Geometry

This page is part of the section “USE basic level. Geometry”. Here are solutions to typical geometry problems: area of a triangle, Pythagorean theorem, area of a circle, volume of a cylinder. All problems are written independently, with neat answers.

Problems with solutions

Problem 1. Area of a triangle

Find the area of a triangle if two of its sides are 3 and 4, and the angle between them is 30°.

The area of a triangle in terms of two sides and the angle between them:

S = \frac{1}{2} \cdot a \cdot b \cdot sin γ = \frac{1}{2} \cdot 3 \cdot 4 \cdot sin 30 °

Substitute sin 30° = 1/2:

S = \frac{1}{2} \cdot 3 \cdot 4 \cdot \frac{1}{2} = 3

Answer: 3.

Problem 2. Pythagorean theorem

In a right triangle the legs are 6 and 8. Find the hypotenuse.

By the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the legs:

c^{2} = a^{2} + b^{2} = 6^{2} + 8^{2} = 36 + 64 = 100

c = \sqrt{100} = 10

Answer: 10.

Problem 3. Area of a circle

Find the area of a circle of radius 5.

The area of a circle is given by S = πR²:

S = π R^{2} = π \cdot 5^{2} = 25 π

Answer: 25π.

Problem 4. Volume of a cylinder

Find the volume of a cylinder if the radius of its base is 2 and the height is 5.

The volume of a cylinder is given by V = πR²h:

V = π R^{2} h = π \cdot 2^{2} \cdot 5 = 20 π

Answer: 20π.

Frequently asked questions

How to find the area of a triangle from two sides and the included angle?

Use the formula S = ½·a·b·sin γ, where a and b are the sides and γ is the angle between them. For example, S = ½·3·4·sin 30° = ½·3·4·½ = 3.

How to apply the Pythagorean theorem?

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs: c² = a² + b². Knowing two elements, you can find the third. For legs 6 and 8: c² = 36 + 64 = 100, c = 10.

How to find the area of a circle?

Use the formula S = πR², where R is the radius. For R = 5: S = π·25 = 25π. The answer can be left with π or approximated with π ≈ 3.14 — but in the USE basic level it is usually left with π.

How to find the volume of a cylinder?

Use the formula V = πR²h, where R is the base radius and h is the height. For R = 2 and h = 5: V = π·4·5 = 20π.