Learn on PengiSaxon Math, Course 2Chapter 4: Lessons 31-40, Investigation 4

Lessons 37: Area of a Triangle, Rectangular Area, Part 2

In this Grade 7 Saxon Math Course 2 lesson, students learn how to calculate the area of a triangle using the formula A = ½bh, understanding that a triangle's area is always half that of a rectangle with the same base and height. The lesson covers identifying the base and perpendicular height (altitude) for right, acute, and obtuse triangles, including cases where the height falls outside the triangle. Students apply the formula through hands-on activities and worked examples involving real measurements in square centimeters.

Section 1

📘 Area of a Triangle

New Concept

To find the area of a triangle with a base of bb and a height of hh, we find half of the product of bb and hh. We show two formulas for finding the area of a triangle.

Area of a triangle=12bh \text{Area of a triangle} = \frac{1}{2}bh
Area of a triangle=bh2 \text{Area of a triangle} = \frac{bh}{2}

What’s next

This is just the foundation. Soon, you'll tackle worked examples on triangle area and also find the area of complex shapes by breaking them into rectangles.

Section 2

Area of a Triangle

Property

To find the area of a triangle, you can use two different but equally cool formulas: Area=12bh\text{Area} = \frac{1}{2}bh or Area=bh2\text{Area} = \frac{bh}{2}.

Examples

  • For a triangle with a base of 10 m and height of 6 m: Area=10⋅62=30 m2\text{Area} = \frac{10 \cdot 6}{2} = 30 \text{ m}^2.
  • For a right triangle with sides of 5 ft and 4 ft: Area=12(5⋅4)=10 ft2\text{Area} = \frac{1}{2}(5 \cdot 4) = 10 \text{ ft}^2.

Explanation

A triangle is really just a rectangle that has been sliced in half. So, we find the area of the full rectangle and divide by two. Easy!

Section 3

Rectangular Area, Part 2

Property

To find the area of a weirdly shaped room, you can divide the shape into smaller, familiar rectangles. Calculate the area of each smaller part, and then add them all up for the total.

Examples

  • An L-shape split into a 10×510 \times 5 and a 3×73 \times 7 rectangle: (10⋅5)+(3⋅7)=50+21=71 ft2(10 \cdot 5) + (3 \cdot 7) = 50 + 21 = 71 \text{ ft}^2.
  • Split into a 12×412 \times 4 and 8×88 \times 8 rectangle: (12⋅4)+(8⋅8)=48+64=112 in2(12 \cdot 4) + (8 \cdot 8) = 48 + 64 = 112 \text{ in}^2.

Explanation

Think of it like building with LEGOs! Find the area of each separate rectangular piece and just add them together for the grand total.

Section 4

Finding Area with Subtraction

Property

See a complex shape as a big rectangle with a piece missing. Find the area of the whole rectangle, then subtract the area of the missing chunk to get your final answer.

Examples

  • A 10×1210 \times 12 rectangle with a 3×43 \times 4 piece missing: (10⋅12)−(3⋅4)=120−12=108 m2(10 \cdot 12) - (3 \cdot 4) = 120 - 12 = 108 \text{ m}^2.
  • A 20×820 \times 8 shape missing a 5×55 \times 5 square: (20⋅8)−(5⋅5)=160−25=135 m2(20 \cdot 8) - (5 \cdot 5) = 160 - 25 = 135 \text{ m}^2.

Explanation

It's the donut method! You find the area of the whole thing as if it were solid, then just subtract the empty hole in the middle.

Book overview

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Chapter 4: Lessons 31-40, Investigation 4

  1. Lesson 1

    Lessons 31: Reading and Writing Decimal Numbers

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  2. Lesson 2

    Lessons 32: Metric System

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  3. Lesson 3

    Lessons 33: Comparing Decimals, Rounding Decimals

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  4. Lesson 4

    Lessons 34: Decimal Numbers on the Number Line

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  5. Lesson 5

    Lessons 35: Adding, Subtracting, Multiplying, and Dividing Decimal Numbers

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  6. Lesson 6

    Lessons 36: Ratio, Sample Space

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  7. Lesson 7Current

    Lessons 37: Area of a Triangle, Rectangular Area, Part 2

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  8. Lesson 8

    Lessons 38: Interpreting Graphs

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  9. Lesson 9

    Lessons 39: Proportions

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  10. Lesson 10

    Lessons 40: Sum of the Angle Measures of a Triangle, Angle Pairs

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  11. Lesson 11

    Investigation 4: Stem-and-Leaf Plots, Box-and-Whisker Plots

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Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Area of a Triangle

New Concept

To find the area of a triangle with a base of bb and a height of hh, we find half of the product of bb and hh. We show two formulas for finding the area of a triangle.

Area of a triangle=12bh \text{Area of a triangle} = \frac{1}{2}bh
Area of a triangle=bh2 \text{Area of a triangle} = \frac{bh}{2}

What’s next

This is just the foundation. Soon, you'll tackle worked examples on triangle area and also find the area of complex shapes by breaking them into rectangles.

Section 2

Area of a Triangle

Property

To find the area of a triangle, you can use two different but equally cool formulas: Area=12bh\text{Area} = \frac{1}{2}bh or Area=bh2\text{Area} = \frac{bh}{2}.

Examples

  • For a triangle with a base of 10 m and height of 6 m: Area=10⋅62=30 m2\text{Area} = \frac{10 \cdot 6}{2} = 30 \text{ m}^2.
  • For a right triangle with sides of 5 ft and 4 ft: Area=12(5⋅4)=10 ft2\text{Area} = \frac{1}{2}(5 \cdot 4) = 10 \text{ ft}^2.

Explanation

A triangle is really just a rectangle that has been sliced in half. So, we find the area of the full rectangle and divide by two. Easy!

Section 3

Rectangular Area, Part 2

Property

To find the area of a weirdly shaped room, you can divide the shape into smaller, familiar rectangles. Calculate the area of each smaller part, and then add them all up for the total.

Examples

  • An L-shape split into a 10×510 \times 5 and a 3×73 \times 7 rectangle: (10⋅5)+(3⋅7)=50+21=71 ft2(10 \cdot 5) + (3 \cdot 7) = 50 + 21 = 71 \text{ ft}^2.
  • Split into a 12×412 \times 4 and 8×88 \times 8 rectangle: (12⋅4)+(8⋅8)=48+64=112 in2(12 \cdot 4) + (8 \cdot 8) = 48 + 64 = 112 \text{ in}^2.

Explanation

Think of it like building with LEGOs! Find the area of each separate rectangular piece and just add them together for the grand total.

Section 4

Finding Area with Subtraction

Property

See a complex shape as a big rectangle with a piece missing. Find the area of the whole rectangle, then subtract the area of the missing chunk to get your final answer.

Examples

  • A 10×1210 \times 12 rectangle with a 3×43 \times 4 piece missing: (10⋅12)−(3⋅4)=120−12=108 m2(10 \cdot 12) - (3 \cdot 4) = 120 - 12 = 108 \text{ m}^2.
  • A 20×820 \times 8 shape missing a 5×55 \times 5 square: (20⋅8)−(5⋅5)=160−25=135 m2(20 \cdot 8) - (5 \cdot 5) = 160 - 25 = 135 \text{ m}^2.

Explanation

It's the donut method! You find the area of the whole thing as if it were solid, then just subtract the empty hole in the middle.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 4: Lessons 31-40, Investigation 4

  1. Lesson 1

    Lessons 31: Reading and Writing Decimal Numbers

    LearnPractice
  2. Lesson 2

    Lessons 32: Metric System

    LearnPractice
  3. Lesson 3

    Lessons 33: Comparing Decimals, Rounding Decimals

    LearnPractice
  4. Lesson 4

    Lessons 34: Decimal Numbers on the Number Line

    LearnPractice
  5. Lesson 5

    Lessons 35: Adding, Subtracting, Multiplying, and Dividing Decimal Numbers

    LearnPractice
  6. Lesson 6

    Lessons 36: Ratio, Sample Space

    LearnPractice
  7. Lesson 7Current

    Lessons 37: Area of a Triangle, Rectangular Area, Part 2

    LearnPractice
  8. Lesson 8

    Lessons 38: Interpreting Graphs

    LearnPractice
  9. Lesson 9

    Lessons 39: Proportions

    LearnPractice
  10. Lesson 10

    Lessons 40: Sum of the Angle Measures of a Triangle, Angle Pairs

    LearnPractice
  11. Lesson 11

    Investigation 4: Stem-and-Leaf Plots, Box-and-Whisker Plots

    LearnPractice