Area & Perimeter

Unit 8: 2D Geometry

Lesson 2: Area + Perimeter

Learning Goals

• To calculate area and perimeter of rectangles, squares, and triangles.

• To find missing lengths when given area and perimeter of rectangles, squares, and triangles.

Minds On: Review

• Assess prior knowledge regarding perimeter and area of basic shapes.

• Example:

  • Rectangle:

  • 
    Dimensions: 3 cm (width) & 5 cm (length)
    Perimeter Calculation:
    $P = 2l + 2w = 2(5) + 2(3) = 10 + 6 = 16 ext{ cm}$
    Area Calculation:
    $A = lw = 5(3) = 15 ext{ cm}^2$

  • Triangle:

  • 
    Side lengths: 9 m, 14 m, and 11 m
    Perimeter Calculation:
    $P = a + b + c = 9 + 14 + 11 = 34 ext{ m}$
    Area Calculation:
    $A = rac{bh}{2} = rac{7(14)}{2} = rac{98}{2} = 49 ext{ m}^2$

Definitions

Perimeter

• Definition: The calculation of the distance around a closed 2D shape.

• Formula: $P = ext{side} + ext{side} + ext{side} + ext{side}$

• Alternately, for rectangles:

  • $P = 2w + 2l$

  • $P = 2(l + w)$

• Practical Implication: To find how far one would travel around a shape.

Area

• Definition: The calculation of the amount of space inside a closed 2D shape, measured in square units.

• Major Calculation:

• For Squares and Rectangles:

  • Count the squares or multiply the sides.

  • Formula: $A = ext{Length} imes ext{Width}$

Squares: Perimeter

Perimeter Formula

• Definition: The perimeter of squares can be found by adding up all sides.

• General Formula: $P = s + s + s + s = 4s$

Squares: Area

Area Formula

• Area of squares is calculated as the total number of squares inside.

• Formula:

  • $A = s imes s$

  • $A = s^2$

Squares: Practice Examples

• Find the perimeter and area of squares:

  • For example:

  • Side length: 50 cm

    • $P = 4(50) = 200 ext{ cm}$

    • $A = 50^2 = 2500 ext{ cm}^2$

  • For side length: 22 m

    • $P = 4(22) = 88 ext{ m}$

    • $A = 22^2 = 484 ext{ m}^2$

Rectangles: Perimeter

Perimeter Calculation

• Definition: The perimeter of a rectangle is calculated by adding up all sides.

• General Formula: $P = L + W + L + W = 2L + 2W$

• Alternatively, $P = 2(L + W)$

Rectangles: Area

Area Formula

• Area of rectangles is calculated as the number of squares inside.

• Formula:

  • $A = ext{Length} imes ext{Width}$

  • $A = LW$

Rectangles: Practice Examples

• Example Problem:

  • Dimensions: 70 cm by 30 cm

  • $P = 2(70) + 2(30) = 140 + 60 = 200 ext{ cm}$

  • $A = 70(30) = 2100 ext{ cm}^2$

Triangles: Perimeter

Perimeter Calculation

• Definition: The perimeter of triangles is calculated by adding up all sides.

• General Formula: $P = a + b + c$

• Types of Triangles:

  • Scalene: All sides different; $P = a + b + c$

  • Equilateral: All sides equal; $P = 3a$

  • Isosceles: Two sides equal; $P = 2b + c$

Triangles: Area

Area Formula

• Area of triangles is calculated similar to rectangles but requires division by 2 (since it is half of a rectangle).

• Formula:

  • $A = rac{b imes h}{2}$

  • Where $b$ is the base and $h$ is the height.

Triangles: Practice Examples

• Example Problem:

  • For a triangle with side lengths of 25 cm, 32 cm, and 25 cm:

  • $P = 25 + 32 + 25 = 82 ext{ cm}$

  • For the area with base of 50 cm and height of 20 cm:
    $A = rac{50 imes 20}{2} = 500 ext{ cm}^2$

Challenge Problems

• Squares:

  • Given perimeter P = 220 cm, find side length.

  • $P = 4s$, therefore $s = 220/4 = 55 ext{ cm}$

  • Given area A = 256 m², find side length.

  • $A = s^2 <br>ightarrow s = ext{√256} = 16 ext{ m}$

• Rectangles:

  • Given perimeter P = 190 cm with one side 45 cm, find width:

  • Solve:
    $190 = 2(45) + 2w <br>ightarrow 190 = 90 + 2w$
    $2w = 100 <br>ightarrow w = 50 ext{ cm}$

  • Area given as 375 m², find length from width of 25 m:

  • Solve:
    $375 = 25L <br>ightarrow L = rac{375}{25} = 15 ext{ m}$

• Triangles:

  • Given perimeter P = 121 m with base 45 m, find missing equal sides:

  • $121 = 2a + 45 <br>ightarrow 2a = 121 - 45 <br>ightarrow 2a = 76 <br>ightarrow a = 38 ext{ m}$

  • Area given as 1575 cm², find height using base of 75 cm:

  • $1575 = rac{75h}{2} <br>ightarrow 3150 = 75h <br>ightarrow h = rac{3150}{75} = 42 ext{ cm}$

Consolidation Vocabulary

• Area

• Perimeter

• Area & Perimeter Formulas for:

  • Rectangle

  • Square

  • Triangle

Additional Practice

IXL Quiz

• Complete exercises regarding Area & Perimeter in math notebook following proper mathematical format and organization!