Pythagorean theorem in 3D | Geometry | 8th grade | Khan Academy

Recent questions

• What is a rectangular prism?

  A rectangular prism is a three-dimensional geometric shape characterized by six rectangular faces, where opposite faces are equal in area. It has length, width, and height dimensions, making it a type of polyhedron. The edges of a rectangular prism meet at right angles, and it can be visualized as a box or a cuboid. The volume of a rectangular prism can be calculated by multiplying its length, width, and height, while the surface area is determined by adding the areas of all six faces. This shape is commonly found in everyday objects, such as books, boxes, and buildings.

• How do you calculate the hypotenuse?

  To calculate the hypotenuse of a right triangle, you can use the Pythagorean theorem, which states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). The formula is expressed as c² = a² + b². For example, if one side measures 1 unit and the other side measures 2 units, you would calculate the hypotenuse as follows: c² = 1² + 2², which simplifies to c² = 1 + 4, resulting in c² = 5. Taking the square root of both sides gives you the hypotenuse, which in this case would be √5 units.

• What is the Pythagorean theorem?

  The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be expressed mathematically as a² + b² = c², where c represents the hypotenuse, and a and b represent the other two sides. This theorem is widely used in various fields, including architecture, engineering, and physics, to solve problems involving right triangles and to calculate distances.

• What is a right pyramid?

  A right pyramid is a three-dimensional geometric figure that has a polygonal base and triangular faces that converge at a single point called the apex. The apex is directly above the center of the base, making the pyramid "right" as opposed to oblique. The base can be any polygon, such as a triangle, square, or pentagon, and the height of the pyramid is the perpendicular distance from the apex to the base. Right pyramids are commonly seen in structures like the Great Pyramid of Giza and are characterized by their symmetrical shape. The volume of a right pyramid can be calculated using the formula V = (1/3) × base area × height.

• How do you find the edge length of a pyramid?

  To find the edge length of a pyramid, particularly when dealing with a right pyramid, you can use the Pythagorean theorem. First, identify the relevant dimensions, such as the height of the pyramid and the dimensions of the base. By forming a right triangle that includes the height and half the base length, you can apply the theorem. For instance, if the height is 1 unit and the base dimensions create a right triangle with one side measuring 1 unit and the other measuring 2 units, you can calculate the hypotenuse. This hypotenuse represents the slant height, and by using the height and the hypotenuse, you can find the edge length using the formula x = √(height² + hypotenuse²). This method provides a clear way to determine the edge length accurately.