😉 6th Grade, Unit 1, All Lessons 1-18 | Area and Surface Area | Illustrative Mathematics
Illustrative Tutor・2 minutes read
Calculate areas of shapes by decomposing them into smaller parts, using formulas for triangles and parallelograms. Understand perfect squares and cubes to find areas and volumes of various shapes accurately.
Insights
- Small squares cover 10, medium squares cover 40, and large squares cover 45 small squares.
- Tiling the plane: Small squares leave gaps, large squares cover the entire plane.
- The area of a square is 1 square unit, small triangle is 0.5, medium triangle is 1.
- The area of a rectangular playground is 78 sq meters, length is 13 meters - width is 6 meters.
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Recent questions
How can you find the area of a polygon?
Decompose into shapes, multiply base by height, add areas.
What are perfect squares and cubes?
Whole number multiplied by itself, three times respectively.
How do you calculate the volume of a cube?
Multiply edge length by itself three times.
What is the surface area of a polyhedron?
Sum of areas of all outside faces.
How can you find the area of a triangle?
Half of base times height, multiply and divide.
Related videos
Summary
00:00
Area Calculations and Shape Comparisons
- Small squares cover 10, medium squares cover 40, and large squares cover 45 small squares.
- Draw three quadrilaterals with an area of 12 square units: 1x12, 2x6, 3x4.
- Tiling the plane: Small squares leave gaps, large squares cover the entire plane.
- Identify true statements about the area of a shape: B and D are correct.
- Two ways to find the area of a shaded region: 60-5-3-10=42 units.
- Compare areas: 7x3/4 rectangle vs. 2.5 square - square has larger area.
- Decompose and recompose a rectangle along the diagonal to create a new shape.
- Area of a square is 1 square unit, small triangle is 0.5, medium triangle is 1.
- Decompose a square into 16 smaller squares, add 4 to the outside - same area.
- Area of a rectangular playground is 78 sq meters, length is 13 meters - width is 6 meters.
- Student's statement about finding the area of a shape with different measurements is incorrect - can be found by multiplying and subtracting empty squares.
- Find the area of each shaded region: A-22, B-28, C-18.
- Find the area of each shaded region: A-28, B-34, C-96, D-40.
- Noah's statement about two plots having the same area is correct.
- Homeowner deciding on tile size for a wall - disagree with statement A, agree with B, C, D.
- Select parallelograms: B and C are parallelograms, A is a trapezoid, D is a pentagon, E is a right triangle.
- Decompose and rearrange a parallelogram to form a rectangle.
- Find the area of a parallelogram: Base 10 cm, height 3 cm - area is 30 sq cm.
- Explain why a quadrilateral is not a parallelogram based on side lengths.
- Find the area of shapes: 2x6 shape is 12 sq units, combined area of two shapes is 19 sq units.
- Find the areas of rectangles with different side lengths.
- Select parallelograms with correct height labeled for the given base.
- Draw a segment showing the height corresponding to the base of a parallelogram.
- Find the area of each parallelogram: A-8 sq units, B-10 sq units, C-8 sq units.
22:46
Calculating Areas of Parallelograms and Triangles
- The area of a parallelogram can be found by multiplying its base by its height, with the height being at a 90-degree angle to the base.
- To find the corresponding height of a parallelogram with a base of 6 units, the height is selected at a 90-degree angle to the base, making it 4 units.
- Calculating the area of parallelograms involves multiplying the base by the height, with specific measurements provided for each case.
- Understanding the properties of parallelograms, such as having two pairs of parallel sides and opposite angles being equal.
- Decomposing shapes into smaller parts to find areas, like dividing a square into smaller squares and triangles.
- Utilizing graph paper to visually represent and calculate areas of parallelograms with fractional heights.
- Applying division to determine missing measurements in parallelograms, like finding the base when given the area and height.
- Calculating the area of complex shapes by breaking them down into simpler components and summing their areas.
- Exploring different strategies to find the area of triangles, such as decomposing them into parallelograms or using copies to form larger shapes.
- Demonstrating how different methods, like creating parallelograms or rearranging shapes, can aid in calculating triangle areas accurately.
44:20
Parallelograms and Triangles: Area Calculations Explained
- The base of a parallelogram is four units, with a height of one and five-tenths units, resulting in an area of six square units.
- To determine which of three triangles has the greatest area, they are transformed into parallelograms with a base and height of 5 units and 4 units, respectively, leading to an area of 20 square units for each.
- A parallelogram with a base of three and five-tenths units and a height of two units has an area of seven square units.
- For a parallelogram with a base of three units and an area of one and eight-tenths square units, the corresponding height is six-tenths of a unit.
- If a parallelogram has an area of twenty and four-tenths square units and a height of four units, the base is five and one-tenth units.
- The formula for the area of a triangle is half of the base times the height, with triangles A, B, D, and F correctly identifying the corresponding height for a given base.
- Triangles A and B have an area of eight square units, calculated by multiplying the base by the height and dividing by two.
- The area of a triangle with a base of six units and a height of four units is determined to be 12 square units.
- Side D can be the base for a triangle, with side G as the corresponding height.
- By decomposing a shape into rectangles and triangles, the area of the entire shape is found to be 18 square units.
- Two parallelograms with bases of four and eight units, each with a height of four and two units, respectively, are shown to have equal areas of sixteen square units.
01:05:27
Calculating Areas and Surface Areas of Shapes
- The base of the first triangle is 10 and 4 tenths inches, with a height of 3 inches.
- The area of each triangle is calculated, resulting in 15 and 6 tenths square inches.
- Adding the areas of the two triangles and the rectangle gives the area of the hexagon as 93 and 6 tenths square inches.
- A base labeled as 'b' and a corresponding height labeled as 'h' are identified for a triangle.
- The area of the triangle is found to be 11 square units.
- Three different triangles with an area of 12 square units are drawn on a grid, each with specific base and height measurements.
- The surface area of a rectangular prism is calculated to be 64 square units.
- The surface area of a trunk is defined as the number of square feet covering all outside faces.
- Two figures are compared for surface area, with both having a total of 22 square units.
- The surface area of a rectangular prism with specific measurements is determined to be 88 square units.
01:27:28
Polyhedra Surface Area Calculations and Nets
- Figures A and B both have triangular faces.
- There are more vertices than edges in these figures.
- Figure D has rectangular faces.
- Figure A has rectangular faces.
- Figure E is a pyramid.
- Figure B is a pyramid.
- There is exactly one face that can be the base for these figures.
- The base of these figures is a triangle.
- Figure A has two identical and parallel faces that can be the base.
- Figure B has three identical faces, none of which are parallel.
- Units that can be used for surface area are square meters, square inches, and square feet.
- Square meters, square inches, and square feet result from multiplying the base by the height, representing a two-dimensional surface area.
- Feet cannot be used for surface area as it is a unit of length, not two-dimensional.
- Centimeters also cannot be used for surface area as it is a unit of length, not two-dimensional.
- Cubic inches cannot be used for surface area as it is a unit of volume, measuring three dimensions.
- To find the area of a polygon, decompose the shape into a large rectangle, a square, and a parallelogram.
- Calculate the area of each shape by multiplying the base by the height.
- Add the areas of the shapes to find the total area of the original polygon, which is 33 square units.
- Jada drew a net for a polyhedron that can be assembled into a triangular prism.
- Jada made mistakes in her area calculation by using 12 centimeters instead of 6 for the triangles and forgetting to multiply half the base by the height.
- The surface area of the polyhedron is 60 square centimeters.
- A cereal box measuring 8 inches by 2 inches by 12 inches has a surface area of 272 square inches.
- The surface area is calculated by finding the area of each side and adding them together.
- Twelve cubes stacked to form a figure have a surface area of 36 square units.
- Removing the top two cubes reduces the surface area to 30 square units.
- Two polyhedra and their nets are provided, with edges labeled with their correct lengths.
- The lengths of the edges in the nets are specified for both polyhedra.
- A square pyramid can be assembled from a given net.
- The surface area of the figure is 56 square units, calculated by adding the area of the base and the outer triangles.
01:48:28
Geometric Shapes: Cubes, Squares, Prisms, Nets
- A cube has a volume of eight cubic centimeters, with each side measuring two centimeters, resulting in a total volume of eight cubic centimeters.
- Perfect squares are numbers that are the product of a whole number multiplied by itself, such as 4, 16, 25, and 144, while numbers like 20 and 125 are not perfect squares.
- Perfect cubes are numbers that result from multiplying a whole number by itself three times, like 8, 27, 64, and 125, while numbers like 3, 9, and 100 are not perfect cubes.
- The area of a square with a side length of four centimeters is 16 square centimeters, and the area of a square with an area of 49 meters squared has a side length of 7 meters.
- The volume of a cube with an edge length of 3 inches is 27 cubic inches, calculated by multiplying the edge length three times.
- Prism A and Prism B have the same volume of 6 cubic inches but different surface areas, with Prism B having a greater surface area of 26 square inches compared to Prism A's 22 square inches.
- A triangular prism can be assembled from a given net, and to find its surface area, one would need the height and base length of the triangle faces.
- The surface area of a triangular prism with specific measurements is calculated to be 4.8 square meters by adding the areas of its faces.
- The volume of a cube with an edge length of 8 inches is 512 cubic inches, while a cube with an edge length of one-third centimeter has a volume of 1.27 cubic centimeters.
- A cube can be assembled from a given net, with each square having a side length of 61 centimeters, resulting in a surface area of 22,326 square centimeters and a volume of 61 cubic centimeters.
