13 Questions
What is the perimeter of a compound shape consisting of a rectangle with length 8cm and width 5cm, and a triangle with base 6cm and height 7cm?
P = 2(8) + 2(5) + 6 + 7 = 39cm
Find the area of a compound shape consisting of a circle with radius 3cm and a rectangle with length 4cm and width 2cm.
A = π(3)² + 4(2) = 29.34cm²
A cylinder has a radius of 5cm and a height of 12cm. A cone has a radius of 5cm and a height of 8cm. What is the total volume of the compound shape?
V = π(5)²(12) + π(5)²(8)/3 = 942.48cm³
A triangle has a base of 8cm and a height of 9cm. A rectangle has a length of 10cm and a width of 6cm. What is the perimeter of the compound shape?
P = 2(10) + 2(6) + 8 + 9 = 44cm
Find the area of a compound shape consisting of a rectangle with length 12cm and width 8cm, and a triangle with base 10cm and height 11cm.
A = 12(8) + (10/2)(11) = 164cm²
What is the volume of a compound shape consisting of a cylinder with radius 4cm and height 15cm, and a cone with radius 4cm and height 7cm?
V = π(4)²(15) + π(4)²(7)/3 = 1130.4cm³
A circle has a radius of 6cm and a rectangle has a length of 8cm and a width of 4cm. What is the perimeter of the compound shape?
P = 2π(6) + 2(8) + 2(4) = 52.57cm
Find the area of a compound shape consisting of a rectangle with length 9cm and width 5cm, and a triangle with base 7cm and height 9cm.
A = 9(5) + (7/2)(9) = 90.5cm²
The width and length of a rectangle are $x+2$ cm and $2x-3$ cm respectively. If a right-angled triangle with a base of $x-1$ cm and a height of $x+4$ cm is removed from the rectangle, what is the area of the resulting compound shape in terms of $x$?
$(x+2)(2x-3) - rac{1}{2}(x-1)(x+4)$
A compound shape consists of a rectangle with a length of $3x+2$ cm and a width of $x-1$ cm, and a triangle with a base of $2x-3$ cm and a height of $x+2$ cm. What is the perimeter of the compound shape in terms of $x$?
$2(3x+2) + 2(x-1) + 2x-3 + x+2 + x-1$
The length and width of a rectangle are $2x+1$ cm and $x-2$ cm respectively. If a semicircle with a radius of $x-1$ cm is removed from the rectangle, what is the area of the resulting compound shape in terms of $x$?
$(2x+1)(x-2) - rac{1}{2}\pi(x-1)^2$
A compound shape consists of a rectangle with a length of $x+3$ cm and a width of $2x-2$ cm, and a square with a side length of $x-1$ cm. What is the perimeter of the compound shape in terms of $x$?
$2(x+3) + 2(2x-2) + 4(x-1)$
The length and width of a rectangle are $x+2$ cm and $x-3$ cm respectively. If a right-angled isosceles triangle with a base of $x-2$ cm is removed from the rectangle, what is the area of the resulting compound shape in terms of $x$?
$(x+2)(x-3) - rac{1}{2}(x-2)^2$
Study Notes
Compound Shapes Algebra
Definition
- Compound shapes are geometric shapes composed of two or more simpler shapes.
- Algebraic representations of compound shapes are used to calculate their perimeters, areas, and volumes.
Notation
- Let's denote the area of a shape as A, perimeter as P, and volume as V.
- Use subscripts to distinguish between different shapes (e.g., A₁, P₂, V₃).
Properties
- Perimeter: The perimeter of a compound shape is the sum of the perimeters of its individual shapes.
- Area: The area of a compound shape is the sum of the areas of its individual shapes.
- Volume: The volume of a compound shape is the sum of the volumes of its individual shapes.
Algebraic Representations
-
Rectangle and Triangle: A rectangle with length l and width w, combined with a triangle with base b and height h, can be represented as:
- A = lw + (bh/2)
- P = 2l + 2w + b + h
-
Circle and Rectangle: A circle with radius r, combined with a rectangle with length l and width w, can be represented as:
- A = πr² + lw
- P = 2πr + 2l + 2w
-
Cylinder and Cone: A cylinder with radius r and height h, combined with a cone with radius r and height h, can be represented as:
- V = πr²h + (πr²h/3)
Example Problems
- Find the area and perimeter of a compound shape consisting of a rectangle with length 5cm and width 3cm, and a triangle with base 4cm and height 6cm.
- Calculate the volume of a compound shape consisting of a cylinder with radius 2cm and height 10cm, and a cone with radius 2cm and height 5cm.
Key Concepts
- Recognize and apply the properties of compound shapes (perimeter, area, volume).
- Use algebraic representations to calculate the perimeter, area, and volume of compound shapes.
- Apply problem-solving strategies to simplify complex compound shapes into individual shapes.
Compound Shapes Algebra
Definition
- Compound shapes are composed of two or more simpler shapes.
- Algebraic representations are used to calculate perimeters, areas, and volumes.
Notation
- Area is denoted as A, perimeter as P, and volume as V.
- Subscripts are used to distinguish between different shapes (e.g., A₁, P₂, V₃).
Properties
- The perimeter of a compound shape is the sum of the perimeters of its individual shapes.
- The area of a compound shape is the sum of the areas of its individual shapes.
- The volume of a compound shape is the sum of the volumes of its individual shapes.
Algebraic Representations
- A rectangle with length l and width w, combined with a triangle with base b and height h:
- A = lw + (bh/2)
- P = 2l + 2w + b + h
- A circle with radius r, combined with a rectangle with length l and width w:
- A = πr² + lw
- P = 2πr + 2l + 2w
- A cylinder with radius r and height h, combined with a cone with radius r and height h:
- V = πr²h + (πr²h/3)
Problem-Solving Strategies
- Simplify complex compound shapes into individual shapes.
- Apply algebraic representations to calculate perimeter, area, and volume.
- Recognize and apply the properties of compound shapes.
Compound Shapes Algebra
Definition and Notation
- Compound shapes are geometric shapes composed of two or more simpler shapes.
- Algebraic representations of compound shapes are used to calculate their perimeters, areas, and volumes.
- Notations: A for area, P for perimeter, and V for volume; use subscripts to distinguish between different shapes (e.g., A₁, P₂, V₃).
Properties of Compound Shapes
Perimeter
- The perimeter of a compound shape is the sum of the perimeters of its individual shapes.
Area
- The area of a compound shape is the sum of the areas of its individual shapes.
Volume
- The volume of a compound shape is the sum of the volumes of its individual shapes.
Algebraic Representations of Compound Shapes
Rectangle and Triangle
- A rectangle with length l and width w, combined with a triangle with base b and height h, can be represented as:
- A = lw + (bh/2)
- P = 2l + 2w + b + h
Circle and Rectangle
- A circle with radius r, combined with a rectangle with length l and width w, can be represented as:
- A = πr² + lw
- P = 2πr + 2l + 2w
Cylinder and Cone
- A cylinder with radius r and height h, combined with a cone with radius r and height h, can be represented as:
- V = πr²h + (πr²h/3)
Example Problems and Key Concepts
- Recognize and apply the properties of compound shapes (perimeter, area, volume).
- Use algebraic representations to calculate the perimeter, area, and volume of compound shapes.
- Apply problem-solving strategies to simplify complex compound shapes into individual shapes.
Learn about the algebraic representations of compound shapes, including perimeters, areas, and volumes. Understand notation and properties of these complex geometric shapes.
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