D. Trigonometric Functions · Trigonometric Ratios, Identities and Laws

Problems in Three Dimensions: Right Triangles

D1.7About 70 minutes

Learning goals

I can sketch a three-dimensional situation and find the right triangles in it.
I can find space diagonals and the angle a line makes with a plane.
I can solve problems with angles of elevation measured from points in different directions.
I can chain two or more right triangles together, solving one to get information for the next.

Words to know

three-dimensional (3-D)
Having length, width and height, like a room or a box, rather than lying flat on a page.

space diagonal
A segment joining two vertices of a box that are not on the same face, passing through the inside of the box.

plane
A flat surface that extends forever, such as the floor or a wall.

angle between a line and a plane
The angle between the line and its shadow (projection) on the plane, straight below it.

horizontal plane
A level surface, such as level ground. Vertical lines (like towers) are perpendicular to it.

Off the page

Architects, surveyors, drone pilots and video-game designers all work in three dimensions. So far, every triangle in this unit lay flat on the page. In 3-D, triangles lean in different directions: one may lie on the floor and another may stand up against a wall. The key skill is seeing the right triangles inside a 3-D shape. Once you find them, the tools are the ones you already know.

Sketching 3-D problems

Draw the ground as a slanted parallelogram, and vertical lines straight up.
Use dashed lines for edges you can't see.
Mark right angles clearly: a vertical tower makes a right angle with every line on the ground through its base.
Shade or colour the triangle you are about to solve.
Solve one triangle at a time. Write each answer on the sketch.

Worked example

Example 1: The longest rod in a room

A room is 5 m long, 4 m wide and 2.5 m high. Find the length of the longest straight rod that fits in the room (the space diagonal), and the angle it makes with the floor.

Right triangle on the floor: the floor diagonal is √(5² + 4²) = √41 ≈ 6.40 m.
This floor diagonal and the 2.5 m vertical edge form a second right triangle, standing up. Its hypotenuse is the space diagonal: d = √(41 + 2.5²) = √47.25 ≈ 6.87 m.
The angle with the floor is the angle between d and the floor diagonal (its shadow on the floor): tan θ = 2.5 ÷ √41, so θ ≈ 21.3°.

The rod is about 6.87 m long and makes an angle of about 21.3° with the floor. In general, a box l × w × h has space diagonal √(l² + w² + h²).

The space diagonal d is the hypotenuse of the shaded right triangle, whose legs are the floor diagonal √41 and the 2.5 m height.

Worked example

Example 2: A tower seen from two directions

A 60 m communications tower stands on level ground. From point P, due west of the tower's foot F, the angle of elevation of the top is 35°. From point Q, due south of F, it is 28°. How far apart are P and Q?

Right △PFT (vertical): FP = 60 ÷ tan 35° ≈ 85.69 m.
Right △QFT (vertical): FQ = 60 ÷ tan 28° ≈ 112.84 m.
On the ground, west and south are perpendicular, so ∠PFQ = 90°. Right △PFQ (horizontal): PQ = √(85.69² + 112.84²) ≈ 141.7 m.

P and Q are about 141.7 m apart. Two vertical triangles fed one horizontal triangle.

Two vertical right triangles (through the tower) and one horizontal right triangle (on the ground). The right angle at F on the ground comes from the directions west and south. (Sketch, not to scale.)

Worked example

Example 3: A pyramid tent

A tent is a square-based pyramid. The base is 3 m by 3 m and the top V is 2 m directly above the centre O of the base. Find the length of each sloping edge and the angle that an edge makes with the ground.

The base diagonal is 3√2 ≈ 4.24 m, so the distance from the centre O to a corner B is half of that: OB = 3√2 ÷ 2 ≈ 2.121 m.
△VOB is right-angled at O (VO is vertical). VB = √(2² + 2.121²) = √8.5 ≈ 2.92 m.
The angle between VB and the ground is ∠VBO: tan θ = 2 ÷ 2.121, θ ≈ 43.3°.

Each edge is about 2.92 m long and makes an angle of about 43.3° with the ground.

Inside the pyramid, the height VO, the half-diagonal OB and the edge VB form a right triangle (shaded).

Just for fun

A cube's space diagonal is always √3 times its edge. So a 10 cm cube can hide a pencil about 17.3 cm long, even though no edge is longer than 10 cm.

Hands-on activity

The longest pencil in a box

You need: a closed cardboard box (a shoebox or cereal box), a ruler or measuring tape, a piece of string, tape and a calculator.

Measure the box's length, width and height in centimetres.
Predict the length of a face diagonal of the bottom and the space diagonal: √(l² + w²) and √(l² + w² + h²).
Open the box. Tape one end of the string in a bottom corner and stretch it to the opposite top corner (the corner not on any face shared with the first). Mark the string and measure it.
Predict the angle the space diagonal makes with the bottom: tan θ = h ÷ √(l² + w²).

What to look for: for a 30 cm × 20 cm × 7 cm cereal box, the space diagonal should be about 36.7 cm and the angle about 11°. Your string should be within about 1 cm of your prediction.

Your turn

Find the space diagonal of a box that is 3 cm by 4 cm by 12 cm.

A cereal box is 30 cm by 20 cm by 7 cm. Find its space diagonal.

Find the angle between a space diagonal of a cube and the base of the cube.

A 15 m flagpole stands on level ground. From A, due east of the pole, the angle of elevation of the top is 40°. From B, due north of the pole, it is 25°. Find AB.

A square-based pyramid has base edges of 10 m and a height of 6 m. Find the slant height of a face (from the top to the midpoint of a base edge) and the angle a face makes with the base.

More practice: warm-up

Find the diagonal of a 6 by 8 rectangle.

Write the formula for the space diagonal of an l × w × h box.

Find the space diagonal of a 2 × 3 × 6 box.

Find the space diagonal of a cube with edge 1.

A vertical pole stands on level ground. What angle does it make with any line on the ground through its foot?

More practice: on your own

A drone hovers 40 m above point D on a level field. Ama stands at A, 70 m from D. Ben stands at B, 50 m from D, with ∠ADB = 90°. Find the angle of elevation of the drone from each person, and the distance AB.

What is the longest straight rod that fits in a room 6 m by 5 m by 3 m?

A square-based pyramid tent has base edges of 2.4 m and sloping edges of 2.0 m. How tall is it?

From A, 60 m due south of the foot of a tower, the angle of elevation of the top is 32°. From B, due west of the foot, it is 24°. Find the tower's height and the distance AB.

A box is 40 cm by 30 cm by 20 cm. Find the angle between a space diagonal and the base.

Challenge

From the top of a 60 m cliff, a boat due south is seen at an angle of depression of 25° and a boat due east at an angle of depression of 18°. How far apart are the boats?

Prove that the space diagonal of an l × w × h box is √(l² + w² + h²).

A tower of unknown height h stands on level ground. From A, due south of its foot, the angle of elevation is 40°; from B, due east of its foot, it is 30°. A and B are 100 m apart. Find h.

Check your understanding

Quiz

What is the space diagonal of a 3 × 4 × 12 box?

The angle between a line and a plane is measured between the line and:

A tower is seen from P (due west) and Q (due south). Why is ∠PFQ = 90° at the foot F?

A room is 5 m × 4 m × 2.5 m. What is its floor diagonal?

In a square-based pyramid, which right triangle gives the length of a sloping edge?

A cube has edges of 10 cm. About how long is its space diagonal?

For parents and teachers

Big idea (D1.7): 3-D problems are solved by finding right triangles that stand at different angles: some lie on the ground, some stand up vertically. Each solved triangle gives a side for the next one.

How to help: A shoebox and a piece of string make 3-D problems much easier to picture. Ask your teen to point to each triangle on a real object before calculating.

Common difficulties:

Drawing the 3-D sketch so that right angles don't look like right angles, and then missing them. Label them.
Using a face diagonal when the space diagonal is needed (or the reverse).
Measuring the angle with a plane from the wrong line: it is always the line's shadow on that plane.