Solids

SOLIDS A solid figure, or solid, is any portion of space bounded by one or more surfaces, plane or curved. These surfaces are called the faces of the solid, and the intersections of adjacent faces are called edges. Let’s have a look at some common solids and their properties. Parallelepiped: A parallelepiped is a solid bounded by three pairs of parallel plane faces. • • • Each of the six faces of a parallelepiped is a parallelogram. Opposite faces are congruent. The four diagonals of a parallelepiped are concurrent and bisect one another. Cuboid: A parallelepiped whose faces are rectangular is called a cuboid. The three dimensions associated with a cuboid are its length, breadth and height (denoted as l, b and h here.) • • • • The length of the three pairs of face diagonals are BF = b2 + h2 , AC = l2 + h2 , and DF = l2 + b2 . The length of the four equal body diagonals AF = l2 + b2 + h2 . The total surface area of the cuboid = 2(lb + bh + hl) Volume of a cuboid = lbh + h2 Diagonal = . 2 2 Note that if the dimensions of the cuboid are not equal, there cannot be a sphere which can be • inscribed in it, i.e. a sphere which touches all the faces from inside. • The radius of the sphere circumscribing the cuboid = Euler’s Formula: the number of faces (F), vertices (V), and edges (E) of a solid bound by plane faces are related by the formula F + V = E + 2 gives here 6 + 8 = 12 + 2. Cube: A cube is a parallelepiped all of whose faces are squares. • • Total surface area of the cube = 6a2 Volume of the cube = a3 • Length of the face diagonal b = 2a • Length of the body diagonal c = 3a • Radius of the circumscribed sphere = • Radius of the inscribed sphere = • Radius of the sphere tangent to edges = 3a 2 a 2 a 2 Prism: A prism is a solid bounded by plane faces, of which two, called the ends, are congruent figures in parallel planes and the others, called side-faces are parallelograms. The ends of a prism may be triangles, quadrilaterals, or polygons of any number of sides. The side- edges of every prism are all parallel and equal. A prism is said to be right, if the side-edges are perpendicular to the ends: In this case the side faces are rectangles. Cuboids and cubes are examples. Curved surface area of a right prism = perimeter of the base × height • Total surface area of a right prism = perimeter of the base × height + 2 × area of the base • Volume of a right prism = area of the base × height • • • Right Circular Cylinder: A right circular cylinder is a right prism whose base is a circle. In the figure given below, the cylinder has a base of radius r and a height of length h. • • • Curved surface area of the cylinder = 2πrh Total surface area of the cylinder = 2πrh + πr2 Volume of the cylinder = πr2h Pyramid: A pyramid is a solid bounded by plane faces, of which one, called the base, is any rectilinear figure, and the rest are triangles having a common vertex at some point not in the plane of the base. The slant height of a pyramid is the height of its triangular faces. The height of a pyramid is the length of the perpendicular dropped from the vertex to the base. In a pyramid with n sided regular polygon as its base, Total number of vertices = n + 1 Perimeter × slant height . Curved surface area of the pyramid = • 2 Perimeter Total surface area of the pyramid = × slant height + area of the base. • 2 Base area Volume of the pyramid = • × height 3 • Tetrahedron: A tetrahedron is a pyramid which has four congruent equilateral triangles as it four faces. The figure below shows a tetrahedron with each face equal to an equilateral triangle of side a. Total number of vertices = 4 The four lines which join the vertices of a tetrahedron to the centroids of the opposite faces meet at a point which divides them in the ratio 3: 1. In the figure. AH: HF = 3: 1. 3 3a2 Curved surface area of the tetrahedron = . • 4 Total surface area of the tetrahedron = 3a2 . • • • • Height of the tetrahedron = • Volume of the tetrahedron = 6a 3 2a3 12 Right Circular Cone: a right circular cone is a pyramid whose base is a circle. In the figure given below, the right circular cone has a base of radius r and a height of length h. • • • • Slant height l = h2 + r2 Curved surface area of the cone = πrl Total surface area of the cone = πrl + πr2 πr2h Volume of the cone = 3 Frustum of a Cone: When a right circular cone is cut by a plane parallel to the base, the remaining portion is known as the frustum. h2 + (R − r)2 . • Slant height l = • Curved surface area of the frustum = π(r + R)l Total surface area = π(r + R)l + π(r2 + R2) πh(r2 + R2 + R r) Volume of the frustum = 3 • • Sphere: A sphere is a set of all points in space which are at a fixed distance from a given point. The fixed point is called the centre of the sphere, and the fixed distance is the radius of the sphere. • • Surface area of a sphere = 4πr2 4 3 Volume of a sphere = πr 3 Spherical Shell: A hollow shell with inner and outer radii of r and R, respectively. • Volume of the shell = 4 π(R3 − r3 ) 3