Surface Area and Volume - Cuboid, Cube, Cylinder, Cone, Sphere

Sep 18, 2026

Uploaded by Mabel

A math learning sheet focused on surface area and volume for common 3D shapes. It defines cuboids, cubes, right circular cylinders, right circular cones, and spheres, then derives key formulas for total surface area, lateral surface area (where applicable), curved surface area, and volume. Expressions are presented using standard variables for length, breadth, height, radius, and slant height, including relations like l^2 = h^2 + r^2 and the cone volume as one-third of a cylinder with the same base and height.

LanguageEnglish

CategoryExam

FormatPDF

Pages6

Reading Time15 min

AccessFree to download

Surface Area and Volume

Cuboid

Cube

Right Circular Cylinder

Right Circular Cone

Sphere

Volume of Solids

Volume and Capacity

Surface Area and Volume Cuboid Cuboid is a three dimensional Shape. Cuboid is made from six rectangular faces, which are placed at right angles. The total surface area of a cuboid is equal to the sum of the areas of its six rectangular faces. Total Surface Area of a Cuboid Consider a cuboid whose length is l cm, breadth is b cm and height h cm. Area offace ABCD = Area of Face EFGH = (l × b)cm2 Area of face AEHD = Area offace BFGC = (b × h)cm2 Area of face ABFE = Area of face DHGC = (l × h)cm2 Total surface area (TSA) of cuboid = Sum of the areas of all its six faces = 2(l × b) + 2(b × h) + 2(l × h) TSA (cuboid)= 2(lb + bh +lh) Lateral Surface Area of a Cuboid Lateral surface area (LSA) is the area of all the sides apart from the top and bottom faces. Lateral surface area of the cuboid = Area of face AEHD + Area offace BFGC + Area of face ABFE + Area of face DHGC = 2(b × h) + 2(l × h) LSA (cuboid) = 2h(l + b) Cube A whose length, breadth and height are all , is called a . It is a threedimensional shape bounded by . It has 12 edges and 8 vertices. Total Surface Area of a cube For cube, length = breadth = height Suppose length of an edge = a Total surface area(TSA) of the cube = 2(a × a + a × a + a × a) TSA(cube) = 2 × (3a2 ) = 6a2 Lateral Surface area of a cube Lateral surface area (LSA) is the area of all the sides apart from the top and bottom faces. Lateral surface area of cube = 2(a × a + a × a) = 4a2 Right Circular Cylinder A right circular cylinder is a closed solid that has two parallel circular bases connected by a curved surface in which the two bases are exactly over each other and the axis is at right angles to the base. Curved Surface area of a right circular cylinder Take a cylinder of base radius r and height h units. The curved surface of this cylinder, if opened along the diameter (d = 2r) of the circular base can be transformed into a rectangle of length 2πr and height h units. Thus, Curved surface area(CSA) of a cylinder of base radius r and height h = 2π × r × h Total surface area of a right circular cylinder Total surface area(TSA) of a cylinder of base radius r and height h = 2π × r × h + area of two circular bases ⇒ TSA = 2π × r × h + 2 × πr2 ⇒ TSA = 2πr(h + r) Right Circular Cone A right circular cone is a circular cone whose axis is perpendicular to its base. Relation between slant height and height of a right circular cone A right-circular cone. The relationship between slant height(l) and height(h) of a right circular cone is: l2 = h2 + r2 (Using Pythagoras Theorem) Where r is the radius of the base of the cone. Curved Surface Area of a Right Circular Cone Consider a right circular cone with slant length land radius r. If a perpendicular cut is made from a point on the circumference of the base to the vertex and the cone is opened up, a sector of a circle with radius lis produced as shown in the ﬁgure below: Label A and B and corresponding b1, b2 ...bn at equal intervals, with O as the common vertex. The Curved surface area(CSA) of the cone will be the sum of areas of the small triangles : 12 × (b1 + [b2. . . bn](b2. . . bn)) × l (b1 + [b2. . . bn](b2. . . bn)) is also equal to circumference of base = 2πr CSA of right circular cone = 12 × (2πr) × l = πrl (On substituting the values) Total Surface Area of a Right Circular Cone Total surface area(TSA) = Curved surface area(CSA) + area of base = πrl + πr2 = πr (l + r) Sphere A sphere is a three-dimensional ﬁgure (solid ﬁgure), which is made up of all points in the space which lie at a constant distance called the radius, from a ﬁxed point called the center of the sphere. Surface area of a Sphere The surface area of a sphere of radius r = 4 times the area of a circle of radius r = 4 × (πr2 ) For a sphere Curved surface area(CSA) = Total Surface area(TSA) = 4πr2 Volume of a Cuboid The volume of a cuboid is the product of its dimensions. Volume of a cuboid

FAQs

What is the total surface area formula for a cuboid?

For a cuboid with length l, breadth b, and height h, TSA = 2(lb + bh + lh).

How do you find the lateral surface area of a cuboid?

LSA for a cuboid equals 2h(l + b), using the side faces excluding the top and bottom.

What is the volume of a right circular cone in terms of cylinder volume?

A right circular cone has volume equal to one-third of a cylinder with the same radius and height: V = (1/3)πr^2h.

Sep 18, 2026

Uploaded by Mabel