Unit 4 Slides

Unit 4: Geometry and Measurement

7th Grade Math

Section 1: Scale Drawings

Example 1.

The following blueprint of a bathroom has a scale of 2 inches = 3 feet. If the
cabinet between the shower and the sink is 1 ½ inches wide on the blueprint,
what is the actual width of the cabinet?

Example 2.

Anderson built a 3-D scale model of the cruise ship where he and his wife got
married. He used the scale of 1 cm = 12 feet. If the cruise ship was 1,134 feet
in length, what was the length of the cruise ship model that he built?

Example 3.

As an architect, Jonathan was hired to design a new building for Dollar Donuts. He is
making a model of the new design on a 15-inch square piece of wood. The dimensions of
the new building measure 45 feet long by 30 feet wide. Which scale should Jonathan use
to make the largest model possible that will still fit on his wood base?

Example 4.

The two drawings above are scale drawings of the same house. If the smaller
drawing on the left measures 2 cm tall, how many centimeters tall does the
larger house on the right measure?

Example 5.

If the area of the smaller rectangle is 10 sq ft, what is the area of the larger
rectangle?

Section 2: Geometric Contruction

Triangles

●Classified by angles:

○Right: triangle has a 90 degree angle

○Acute: triangle has three angles that measure less than 90 degrees

○Obtuse: triangle has an angle measuring greater than 90 degrees

●Classified by sides:

○Equilateral: All sides have the same measure

○Scalene: None of the sides have the same measure

○Isosceles: Two sides have the same measure

Example 1.

How many triangles exist with the given angle measures?

10°, 50°, 120°

Example 2.

How many triangles exist with the given angle measures?

42°, 51°, 85°

Triangle Inequality Theorem

The sum of any two sides must be greater than the third side

Example 3.

How many triangles exist with the given side lengths?

17 cm, 17 cm, 35 cm

Example 4.

How many triangles exist with the given side lengths?

3 feet, 3 feet, 3 feet

Quadrilateral Rules:

●A quadrilateral is a polygon with four sides and four vertices.

●The sum of the measures of the interior angles of a quadrilateral always
equals 360°.

●In a parallelogram, adjacent angles are supplementary, or add up to 180°.

Classification of Quadrilateral:

●Trapezoid - one pair of opposite sides that are parallel.

●Isosceles Trapezoid - two of the opposite sides are parallel, the two other sides are
equal, and the two ends of each parallel side have equal angles.

●Parallelogram - both pairs of opposite sides are parallel.

●Rhombus - four sides of equal length.

●Rectangle - each angle is a right angle.

●Square - four sides have equal length, and each angle is a right angle.

●Kite - exactly one pair of congruent opposite angles, and two pairs of congruent adjacent
sides.

Classification of Quadrilateral:

●Rectangle - each angle is a right angle.

●Square - four sides have equal length, and each angle is a right angle.

●Kite - exactly one pair of congruent opposite angles, and two pairs of congruent
adjacent sides.

Section 3: Cross Sections of 3d
Figures

Vocabulary:

●Cross section: The two-dimensional figure obtained by a solid's
intersection with a plane

Example 1.

Shown below is the cross-section of a plane intersecting a rectangular prism,
where face ABCD is a square. Which two-dimensional shape depicts the
cross-section shown below?

Example 2.

Shown below is the cross-section of a plane intersecting a cube, where face
ABCD is a square. Which two-dimensional shape depicts the cross-section
shown below?

Example 3.

Shown below is the cross-section of a plane intersecting a pyramid, where face
ABCD is a square. Which two-dimensional shape depicts the cross-section
shown below?

Example 4.

Which shape would be formed from the diagonal cross-section of a cylinder?

Example 5.

Which shape would be formed from the horizontal cross-section of a cylinder?

Example 6.

Which shape would be formed from the vertical cross-section of a cylinder?

Example 7.

Which shape would be formed from the diagonal cross-section of a cone?

Example 8.

Which shape would be formed from the horizontal cross-section of a cone?

Example 9

Which shape would be formed from the vertical cross-section of a cone,
passing through its top vertex?

Example 10.

The rectangular prism below has a length of 7 inches, a height of 6 inches, and
a width of 5 inches. A horizontal cross-section of the prism is shown. What is
the area of the cross-section shown?

Example 11.

The cube below has a side length of 7 inches. A vertical cross-section of the
cube is shown. What is the perimeter of the cross-section shown?

Example 12.

The base of the rectangular pyramid below has a length of 6 inches and a width
of 4 inches. The height of the rectangular pyramid is 8 inches. A vertical
cross-section of the pyramid parallel to the length of the base and passing
through the top vertex is shown. What is the area of the cross-section shown?

Section 4: Circles

Vocabulary:

●Diameter: the length of a line segment from one side of a circle to the
other, passing through the center of the circle

●Radius:the length of a line segment from the center of a circle to its edge;
the radius is half the value of the diameter

●Circumference:the distance around a circle

Formulas:

●Circumference (C) = 2πr or , πd where is the radius and is the diameter

●Area (A) = πr2, where is the radius

Example 1.

A circle has a radius of 14 inches. What is the circumference of the circle? Use
3.14 for π.

Example 2.

A circle has a circumference of 44 cm. What is the area of the circle? Use 3.14
for π.

Section 5: Angles

Vocabulary:

●Complementary angles: sum to 90°

●Supplementary angles: sum to 180°

●Adjacent Angles: angles that share a vertex and a side

●Vertical Angles: are the opposite angles formed by two intersecting lines

Example 1.

In the picture below, line AC is intersected by two rays at point O. If ∠AOD and
∠DOC are supplementary angles, then what is the measure of ∠AOD?

Example 2.

In the picture below, line XZ is intersected by two rays at point Q. If ∠XQY and
∠WQX are complementary angles, then what is the measure of ∠XQY?

Example 3.

If the measure of ∠W equals 140°, then what is the measure of ∠Z?

Example 4.

What is the measure of ∠MON?

Example 5.

If the measure of A is ∠25° and complementary to ∠B, what is the measure of
∠B?

Example 6.

If the measure of ∠A is 45° and supplementary to ∠B, what is the measure of
∠B?

Example 7.

If the measure of A equals 165°, then what is the measure of B?

Example 8.

If the measure of A is 40°, what is the measure of B?

Section 6: Area

Formulas

●Area of a Rectangle = length × width

●Area of a Parallelogram = base × height

●Area of a Triangle = ½ × base × height

Example 1.

Find the area of the rectangle below.

Example 2.

Find the area of the triangle below.

Example 3.

Find the area of the parallelogram.

Example 4.

Find the area of the following shape.

Example 5.

Zane designed the T-shirt logo shown below. If x = 4 inches and y = 8 inches,
what is the area of the logo?

Section 7: Surface Area and Volume

Vocabulary:

●Surface Area: of a solid is the measure of the number of square units
needed to cover the outside of the solid. In other words, it is the area of the
surface of a solid

●Volume: of a solid is the number of cubic units contained in the solid

Formulas for Rectangular Prisms:

●Surface Area: 2lw+2lh+2hw

●Volume: lwh

Example 1.

Find the surface area and volume of a box with a length of 5 cm, a width of 4
cm, and a height of 6 cm.

Formulas for Triangular Prisms:

●Surface Area: bh+lH+mH+bH

●Volume: ½ (bhH)

Example 2.

Find the surface area and volume of a isosceles triangular prism with a base of
6 in, a base height of 4 in, a side length of 5 in, and a prism height of 8 in.

Formulas for Cylinders

●Surface Area: 2πr2+ 2πrh

●Volume: 2πr2h

Example 3.

Find the surface area and volume of a cylinder with a radius of 3 in and a height
of 7 in.

Formulas for Triangular Pyramids:

●Surface Area: B+L

●Volume: ⅓ (BH)

Example 4.

Find the approximate surface area and volume of the triangular pyramid below. The base
(b) is an equilateral triangle and it measures 6 inches, and the height (h) measures
approximately 5.2 inches. The height (H) of the triangular pyramid is 8 inches and the
slant height of each lateral triangular face (S) is approximately 8.19 inches.

Formulas for Rectangular Pyramids:

●Surface Area: B+L

●Volume: ⅓ (Bh)

Example 5.

Find the surface area and volume of the pyramid above if l = 36 mm, w = 20
mm, s1 = 26 mm, s2 = 30 mm, and h = 24 mm.