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Questions and Answers
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?
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.
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?
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?
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?
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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.
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.
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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?
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?
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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?
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?
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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.
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.
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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 $x$?
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 $x$?
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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 $x$?
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 $x$?
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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 $x$?
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 $x$?
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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 $x$?
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 $x$?
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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 $x$?
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 $x$?
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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.
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Description
Learn about the algebraic representations of compound shapes, including perimeters, areas, and volumes. Understand notation and properties of these complex geometric shapes.
