Geometry and trigonometry word problems worksheet with diagrams and calculations.

Let’s solve each problem one by one. We’ll use trigonometry — specifically, sine, cosine, and tangent — to find missing sides or angles in right triangles. Remember:

- Sine (sin) = opposite / hypotenuse
- Cosine (cos) = adjacent / hypotenuse
- Tangent (tan) = opposite / adjacent

We’ll also use the Pythagorean theorem when needed:
a² + b² = c², where c is the hypotenuse.

---

Problem 1:

> A ladder leans against a wall at an angle of 65° with the ground. The base of the ladder is 8 feet from the wall. How long is the ladder?

This is a right triangle:
- Angle with ground = 65°
- Adjacent side (ground) = 8 ft
- Hypotenuse = ladder length (what we’re solving for)

Use cosine:
cos(65°) = adjacent / hypotenuse = 8 / L
→ L = 8 / cos(65°)

Calculate cos(65°): ≈ 0.4226
L = 8 / 0.4226 ≈ 18.93 → Round to nearest tenth: 18.9 ft

✔ Check: Makes sense — ladder should be longer than 8 ft since it’s leaning.

---

Problem 2:

> From a point on the ground 75 meters from the base of a tower, the angle of elevation to the top is 32°. Find the height of the tower.

Right triangle:
- Angle = 32°
- Adjacent = 75 m
- Opposite = height (h)

Use tangent:
tan(32°) = h / 75
→ h = 75 × tan(32°)

tan(32°) ≈ 0.6249
h = 75 × 0.6249 ≈ 46.87 → Round to nearest meter: 47 m

✔ Check: Reasonable height for a tower from 75m away at 32°.

---

Problem 3:

> A kite string is 100 meters long and makes a 40° angle with the ground. How high is the kite?

Hypotenuse = 100 m
Angle = 40°
Opposite = height (h)

Use sine:
sin(40°) = h / 100
→ h = 100 × sin(40°)

sin(40°) ≈ 0.6428
h = 100 × 0.6428 = 64.28 → Round to nearest meter: 64 m

✔ Check: Less than 100m — makes sense.

---

Problem 4:

> A ramp rises 3 feet over a horizontal distance of 12 feet. What is the angle of elevation?

Opposite = 3 ft
Adjacent = 12 ft
Find angle θ

Use tangent:
tan(θ) = 3/12 = 0.25
→ θ = arctan(0.25)

arctan(0.25) ≈ 14.04° → Round to nearest degree: 14°

✔ Check: Small rise over long run → small angle — correct.

---

Problem 5:

> A guy wire is attached to the top of a 20-foot pole and anchored 15 feet from the base. Find the length of the wire.

This is a right triangle:
- Legs: 20 ft and 15 ft
- Hypotenuse = wire length

Use Pythagorean theorem:
c² = 20² + 15² = 400 + 225 = 625
c = √625 = 25 → Exactly 25 feet

✔ Check: Classic 3-4-5 triangle scaled by 5 → 15-20-25 — perfect.

---

Problem 6:

> From the top of a cliff 100 meters high, the angle of depression to a boat is 25°. How far is the boat from the base of the cliff?

Angle of depression = 25° → same as angle of elevation from boat to top.

So:
- Opposite = 100 m (height)
- Adjacent = distance to boat (d)
- Angle = 25°

Use tangent:
tan(25°) = 100 / d
→ d = 100 / tan(25°)

tan(25°) ≈ 0.4663
d = 100 / 0.4663 ≈ 214.45 → Round to nearest meter: 214 m

✔ Check: Farther than height — makes sense for shallow angle.

---

Problem 7:

> A tree casts a shadow 20 feet long when the sun’s angle of elevation is 50°. How tall is the tree?

Shadow = adjacent = 20 ft
Angle = 50°
Height = opposite = h

Use tangent:
tan(50°) = h / 20
→ h = 20 × tan(50°)

tan(50°) ≈ 1.1918
h = 20 × 1.1918 ≈ 23.836 → Round to nearest foot: 24 ft

✔ Check: Taller than shadow — yes, because angle > 45°.

---

Problem 8:

> A plane flies at an altitude of 3000 meters. The pilot sees a runway at an angle of depression of 10°. How far horizontally is the plane from the runway?

Same as angle of elevation from runway to plane = 10°
Opposite = 3000 m
Adjacent = horizontal distance (d)

Use tangent:
tan(10°) = 3000 / d
→ d = 3000 / tan(10°)

tan(10°) ≈ 0.1763
d = 3000 / 0.1763 ≈ 17016.45 → Round to nearest meter: 17,016 m

✔ Check: Very far — makes sense for small angle from high up.

---

Problem 9:

> A 12-foot ladder reaches 10 feet up a wall. What angle does it make with the ground?

Hypotenuse = 12 ft
Opposite = 10 ft
Find angle θ

Use sine:
sin(θ) = 10 / 12 ≈ 0.8333
→ θ = arcsin(0.8333) ≈ 56.44° → Round to nearest degree: 56°

✔ Check: Steep angle — makes sense since it reaches high relative to its length.

---

Problem 10:

> A person stands 50 feet from a building. The angle of elevation to the top is 35°. If the person’s eyes are 5 feet above ground, how tall is the building?

First, find height from eye level to top:
tan(35°) = h / 50 → h = 50 × tan(35°)
tan(35°) ≈ 0.7002 → h ≈ 35.01 ft

Add eye height: total height = 35.01 + 5 = 40.01 → Round to nearest foot: 40 ft

✔ Check: Reasonable — slightly taller than the calculated part due to eye height.

---

Final Answer:
1. 18.9 ft
2. 47 m
3. 64 m
4. 14°
5. 25 ft
6. 214 m
7. 24 ft
8. 17,016 m
9. 56°
10. 40 ft