📐 geometry

1. **Problem statement:** (a) Find the height of pyramid VPQR inside the frustum.

1. The problem is to verify or understand why the answer is 1155 square meters. 2. Typically, such a problem involves finding the area of a shape, often a rectangle or composite fi

1. **Problem statement:** Find the total surface area of a building with a floor plan shaped like an irregular polygon (a "U" shape) and a flat roof. The building height is 8 m. 2.

1. **State the problem:** We need to find the total surface area of a building that is 8m tall, with a floor plan shaped like a large rectangle with a smaller rectangular cutout at

1. Énoncé du problème : On a un triangle ABC avec E sur AB et F sur AC, et BC parallèle à EF.

1. **State the problem:** We have an isosceles triangle PQR with base RQ = 22 cm and height from P to midpoint M of RQ equal to 10 cm. We need to find the length of side PR. 2. **R

1. **State the problem:** Kayden has 50 m of railing but needs to know how much more railing is required to go all the way around the roof garden, which has a composite shape with

1. **State the problem:** We have a rectangular pyramid with base ABCD where AB = 7 cm and BC = 9 cm. The apex is T, and all edges TA, TB, TC, and TD are 12 cm. M is the midpoint o

1. **Stating the problem:** We are given a quadrilateral E-D-C-B with an additional point A forming a triangle A-D-B. The angles are \(\angle A = 80^\circ\), \(\angle E = 70^\circ\

1. **State the problem:** We need to find the total area of a sign made from a semicircle on top of an isosceles triangle. The base of the semicircle and the top side of the triang

1. **Problem Statement:** We have a circle with two points A and B on its circumference. Line AB is not a diameter. We need to order the following parts of the circle by their area

1. **Problem statement:** We need to find the measure of angle $S$ in triangle $STR$ where side $TR=4.7$ mi, side $SR=5.1$ mi, and angle $T=77^\circ$. 2. **Formula used:** We will

1. **Stating the problem:** We have a semicircle with diameter 28 cm. Inside it, two curved shapes meet at the top center, forming a shaded area enclosed by two arcs meeting at the

1. **Problem Statement:** Find the area of a regular heptagon and a regular octagon given the perimeter $P=12$. 2. **Formula for area of a regular polygon:**

1. **Stating the problem:** We have a triangle with a base of length 4 and angles 25.71° and 51.43°. We want to find the height $h$ perpendicular to the base and verify the calcula

1. **Problem Statement:** Calculate the area of a regular pentagon with side length $s=2.4$ by dividing it into 5 equal isosceles triangles. 2. **Perimeter Calculation:** The perim

1. **Problem statement:** We are given a cube ABCDA'B'C'D' with vertices A and C' and the midpoint of edge DD' lying on a plane section. The area of this section is given as $50\sq

1. **Problem statement:** Given triangle ABC with angles at B and C as 72° and 36° respectively, and points and lines as described, we need to: - Write the angles BCA and ACB.

1. **Stating the problem:** We have a triangle with points A, B, C and points D, E on sides AC and BC respectively. Given that $FE=3$, $EC=ED$, $AB=AD=5$, and $AF \parallel DE$, we

1. **Problem statement:** Find the volume of the greatest cylinder that can be inscribed in a cone with height $h$ and semi-vertical angle $30^\circ$. The goal is to show that this

1. **Problem statement:** We have an isosceles triangle with two equal sides each measuring 6.2 mm and an angle of 58° opposite the side of length $w$. We need to find the length $