# Lesson Worksheet: The Pythagorean Theorem in 3D Mathematics • 8th Grade

In this worksheet, we will practice using the Pythagorean theorem to solve problems in three dimensions.

Q1:

is a cube. Determine the lengths and . • A,
• B,
• C,
• D,

Q2:

is a cube. Determine the lengths and . • A,
• B,
• C,
• D,

Q3:

is a cube. Determine the lengths and . • A,
• B,
• C,
• D,

Q4:

is a cube. Determine the lengths and . • A,
• B,
• C,
• D,

Q5:

If is a right triangular pyramid; the length of its edge , and its base is right angled at , where and , find the height of the pyramid rounded to the nearest hundredth.

Q6:

Let be an equilateral triangle of side 96, and a perpendicular to the plane of length 48. Determine the length of the perpendicular from to .

Q7:

is a triangular pyramid in which and . Draw . If is perpendicular to the plane , , and , determine the length of .

• A69.6
• B41
• C
• D

Q8:

, , and are mutually orthogonal. Given , , and a point on . Find the length for which is perpendicular to the plane .

• A
• B
• C
• D

Q9:

is a regular pyramid whose base is an equilateral triangle whose side length is 32 cm. If the length of its lateral edge is 88 cm, find the height of the pyramid to the nearest hundredth.

Q10:

A pyramid is on an equilateral triangular base of 21 cm and is 23 cm high. How long, to the nearest hundredth, is the pyramid’s lateral edge?

Q11:

is an inclined prism, and is a square. Draw with on . Given that , , and , find the length of .

• A12
• B20.6
• C
• D29.4

Q12:

In the figure, lies in the plane , and is perpendicular to . Given that and , find the length of . Q13:

is a triangle with and . is drawn perpendicular to the plane of , and the perpendicular to from drawn to meet it at . If , determine the length of and the angle between and the plane of .

• A19.92,
• B19.92,
• C11.5,
• D11.5,
• E30.43,

Q14:

Triangle is right angled at , and is orthogonal to the plane . A perpendicular is drawn from on . The area of is 1,134, , and . Let be the angle between and the plane . Find to the nearest thousandth.

Q15:

In the figure, suppose that , plane , and . Find the length of . • A
• B
• C
• D25.24

Q16:

In triangle , and . is a normal to the plane , and the foot of the perpendicular from to . If , determine the length of .

• A
• B
• C
• D

Q17:

Rectangle has and . Suppose perpendiculars and both of length 27. What is the area of ?

• A900
• B675
• C1,125
• D1,095.7

Q18:

Determine the surface area of the shown rectangular prism, rounding the result to the nearest tenth. Q19:

A piece of paper in the shape of a circular sector having a radius of 72 cm and an angle of is folded in a way so that the points and meet to form a circular cone of the greatest possible area. Determine the cone’s height to the nearest hundredth. Q20:

Given that is a cube whose edge length is cm and is the midpoint of , find the area of the rectangle . • A cm2
• B72 cm2
• C101.82 cm2
• D90 cm2

Q21:

is a rectangular parallelepiped whose three dimensions are , , and . Determine the area of the rectangle . Q22:

Identify a pair of points between which a diagonal of the given rectangular prism can be drawn. • A and
• B and
• C and
• D and
• E and

Q23:

In the figure, is perpendicular to the plane , which contains the points , , , and . If and , find the area of . • A1,386
• B3,272.5
• C3,060
• D1,530

Q24:

A right circular cone has height 90 cm and slant height 106 cm. Find the circumference and area of the base in terms of .

• Acircumference , area
• Bcircumference , area
• Ccircumference , area
• Dcircumference , area

Q25:

A sheet of paper in the shape of a sector of radius 29 cm and area cm2 is folded into a right cone, by gluing together the radii and . What is the height of the cone? Recall that the sector area is given by half the product of its radius and the length of its arc. • A43.96 cm
• B cm
• C cm
• D33.04 cm