📐 geometry

1. The problem is to find the remaining interior angle of a polygon when the sum of the interior angles is 540 degrees. 2. The formula for the sum of interior angles of a polygon w

1. **State the problem:** We have a triangle sharing a side with an irregular pentagon. We know four interior angles of the pentagon: 61°, 113°, 22°, and 108°. We want to find the

1. **State the problem:** We need to find the radius $r$ of a cylindrical glass that holds 150 ml (which equals 150 cm³) of juice up to a height of 8 cm. 2. **Formula used:** The v

1. **State the problem:** We have a cylindrical glass filled with juice to a height of 8 cm, and the volume of juice is 150 cm³. We need to find the radius $r$ of the glass.

1. **State the problem:** Calculate the shaded area in the semicircle diagram where the radius $r=25$ cm and $OM=18$ cm. 2. **Identify the shaded area:** The shaded area is the seg

1. **State the problem:** We have a semicircle with center O and radius 25 cm. M is the midpoint of chord AB, and OM = 18 cm. We need to (a) show that angle AOB = 87.9° correct to

1. **State the problem:** We need to find the total surface area of a cylinder with height $h=7$ m and radius $r=4$ m. 2. **Formula for total surface area of a cylinder:**

1. **State the problem:** We have a cylinder net consisting of two circles each with radius 5 mm and a rectangle with height 3 mm and length $x$.

1. **State the problem:** We have two triangles, ONM and KML, with NO parallel to KL. Given lengths are KL = 21, NO = 14, ML = 24, and KM = 15. We need to find the length of NM. 2.

1. **State the problem:** We have an L-shaped polygon with given side lengths 4 cm, 6 cm, and 10 cm, and an unknown top horizontal side labeled as [Box] cm. The total perimeter is

1. **Problem statement:** We have quadrilateral ABCD with given sides |AD| = 4 cm, |DC| = 6 cm, and angles \(\angle ADC = 60^\circ\), \(\angle CAB = 30^\circ\), and \(\angle ABC =

1. **Problem statement:** Given a prism ABC.A'B'C' with M as the midpoint of BB'. Given vectors \(\overrightarrow{AC} = \vec{a}, \overrightarrow{BC} = \vec{b}, \overrightarrow{AA'}

1. **State the problem:** We have two parallel lines cut by a transversal, creating angles of 72° and y° as shown. 2. **Identify the relationship:** When two parallel lines are cut

1. **Calculate the area of each shape** (i) L-shaped polygon made of rectangles:

1. **Stating the problem:** We are given a zigzag line with 5 points labeled 1 through 5 and some angles: $m\angle 1 = 88^\circ$, $m\angle 3 = 81^\circ$, and $m\angle 5 = 64^\circ$

1. The problem is to find the hypotenuse $a$ of a right triangle where the legs are $\sqrt{10}$ and $\sqrt{6}$.\n\n2. According to the Pythagorean theorem, for a right triangle wit

1. **Stating the problem:** We are given a zigzag line with 5 segments and several angles: 88°, 92°, 81°, and 64°. We need to find the measures of angles $m\angle 1$, $m\angle 2$,

1. The problem asks for the area of a shape denoted as $S$. 2. To find the area, we need to know the shape and its dimensions or a formula that applies to $S$.

1. **State the problem:** We have a right triangle with legs of lengths $\sqrt{10}$ and $\sqrt{6}$, and we want to find the length of the hypotenuse. 2. **Formula used:** In a righ

1. **State the problem:** We have a right triangle with legs of lengths 8 and 6, and we want to find the length of the hypotenuse $a$. 2. **Formula used:** In a right triangle, the

1. **Stating the problem:** We are given a right triangle with side lengths 3, 4, and 5. We need to find the area $S$ of this triangle.