exam-Style Questions - Basics of Algebra2.1.12End of Topic Test - Expanding & Rearranging
2.2

Manipulating Algebra

2.2.1How to Factorise2.2.2Factorising Quadratics2.2.3Algebraic Fractions - Addition & Subtraction2.2.4Algebraic Fractions - Multiplication & Division2.2.5Diagnostic Misconceptions - Common Denominators2.2.6Factorising Quadratics (tt)2.2.7Simplify algebraic fractions - cancelling factors2.2.8Simplify algebraic fractions - factorisation
2.3

Proofs & Functions

2.3.1Identities & Proofs2.3.2Exam-Style Questions - Algebraic Proof2.3.3End of Topic Test - Algebra & Proofs2.3.4Functions2.3.5Combine Functions2.3.6Composite Functions2.3.7Inverse Functions2.3.8Functions (tt)
2.4

Straight Line Graphs

2.4.1Straight Line Graphs2.4.2y = mx + c2.4.3Grad + YIntercept2.4.4Exam-Style Questions - Straight Line Graphs2.4.5End of Topic Test - Functions & Straight Lines2.4.6Diagnostic Misconceptions - The x Coefficient
2.5

Common Graphs

2.5.1Shape of Quadratic Graphs2.5.2Roots of a Quadratic Graph2.5.3Completing Squares2.5.4Turning Points2.5.5Common Functions 12.5.6Common Functions 22.5.7Trigonometric functions2.5.8End of Topic Test - Common Graphs
2.6

Transformations & Tangents

2.6.1Transformations2.6.2Tangent on a Circle
2.7

Properties of Graphs

2.7.1Estimate Gradient2.7.2Estimate Area2.7.3Represent Gradient & Area2.7.4Distance-Time Graph2.7.5Velocity-Time Graph2.7.6End of Topic Test - Transformations & Graphs
2.8

Solving Equations

2.8.1Solve Linear Equations2.8.2Linear Equations with Fractions2.8.3Solve Linear Equations Graphically2.8.4Quadratic ±2.8.5Factorise Quadratics2.8.6Quadratic Formula2.8.7Complete the Square2.8.8Solve Quadratics Graphically2.8.9End of Topic Test - Solving Equations2.8.10Simultaneous Equations2.8.11Quadratic Simultaneous Equations2.8.12Simultaneous Equations - Graphs2.8.13Simultaneous Equations - Word Problems2.8.14Approximate Solutions - Number of Iterations2.8.15Approximate Solutions - Iterations to 1 DP2.8.16Approximate Solutions - Iteration Formula2.8.17Exam-Style Questions - Simultaneous Equations2.8.18End of Topic Test - Simultaneous Equations2.8.19Diagnostic Misconceptions - Solving Quadratics
2.9

Inequalities

2.9.1Solving Inequalities2.9.2Solving Double Inequalities2.9.3Solving Quadratic Inequalities2.9.4Representing Inequalities2.9.5Exam-Style Questions - Inequalities2.9.6End of Topic Test - Inequalities
2.10

Sequences

2.10.1Nth Term Rules2.10.2Recurrence Relationships2.10.3Find Nth Term in Linear Sequence2.10.4Check If a Number is In a Sequence2.10.5Exam-Style Questions - Sequences2.10.6Arithmetic & Geometric Progressions2.10.7Key Sequences2.10.8Quadratic Sequences2.10.9Find Nth Term in Quadratic Sequence2.10.10Exam-Style Questions - Geometric Series2.10.11End of Topic Test - Sequences2.10.12Nth Term Rules (tt)
3

Ratio

3.1

Ratios in Practice

3.1.1Converting Between Units3.1.2Formula Triangles3.1.3Scale Factors3.1.4Ratios in Shapes3.1.5Diagnostic Misconceptions - Writing Ratios
3.2

Manipulating Ratios

3.2.1Simplifying Ratios3.2.2Turn Ratios into Equations3.2.3Ratio Notation & Reduction3.2.4End of Topic Test - Units & Ratios3.2.5Part : Part Ratio3.2.6Part : Whole Ratio3.2.7Two Ratios3.2.8Differences in Ratios3.2.9Fractions in Ratios3.2.10Numbers as Fractions3.2.11Ratios to Fractions3.2.12Exam-Style Questions - Ratios3.2.13End of Topic Test - Fractions & Ratios3.2.14Diagnostic Misconceptions - Combining Ratios
3.3

Percentage & Interest

3.3.1Convert Percentages to Decimals3.3.2Convert Percentages to Fractions3.3.3Find the Percentage3.3.4More/Less Percentage3.3.5Price from Percentage Change3.3.6Calculate Percentage Change from Price3.3.7End of Topic Test - Percentages3.3.8Profit & Interest & Discount3.3.9Original Price after Discounts3.3.10Simple Interest3.3.11Compound Growth & Decay3.3.12Compound Interest3.3.13Compound Interest - After How Many Days?3.3.14End of Topic Test - Percentages in Finance
3.4

Proportion

3.4.1Proportion3.4.2Direct Proportion3.4.3Inverse Proportion3.4.4Exam-Style Questions - Proportionality
3.5

Gradient

3.5.1Gradient on a Straight Line3.5.2Direct & Inverse Proportions as Graphs3.5.3Gradient of a Curve3.5.4Positive & Negative Gradients3.5.5End of Topic Test - Proportion & Gradient
4

Geometry

4.1

Introduction to Geometry

4.1.1Geometry Basics4.1.2Geometry Basics 24.1.3Regular Polygons4.1.4Triangles & Angles4.1.5Midpoint4.1.6Bisectors & Perpendiculars4.1.7Locus4.1.8Angle rules4.1.9Parallel Angle Rules4.1.10Angles in a Polygon4.1.11Angles in a Triangle4.1.12Exam-Style Questions - Angles in Shapes4.1.13End of Topic Test - Geometry
4.2

Triangles & Quadrilaterals

4.2.1Types of Triangle4.2.2Angles & Triangles4.2.3Congruence4.2.4Similarity4.2.5Quadrilaterals
4.3

Transformations

4.3.1Types of Transformations4.3.2Invariant Points4.3.3Effects of Transformations4.3.4Exam-Style Questions - Reflection
4.4

Circle Basics

4.4.1Properties of a Circle 14.4.2Chords4.4.3Properties of a Circle 24.4.4Area of a Circle4.4.5Circumference of a Circle4.4.6Fractions of Circles4.4.7Exam-Style Questions - Circle Formulae4.4.8End of Topic Test - Triangles & Circles
4.5

Circle Theorems

4.5.1Perpendicular Bisector of a Chord4.5.2Angle at the Centre4.5.3Angle of a Semicircle4.5.4Opposite Angles in Cyclic Quadrilaterals4.5.5Tangent Lengths4.5.6Angle in the Same Segment4.5.7Alternate Segment Theorem4.5.8Exam-Style Questions - Circle Theorems4.5.9Diagnostic Misconceptions - Circumference & Radius
4.6

Measurements & Units

4.6.1Length4.6.2Area4.6.3Volume4.6.4Capacity4.6.5Time4.6.6Money4.6.7Mass4.6.8Bearings4.6.9Scale Factor & Enlargement4.6.10Exam-Style Questions - Enlargements4.6.11Diagnostic Misconceptions - Hours
4.7

Calculating Area

4.7.1Area of 2D Shapes4.7.2Exam-Style Questions - Area4.7.3End of Topic Test - Circle Theorems, Units & Area
4.8

Triangle Formulae

4.8.1Pythagoras' Theorem4.8.2Pythagoras in 3D4.8.3Trigonometry4.8.4Exact Values4.8.5Sine Rule4.8.6Cosine Rule4.8.7Area of a Triangle4.8.8Exam-Style Questions - Trigonometry & Area4.8.9Diagnostic Misconceptions - Bearings from Angles4.8.10Diagnostic Misconceptions - Pythagoras' Theorem
4.9

3D Shapes

4.9.1Types of 3D Shape4.9.2Plans & Elevations4.9.3Nets4.9.4Volume of 3D Shapes4.9.5Volume of a Cuboid4.9.6Volume of a Cylinder4.9.7Volume of a Sphere4.9.8Volume of a Hemisphere4.9.9Volume of a Cone4.9.10Volume of a Pyramid4.9.11Surface Area & Composite Shapes4.9.12Surface Area of a Cuboid4.9.13Surface Area of a Cylinder4.9.14Surface Area of a Sphere/Hemisphere4.9.15Surface Area of a Cone4.9.16Exam-Style Questions - Volume
4.10

Vectors

4.10.1Vectors Introduction4.10.2Vectors4.10.3Translation4.10.4Scalar4.10.5Exam-Style Questions - Vectors4.10.6End of Topic Test - Trigonometry, Shapes & Vectors4.10.7Diagnostic Misconceptions - Column Vectors
5

Probability

5.1

Introduction to Probability

5.1.1Randomness & Fairness5.1.2Sum of Probabilities5.1.3Relative & Expected Frequency5.1.4Frequency Tables5.1.5Using Frequency Trees5.1.6Complete Frequency Trees
5.2

Representing Probabilities

5.2.1Probability5.2.2Probability Scale5.2.3Sample Space Diagrams5.2.4Sets5.2.5Venn Diagrams5.2.6Probabilities in Venn Diagrams5.2.7Exam-Style Questions - Venn Diagrams
5.3

Probability Calculations

5.3.1Theoretical Probabilities5.3.2Multiple Events5.3.3Probability Calculations5.3.4Dependent Events5.3.5Independent Events5.3.6Probability Tree Diagram5.3.7Complex Tree Diagrams5.3.8Conditional Probability5.3.9Conditional Probability From a Two Way Table5.3.10Conditional Probability Venn Diagrams5.3.11Conditional Probability Tree Diagrams5.3.12Exam-Style Questions - Probability5.3.13End of Topic Test - Probability5.3.14Diagnostic Misconceptions - Denominator
6

Statistics

6.1

Introduction to Statistics

6.1.1Statistics Basics6.1.2Statistics Key Terms6.1.3Frequency Tables6.1.4Exam-Style Questions - Statistics & Probability6.1.5Diagnostic Misconceptions - Median6.1.6Diagnostic Misconceptions - Mode6.1.7Diagnostic Misconceptions - Mean
6.2

Charts & Diagrams

6.2.1Bar Charts6.2.2Pie Charts6.2.3Stem & Leaf Diagrams6.2.4Back to Back Stem & Leaf Diagrams6.2.5End of Topic Test - Statistics6.2.6Pictograms6.2.7Time Series Graphs6.2.8Histograms6.2.9Histograms 26.2.10Frequency Polygons6.2.11Cumulative Frequency6.2.12Cumulative Frequency 26.2.13Scatter Graphs6.2.14Scatter Graphs - Predictions6.2.15Correlation & Causation6.2.16Exam-Style Questions - Cumulative Frequency
6.3

Comparing Datasets

6.3.1Comparing Datasets6.3.2End of Topic - Datasets

Practice questions on Estimate Area

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

  1. 1
    Will the triangle underneath the graph between x = 3 & x = 6 be an under or overestimate of the true area?Multiple choice
  2. 2
    Will the triangle underneath the graph between x = 0 & x = 3 be an under or overestimate of the true area?Multiple choice
  3. 3
    Will the trapezium approximating the graph between x = 6 & x = 9 be an under or overestimate of the true area?Multiple choice
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Estimate Gradient

Represent Gradient & Area