C. Trigonometric Functions · Trigonometry of Acute Triangles

Right Triangles in Three Dimensions

C1.1 C1.2About 70 minutes

Learning goals

I can find right triangles inside 3-D objects such as boxes, rooms and towers.
I can draw a 3-D situation so that each right triangle is easy to see, then redraw each triangle flat.
I can solve 3-D problems by solving one right triangle after another.
I can find the angle between a line and a flat surface (such as a floor or the ground).

Words to know

three-dimensional (3-D)
Having length, width and height. Problems with points on the ground in different directions plus a height are 3-D.

space diagonal
A segment joining two corners of a box that are not on the same face.

plane
A flat surface that goes on forever, such as the plane of the floor or the ground.

angle between a line and a plane
The angle between the line and its "shadow" (projection) straight down onto the plane.

Triangles hiding in 3-D

Curriculum expectation C1.2 asks for problems with two right triangles in two dimensions. Real life is three-dimensional, though, and the same tools work there too. This lesson goes one step further than the expectation, so treat it as enrichment: the right triangles are no longer in the same flat plane. One is on the floor or the ground; the next stands up out of it.

The method does not change: find a right triangle you can solve, use its answer as a side of the next one.

Drawing 3-D problems

Draw the ground (or floor) as a slanted parallelogram so it looks flat.
Draw vertical lines straight up and down the page.
Mark right angles carefully. A vertical tower makes a right angle with every line on the ground through its base, even if it does not look like 90° on paper.
When you solve a triangle, redraw it flat on the side of your page with its true right angle.

Worked example

Example 1: The longest pole that fits in a room

A storage room is 5.0 m long, 4.0 m wide and 2.6 m high. What is the longest straight pole that fits in the room (from one bottom corner to the opposite top corner)? What angle does it make with the floor?

Label the floor ABCD and the top corner above C as G. The pole is the space diagonal AG.
Triangle 1 (on the floor, right angle at B): AC = √(5.0² + 4.0²) = √41 ≈ 6.403 m.
Triangle 2 (standing up, right angle at C): AG = √(AC² + 2.6²) = √(41 + 6.76) ≈ 6.91 m.
The angle with the floor is the angle between AG and its shadow AC: tan θ = 2.6/AC, so θ = tan⁻¹(2.6 ÷ 6.403) ≈ 22°.

The longest pole is about 6.91 m long and it makes an angle of about 22° with the floor.

Example 1. The floor diagonal AC (dashed) is the shared side: it is the hypotenuse of the floor triangle and a leg of the upright triangle ACG.

Worked example

Example 2: A tower seen from two directions

A communications tower is 60 m tall. From point A, due south of its base F, the angle of elevation to the top T is 40°. From point B, due east of F, the angle of elevation is 32°. How far apart are A and B?

There are three right triangles: △AFT and △BFT stand up; △AFB lies on the ground with the right angle at F (south and east are perpendicular).
△AFT: FA = 60 ÷ tan 40° ≈ 71.51 m.
△BFT: FB = 60 ÷ tan 32° ≈ 96.02 m.
△AFB (ground): AB = √(FA² + FB²) ≈ √(71.51² + 96.02²) ≈ 119.7 m.

A and B are about 120 m apart.

Example 2. The shaded triangle AFB lies flat on the ground with a right angle at F. The tower FT is vertical, so ∠TFA and ∠TFB are also right angles even though they do not look like 90° in a 3-D sketch.

Worked example

Example 3: Surveying a height you cannot walk to

A surveyor stands at A, across a ravine from a tower whose base is F. She walks 100 m from A to B along a line perpendicular to AF. From B, the horizontal angle between B→A and B→F is ∠ABF = 55°. Back at A, the angle of elevation to the top of the tower is 28°. How tall is the tower?

Ground triangle ABF: right angle at A (AB ⟂ AF).
AF is opposite the 55° angle at B and AB = 100 m is adjacent: AF = 100 tan 55° ≈ 142.81 m.
Upright triangle AFT: right angle at F, angle of elevation 28° at A.
Height FT = AF tan 28° ≈ 142.81 × 0.5317 ≈ 75.9 m.

The tower is about 76 m tall.

Just for fun

A funny-but-true fact: "Will this couch fit through the door?" is a 3-D trigonometry problem. Movers tilt furniture on a diagonal because a door's diagonal is longer than its width or its height.

Hands-on activity

Measure a space diagonal

You need: a shoebox or any closed cardboard box, a ruler, a long piece of string or a thin wooden skewer (grown-up help to poke a small hole), a protractor and a calculator.

Measure the box's length, width and height.
Predict the floor diagonal and the space diagonal (Example 1), and the angle the space diagonal makes with the bottom.
Check the floor diagonal with the ruler across the bottom of the box.
Check the space diagonal: with an adult, poke a tiny hole in two opposite corners (one bottom corner and the top corner farthest from it), push string through both holes so it runs straight across the inside of the box, mark the string at each hole, then take it out and measure between the marks.

What to look for: measured diagonals within about 0.5 cm of the predictions. The space diagonal should be longer than every edge and every face diagonal.

Your turn

A box is 30 cm by 20 cm by 15 cm. Find the length of its space diagonal.

For the box in question 1, find the angle between the space diagonal and the 30 cm by 20 cm base.

A cube has edges of 10 cm. Find the angle between a space diagonal and the base.

A flagpole is 12 m tall. From a point due west of its base the angle of elevation is 50°, and from a point due north of its base the angle of elevation is 35°. How far apart are the two points?

A ramp in a parking garage is 20 m long and rises 2.5 m. It runs diagonally across a section of the floor. What angle does the ramp make with the floor?

A surveyor at A walks 60 m along a line perpendicular to AF (F is the base of a tree) to point B. ∠ABF = 40°. At A, the angle of elevation to the treetop is 22°. How tall is the tree?

More practice: warm-up

How many right triangles are needed to find the space diagonal of a box?

Find the diagonal of a 6 by 8 rectangle.

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

Find √(10² + 24²).

tan θ = 1. Find θ.

More practice: on your own

A tent is shaped like a square-based pyramid. The base is 2.4 m by 2.4 m and the top is 1.8 m directly above the centre of the base. Find the length of a slanted edge from a base corner to the top.

For the tent above, find the angle between a slanted edge and the ground.

A room is 6.0 m by 4.5 m by 2.7 m. A cable is run straight from one bottom corner to the opposite top corner. How long is it?

Ben says the angle between the space diagonal of a box and the floor is tan⁻¹(height ÷ length). Explain his mistake.

A drone is 40 m above a point P on a field. Alex stands 30 m north of P and Bea stands 50 m east of P. Find the angle of elevation from each person to the drone, and the distance between Alex and Bea.

Challenge

A box has a square base of side s and height h. Its space diagonal makes a 30° angle with the base. Find h in terms of s.

From a point A due south of a tower, the angle of elevation to the top is 45°. From point B, 100 m due east of A, the angle of elevation is 30°. Find the height of the tower. (Hint: FA = h, FB = h/tan 30° = h√3, and ∠FAB = 90°.)

A ladder 7.0 m long leans against a wall. Its foot is 1.5 m from the wall, but the ladder is also turned so its foot is 1.2 m to the side (along the wall) of the point directly below its top. How high up the wall does it reach?

Check your understanding

Quiz

A room is 5.0 m by 4.0 m by 2.6 m. Its floor diagonal is about:

In the same room, the space diagonal is about:

The angle between a box's space diagonal and its base is found with:

Point A is due south of a tower's base F and point B is due east of F. What is ∠AFB?

In 3-D sketches, a vertical tower makes what angle with a line on the ground through its base?

For parents and teachers

Big idea: This lesson extends C1.1 and C1.2 (right triangles and two-triangle problems) into three dimensions. The Ontario expectation C1.2 names two dimensions; 3-D problems are included here because they are common in applications (and in later courses) and use exactly the same skills. If your teen's school course does not include 3-D, this lesson can be treated as enrichment.

How to help: A real box on the table helps enormously. Ask your teen to put a finger on each right angle before they calculate.

Common difficulties:

Not seeing right angles in a 3-D sketch because they don't look like 90° on paper.
Using the length of a box instead of the floor diagonal for the angle with the floor.
Rounding the shared floor diagonal too early.