Geometry Solved Questions and Answers (original) (raw)

Last Updated : 22 Apr, 2026

Geometry is the branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, surfaces, and solids in space.

**Geometry questions and answers are provided below for you to learn and practice.

**Question 1: In a circle with a radius of 10 cm, find the length of a chord that is 6 cm from the center.

**Solution:

Let the radius r = 10cm and the distance from the center to the chord d = 6.
Use the formula for the chord length:

L = 2\sqrt{r^2 - d^2}

L = 2\sqrt{10^2 - 6^2}

= 2\sqrt{100 - 36}

= 2√64

= 2 × 8

= **16 cm

**Question 2: Find the sum of the interior angles of a polygon with 8 sides.

**Solution:

Sum of interior angles of an n-sided polygon = (n−2) × 180∘
Sum = (8 − 2) × 180 = 6 × 180 = 1080 ∘

**Question 3: Find the measure of an angle if five times its complement is 10° less than twice its supplement.

**Solution:

According to the problem:
5 × (90∘ − x) = 2 × (180∘ − x) − 10∘

Now, solve the equation:
450∘ − 5x = 360∘ − 2x − 10∘
450∘ − 350∘ = −2x + 5x
100∘ = 3x
**x = 33.33 ∘

**Question 4: In a triangle ΔXYZ, if 3∠X = 4∠Y = 5∠Z, then find the value of ∠X.

**Solution:

Let: 3∠X = 4∠Y = 5∠Z = k

From this, we can express the angles as:

The sum of the angles in a triangle is 180∘:

∠X + ∠Y+ ∠Z = 180∘

Substituting the expressions for ∠X, ∠Y, and ∠Z:

k/3 + k/4 + k5 = 180∘

20k + 15k + 12k/60 ​= 180∘

47k = 10800

k = 229.79∘

∠X = k/3 ​= 229.79/3 = **76.6.

**Question 5: What is the formula for calculating the volume of a cylinder, and how do you apply it to a cylinder with a radius of 7 cm and a height of 10 cm?

**Solution:

To calculate the volume of a cylinder, you can use the formula:

Volume = πr2h

r = 7

h =10

volume = π(7 × 7) × 10

**490π cm 3

**Question 6: A triangle has sides of lengths 12 cm, 12 cm, and 9 cm. Calculate the area of the triangle.

**Solution:

A triangle has a base b=10 and two equal sides of 12 cm each.

we can use **Heron's formula.
The formula for the area of a triangle with sides of lengths a, b, and c is:

A = \sqrt{s(s - a)(s - b)(s - c)}

Where:
s = a + b + c/2

\frac{12 + 12 + 9}{2} = \frac{33}{2} = 16.5\ \text{cm}

A = \sqrt{16.5(16.5 - 12)(16.5 - 12)(16.5 - 9)}

A = \sqrt{16.5 \times 4.5 \times 4.5 \times 7.5}

16.5 × 4.5 = 74.25
74.25 × 4.5 = 333.125
333.125 × 7.5 = 2498.4375

Now, take the square root:

A = √2498.4375 ​≈ 49.98cm2

The area of the triangle is approximately **50 cm².

**Question 7: The distance between the centers of two circles with radii 8 cm and 5 cm is 20 cm. What is the length of the traverse common tangent to the circles?

**Solution:

Length of traverse common tangent = √[(Distance between their centres)2-(r1 + r2)2]

= √[(20)2 - (8 + 5)2]

= √(400 - 169)

= √ (231)

= 15.2 cm

**Question 8: If each interior angle of a regular polygon is 120∘120^\circ120∘, what is the number of sides of the polygon?

**Solution:

**Interior angle = 120∘
**Exterior angle = 180∘ − 120∘ = 60∘
**Number of sides of polygon = 360∘/exterior angle = 360∘/60∘ = 6

The number of sides of the polygon is **6.

Also Check:

**Geometric Shapes in Maths****.**