We included **HMH Into Math Grade 7 Answer Key PDF** **Module 11 Lesson 1 Describe and Analyze Cross Sections of Prisms and Pyramids **to make students experts in learning maths.

## HMH Into Math Grade 7 Module 11 Lesson 1 Answer Key Describe and Analyze Cross Sections of Prisms and Pyramids

I Can describe and analyze the cross sections of pyramids and prisms of all types, with or without a diagram.

**Spark Your Learning**

Brittany bought a candle in the shape of a square pyramid. She wants to know what the inside of the candle looks like. She wonders what figures would be formed if she slices the candle. How could Brittany slice the candle to make a figure with a square face? A triangular face?

Answer:

Triangles of different sizes as shown in the below figure

Explanation:

Slice the candle as shown above perpendicular to its base, through the vertex,

then it forms a triangular faces.

**Turn and Talk** How can you move the slice of the candle to get a smaller or larger square face?

Answer:

Explanation:

A cross-section shown above made parallel or perpendicular to the base to get a smaller or larger square face.

**Build Understanding**

A cross-section is an inside view produced by making a cut or slice. In this lesson, only cuts made parallel or perpendicular to the base will be shown.

Question 1.

Analyze the shipping box shown, which is in the shape of a rectangular prism with two square faces. What are the figures formed when slicing the box from different directions?

A. The box is a prism. What polygon describes the two-dimensional bases of this prism? What polygon describes the other faces?

Answer:

Square face by cutting parallel to base

Rectangle face by cutting horizontal to base

Explanation:

A prism is a polyhedron whose bottom and top faces are congruent polygons and faces are parallelograms, so when the two dimensional cross-section of the prism takes place we get rectangle polygon to describe the other faces.

B. Suppose you slice the prism parallel to its base as shown. What two-dimensional figure is the cross-section?

Answer:

Square

Explanation:

A prism is a polyhedron whose bottom and top faces are congruent polygons and faces are parallelograms, when the prism parallel to its base as shown is cross-section we get square.

C. Suppose you slice the prism perpendicular to its base as shown. What two-dimensional figure is the cross-section?

Answer:

Rectangle

Explanation:

When the cross-section takes perpendicular to its base, then the polygon is rectangle.

D. Consider the pyramid shown. Identify the two-dimensional base and faces. What is the name of this pyramid?

Answer:

A triangular pyramid is a pyramid having a triangular base.

The tetrahedron is a triangular pyramid.

Explanation:

A tetrahedron is a triangular pyramid, with four triangular faces six edges and four corners.

E. Suppose you slice the pyramid shown parallel to its base. What two-dimensional figure is the cross-section?

Answer:

Rectangle

Explanation:

When the cross-section takes parallel to its base, then the polygon is rectangle.

F. Suppose you slice the pyramid shown perpendicular to its base, through the vertex. What two-dimensional figure is the cross-section?

Answer:

Triangle

Explanation:

When the cross-section takes perpendicular to its base through the vertex,

then the polygon is triangle.

G. Suppose you slice the pyramid perpendicular to its base through one edge. What shape is the cross-section?

Answer:

Quadrilateral

Explanation:

A quadrilateral is a polygon having four sides, four angles, and four vertices.

When the cross-section takes perpendicular to its base through one edge,

then the polygon is quadrilateral.

**Step It Out**

Question 2.

Rajesh is doing a research project on the Pentagon, which is located near Washington, D.C. Rajesh sketched a pentagonal prism to help him model the Pentagon.

A. When Rajesh slices the pentagonal prism parallel to the bases, the result is a plane figure that has the same shape as the (bases / faces connecting the bases). So the figure is a ____________.

Answer:

Pentagon

Explanation:

A pentagon is a polygon with five sides and five angles.

When pentagon is cross-section parallel to the prism we get pentagon.

B. When Rajesh slices the pentagonal prism perpendicular to the bases, the result is a figure of the same type as the (bases / faces connecting the bases). So the figure of this cross section is a ____________.

Answer:

Rectangle

Explanation:

When the pentagonal prism is cross-sectioned perpendicular to the bases,

then the result is rectangle with two sides are equal and parallel to each other.

C. Now consider a hexagonal pyramid. The two-dimensional figure that results from slicing the hexagonal pyramid shown parallel to the base is a _____________.

Answer:

Hexagon

Explanation:

When hexagonal pyramid is sliced we get the hexagon only.

As, hexagon has 6 vertices and 6 faces.

D. The figure that results from slicing the hexagonal pyramid perpendicular to the base but not through the vertex, as shown, is a ___________.

Answer:

Parallelogram

Explanation:

When the hexagonal pyramid sliced perpendicular to the base but not through the vertex,

we get parallelogram, as the two sides of the figure is parallel to each other.

**Turn and Talk** What other shape can be produced by slicing a hexagonal pyramid perpendicular to the base? Explain.

Answer:

Rectangle and Square shape can be produced by slicing a hexagonal pyramid perpendicular to the base.

Explanation:

Both shapes have equal opposite sides which are made of parallel lines, both shapes have equal diagonals, and they both have the property that their sides meet each other at 90 degrees.

**Check Understanding**

Question 1.

A triangular prism is sliced parallel to its base. What two-dimensional figure is the cross section? Use a sketch to support your answer.

Answer:

Square

Explanation:

As shown in the above figure we get two dimensional figure square.

Because square has 4 equal sides.

Question 2.

What cross-section is made when a hexagonal pyramid like the one in Task 2 is sliced perpendicular to its base, through its vertex?

Answer:

Rectangle shapes

Explanation:

In rectangle opposite sides are parallel to each other.

When hexagonal pyramid is sliced perpendicular to its base, through its vertex we get rectangle.

Question 3.

Suppose the box of cereal shown is sliced parallel to its base. What is the resulting cross-section?

Answer:

Rectangle

Explanation:

Given cereal box is in rectangle shape.

When we cross-section the box at the base we get Rectangle.

Question 4.

Compare the cross-sections made from slicing the cereal box parallel to its base and slicing it perpendicular to its base.

Answer:

Rectangle

Explanation:

When the cross-section made from slicing the cereal box parallel to its base and slicing it perpendicular to its base, we get rectangle only.

**On Your Own**

For Problems 5-7, use the picture of the hotel with a roof in the shape of a regular pentagonal pyramid.

Question 5.

**Use Structure** Describe how the roof can be sliced to make a cross-section in the shape of a pentagon.

Answer:

Sliced parallel to base.

Explanation:

the roof can be parallel to base sliced to make a cross-section in the shape of a pentagon.

Question 6.

**Use Structure** Describe how the roof can be sliced to make a cross-section in the shape of a triangle.

Answer:

Sliced perpendicular to base

Explanation:

the roof can be perpendicular to base when sliced to make a cross-section in the shape of a triangle.

Question 7.

**Reason** Describe how the location of a slice parallel to the base can affect the size of the resulting pentagon. Explain your reasoning.

Answer:

Explanation:

larger cross section can make the larger size of pentagon and the smaller for small cross section, location of a slice parallel to the base can affect the size of the resulting pentagon

**For Problems 8-13, identify the shape of the two-dimensional cross-section shown.**

Question 8.

perpendicular to base of a rectangular prism

Answer:

Rectangle

Explanation:

When we cross-section a rectangular prism perpendicular to base,

we get the two-dimensional figure as rectangle with opposite sides are equal.

Question 9.

perpendicular to base not through vertex of a rectangular pyramid

Answer:

parallelogram, Trapezium

Explanation:

In Parallelogram the opposite sides are parallel to each other.

Trapezium has parallel sides at the top and bottom of the shape.

When the cross-section of a rectangular pyramid is done perpendicular to base and not through vertex, we get parallelogram or trapezium.

Question 10.

perpendicular to base and through vertex of a square pyramid

Answer:

Triangle shape

Explanation:

When the cross-section of a square pyramid is done perpendicular to base and through vertex, we get triangle which has three edges and three vertices.

Question 11.

parallel to base of a pentagonal prism

Answer:

pentagon shape

Explanation:

When the cross-section of a pentagonal prism is done parallel to base, we get pentagon.

A pentagon is a polygon with five sides and five angles.

Question 12.

parallel to base of a hexagonal prism

Answer:

Hexagon shape

Explanation:

When the cross-section of a hexagonal prism is done parallel to base, we get hexagon.

A hexagon has six vertices and six faces.

Question 13.

parallel to base of a square pyramid

Answer:

Square shape

Explanation:

When the cross-section of square pyramid is done parallel to base, we get square.

In square four sides are equal.

**I’m in a Learning Mindset!**

What methods are most effective for analyzing cross-sections of prisms and pyramids?

Answer:

A plane at an angle relative to the base that intersects the base of a rectangular pyramid will make a cross section that has four sides.

**Lesson 11.1 More Practice/Homework**

**For Problems 1-3, describe the two-dimensional figure that results from slicing the given three-dimensional figure.**

Question 1.

Slice parallel to the base

Answer:

Square

Explanation:

When we slice the given shape parallel to the base, we get square only.

A square is a four sided regular polygon with four equal sides and their angles are 90° each.

Question 2.

Slice perpendicular to the base, through the vertex

Answer:

Triangle

Explanation:

When wee slice the given shape perpendicular to the base through vertex, we get triangle.

A triangle is a polygon with 3 sides and 3 corners or vertex.

Question 3.

Slice perpendicular to the base, not through the vertex

Answer:

Quadrilateral

Explanation:

When we slice the given shape perpendicular to the base and not through the vertex,

we get quadrilateral with four sides, four vertex and four angles.

**For Problems 4-6, tell whether the slice must be parallel or perpendicular to the base to make the given cross-section.**

Question 4.

A slice of a pentagonal pyramid results in a pentagon.

Answer:

Parallel to base

Explanation:

When we slice a pentagonal pyramid parallel to the base the result is pentagon only.

Question 5.

A slice of a triangular prism results in a triangle.

Answer:

perpendicular to base

Explanation:

When we slice a triangular prism perpendicular to the base the result is triangle only.

Question 6.

A slice of a hexagonal pyramid results in a triangle.

Answer:

perpendicular to the base for top edge

Explanation:

When we slice a hexagonal pyramid perpendicular to the base the result is triangle only.

Question 7.

**Use Repeated Reasoning** A sphere has a radius of 12 inches.

A. Describe the cross-section formed by slicing the sphere through the sphere’s center.

Answer:

Radius of circle is 12 inches

Explanation:

Cross-section passing through the center of the sphere is a circle having the radius equal to the radius of the sphere.

B. Describe the cross-sections formed by slicing the sphere many times, each time farther from the center of the sphere.

Answer:

Circle has different size of radius

Explanation:

Cross-section formed as a plane intersects the interior of the sphere and outside the center are circles with different radius.

Question 8.

**Use Structure** What cross-sections might you see when slicing a cone that you would not see when slicing a pyramid or a prism?

Answer:

An Ellipse, Hyperbola and Parabola.

Explanation:

When we slice a pyramid we get the polygonal shapes, but when slice the cone we get ellipse, hyperbola and parabola because their sides are not polygons and they have curved surfaces known as non-polyhedrons.

**Test Prep**

Question 9.

Select all of the following three-dimensional figures that could have a cross-section of a triangle when sliced parallel to the base, perpendicular to the base through the vertex, or perpendicular to the base not through the vertex.

(A) cube

(B) triangular prism

(C) rectangular prism

(D) triangular pyramid

(E) pentagonal prism

Answer:

cube, rectangular prism, rectangular prism, triangular pyramid, pentagonal prism

Explanation:

cube, rectangular prism, rectangular prism, triangular pyramid, pentagonal prism

could have a cross-section of a triangle when sliced parallel to the base, perpendicular to the base through the vertex

Question 10.

Match each description of slicing a three-dimensional figure with the resulting two-dimensional figure shown.

Answer:

Explanation:

When we slice a square pyramid parallel to the base we get square.

When we slice a non-square rectangular prism parallel to base we get rectangle.

When wee slice a square pyramid perpendicular to base through the vertex we get triangle.

When we slice a pentagonal prism parallel to base we get pentagon.

Question 11.

Describe the cross-sections made by slicing a hexagonal prism parallel to and perpendicular to its base.

Answer:

Hexagon

Parallelogram

Explanation:

When the cross-sections made by slicing a hexagonal prism parallel to the base we get hexagon and perpendicular to its base we get parallelogram.

**Spiral Review**

Question 12.

Draw a quadrilateral with two pairs of parallel, congruent sides and four right angles.

Answer:

Explanation:

The above quadrilateral has two pairs of parallel lines, congruent sides and four right angles.

Question 13.

Find the area of the composite figure.

Answer:

108 sq in

Explanation:

Area = the given figure

Area = A1 – A2

A1 = 12 x 12 = 144 sq in

A2 = 6 x 6 = 36 sq in

Area = 144 – 36 = 108 sq in