CONNECT: Volume, Surface Area - 2. SURFACE AREAS OF SOLIDS - Units and Formulae

Sep 18, 2026

Uploaded by Dipper

Surface areas of solids explains how to calculate the total outside area of 3D shapes by finding the area of each face and summing them, with emphasis on using square units. Worked examples cover a rectangular prism by decomposing faces, then derive cylinder surface area using a rolled A4 rectangle interpretation and formulas for open versus closed ends. Cone surface area is handled with slant height, formulas for open and closed cases, and an example using Pythagoras to compute the missing slant height.

LanguageEnglish

CategoryExam

FormatPDF

Pages6

Reading Time15 min

AccessFree to download

Surface areas of solids

Surface area of prisms

Surface area of cylinders

Surface area of cones

CONNECT: Volume, Surface Area 2. SURFACE AREAS OF SOLIDS If you need to know more about plane shapes, areas, perimeters, solids or volumes of solids, please refer to CONNECT: Areas, Perimeters – 1. AREAS OF PLANE SHAPES; CONNECT: Areas, Perimeters – 2. PERIMETERS OF PLANE SHAPES and CONNECT: Volume, Surface Area – 1. VOLUMES OF SOLIDS. You may also need to review Pythagoras’ Theorem – if so, please refer to CONNECT: Pythagoras’ Theorem The surface area of a 3D shape is “the total area of the outside” of the shape (De Klerk, 2007, p. 129) . So, to work out the surface area of a prism, you need to work out the area of each face and add these areas together. Before we calculate the surface area of an example, have a think about the units we will use to measure surface area. Example: Calculate the surface area of this rectangular prism: If we draw in the edges that are blocked from sight by the other faces, we can see the shapes of all the faces: The faces are all rectangles and so each of their areas is A = l x w units2 . For each of the top and bottom faces, the area is 3cm x 4cm = 12cm2 . For each of the left and right faces, the area is 4cm x 12cm = 48cm2. For each of the front and back faces, the area is 3cm x 12cm = 36cm2 . So the total surface area is 12 x 2 + 48 x 2 + 36 x 2 cm2 that is, 192cm2. The units are square units because we are measuring area. Surface area of a cylinder Take an ordinary A4 sheet of paper. Now roll it into a cylinder so that the edges just meet. Do you realise that you have used a rectangle to make the curved surface of a cylinder? Place your cylinder so that it stands on its circular base. The length of the rectangle is now the circumference of the circle (base) and the width of the rectangle is now the height of the cylinder.(Or vice versa, depending on which way you rolled the cylinder, but that will not make any difference to our formula.) l So the area of the curved surface of the cylinder is 􀜣 = 2􀟨􀝎 × ℎ units2 . (2􀟨􀝎 is the length of the circumference , which is the same as l, the length of the rectangle. ) If the cylinder is open at both ends, the total surface area of the cylinder is just the area of the curved surface. If it is closed at one end, we need to add the area of that circle and if it is closed at both ends, we need to add the area of both circles. So, the total surface area of a cylinder could be any of the following: 􀜣 = 2􀟨􀝎 × ℎ units2 ← open at both ends OR 􀜣 = 2􀟨􀝎 × ℎ + 􀟨􀝎2 units2 ← open at one end OR 􀜣 = 2􀟨􀝎 × ℎ + 2 × 􀟨􀝎2 units2 ← closed at both ends. Example: Find the surface area of the closed cylinder over the page. r=6cm h=10.3cm We need to use the formula 􀜣 = 2􀟨􀝎 × ℎ + 2 × 􀟨􀝎2 units2 because the cylinder is closed. So, A = 2 x 􀟨 x 6 x 10.3 + 2 x 􀟨 x 62 cm 2 Using a calculator we obtain 614.495 523 cm2 , that is, approximately 614.496cm2 . Surface Area of a Cone The curved surface area of a cone is a little more involved to calculate. We need to know more than just our usual measurements – the radius of the circular base and the perpendicular height of the cone. We also need to know the slant height (s) of the cone. Diagram retrieved January 31, 2013, from [http://www.mathsteacher.com.au/year10/ch14_measurement/18_cone/20cone.htm](http://www.mathsteacher.com.au/year10/ch14_measurement/18_cone/20cone.htm) If we know the slant height of the cone, we can use the formula 􀜣 = 􀟨􀝎􀝏 units 2 to find the area of the curved surface. So for an open cone, this formula gives the total surface area of the cone. But if we have a closed cone, we need to add the area of the circle as well. So, the total surface area of a cone will be: 􀜣 = 􀟨􀝎􀝏 units 2 ← open cone 􀜣 = 􀟨􀝎􀝏 + 􀟨􀝎2 units 2 ← closed cone Example: Find the surface area of the following closed cone: Curved surface area = 􀟨􀝎􀝏 cm2 = 􀟨 x 5.1 x 8.1 cm2 = 129.7791925…cm2 Area of base = 􀟨􀝎2 cm2 = 􀟨 x 5.12 cm2 = 81.71282492…cm2 So the total area = 129.77919

FAQs

How do you calculate the surface area of a prism?

Compute the area of each rectangular face and add them together. Use square units because you are measuring area.

What formulas are used for the surface area of a cylinder?

Use the curved surface area formula based on circumference and height, then add base-circle areas depending on whether the cylinder is open at both ends, open at one end, or closed at both ends.

How do you find the surface area of a cone when slant height is missing?

Use right-triangle relationships with the radius and perpendicular height to apply Pythagoras’ Theorem, find the slant height, then substitute into the cone’s curved-surface and base-area formulas.

Sep 18, 2026

Uploaded by Dipper