Right Triangle Worksheets - Math Monks - Free Printable

Let’s solve each problem one by one, step by step. We’ll use right triangle trigonometry — specifically sine, cosine, and tangent — depending on what sides and angles we’re given.

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Problem 1:

Anuj is flying a kite with a string of 250 meters, making an angle of 60° with the ground. Find how high the kite is (to nearest foot).

Wait — units! The string is in meters, but answer needs to be in feet.

First, find height in meters using sine:

In a right triangle:
- Hypotenuse = string = 250 m
- Angle with ground = 60°
- Height (opposite side) = ?

→ sin(θ) = opposite / hypotenuse
→ sin(60°) = height / 250
→ height = 250 × sin(60°)

sin(60°) ≈ 0.8660

height = 250 × 0.8660 = 216.5 meters

Now convert meters to feet:
1 meter ≈ 3.28084 feet

216.5 × 3.28084 ≈ let’s calculate:

200 × 3.28084 = 656.168
16.5 × 3.28084 ≈ 54.13386
Total ≈ 656.168 + 54.13386 = 710.30186 feet

Rounded to nearest foot → 710 feet

✔ Check: 250 * sin(60) = 216.5 m → 216.5 * 3.28084 ≈ 710.3 → yes, rounds to 710.

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Problem 2:

Ladder is 30 ft long, reaches 27 ft up building. What is the angle with the vertical?

Note: It says “angle of the vertical” — meaning angle between ladder and the wall (vertical), not the ground.

So, imagine:
- Ladder = hypotenuse = 30 ft
- Height up wall = adjacent side to the angle with vertical = 27 ft
- We want angle θ between ladder and wall (vertical)

Use cosine:

cos(θ) = adjacent / hypotenuse = 27 / 30 = 0.9

θ = cos⁻¹(0.9)

Using calculator: cos⁻¹(0.9) ≈ 25.84 degrees

But wait — is this the angle with the vertical? Yes, because we used the side along the wall (adjacent to θ) and hypotenuse.

So angle with vertical ≈ 25.8°

We can round to nearest tenth or whole number? Problem doesn’t specify — let’s keep one decimal: 25.8°

✔ Check: If angle with vertical is 25.8°, then cos(25.8°) ≈ 0.9 → 30 * 0.9 = 27 → correct.

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Problem 3:

Tower casts shadow 60 ft long. Sun’s angle of elevation = 65°. How tall is tower?

Angle of elevation is from ground to sun → so at end of shadow, looking up at top of tower.

Right triangle:
- Shadow = adjacent side = 60 ft
- Tower height = opposite side = ?
- Angle = 65°

Use tangent:

tan(θ) = opposite / adjacent
tan(65°) = height / 60
height = 60 × tan(65°)

tan(65°) ≈ 2.1445

height = 60 × 2.1445 ≈ 128.67 ft

Round to nearest foot? Probably → 129 ft

✔ Check: 60 * tan(65) ≈ 60 * 2.1445 = 128.67 → rounds to 129.

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Problem 4:

You stand 70 ft from hot air balloon. Angle of elevation to top = 28°. Find height.

Again, right triangle:
- Distance from you to balloon base = adjacent = 70 ft
- Height = opposite = ?
- Angle = 28°

tan(28°) = height / 70
height = 70 × tan(28°)

tan(28°) ≈ 0.5317

height = 70 × 0.5317 ≈ 37.219 ft

Round to nearest foot? → 37 ft

✔ Check: 70 * 0.5317 ≈ 37.2 → yes.

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Problem 5:

Tripti walks 200 ft from tree. Angle from eyes to top = 35°. Eye level = 5 ft above ground. Find total tree height.

Two parts:
1. Height from eye level to top of tree
2. Add 5 ft for eye level

Right triangle:
- Adjacent = 200 ft
- Opposite = height above eyes = ?
- Angle = 35°

tan(35°) = opposite / 200
opposite = 200 × tan(35°)

tan(35°) ≈ 0.7002

opposite = 200 × 0.7002 = 140.04 ft

Add eye level: 140.04 + 5 = 145.04 ft

Round to nearest foot → 145 ft

✔ Check: 200 * tan(35) ≈ 140.04 + 5 = 145.04 → yes.

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Problem 6:

Flag pole = 80 ft tall. Rope tied from ground, makes 45° with ground. How long is rope?

Rope is hypotenuse. Pole is opposite side to 45° angle.

sin(45°) = opposite / hypotenuse = 80 / rope_length

rope_length = 80 / sin(45°)

sin(45°) ≈ 0.7071

rope_length = 80 / 0.7071 ≈ 113.14 ft

Alternatively, since it’s 45°, it’s an isosceles right triangle → legs equal → both 80 ft? Wait no!

Wait — if angle with ground is 45°, and pole is vertical, then yes — the two legs are equal only if angle at ground is 45° and pole is perpendicular.

Actually, in this case:

- Pole = opposite = 80 ft
- Angle at ground = 45°
- So adjacent (ground distance) should also be 80 ft? Because tan(45)=1 → opposite/adjacent=1 → adjacent=80

Then hypotenuse (rope) = √(80² + 80²) = √(12800) = 80√2 ≈ 80 × 1.4142 ≈ 113.14 ft

Same as before.

So rope length ≈ 113 ft (rounded to nearest foot)

✔ Check: 80 / sin(45) = 80 / 0.7071 ≈ 113.14 → yes.

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Final Answers:

1. 710 feet
2. 25.8 degrees
3. 129 feet
4. 37 feet
5. 145 feet
6. 113 feet

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Final Answer:
1. 710
2. 25.8
3. 129
4. 37
5. 145
6. 113