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Geometry and Measurement - Volume and Surface Area of 3D Shapes (Prisms and Cylinders)

Grade 7IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A prism is a 3D solid with two identical, parallel ends called bases and flat rectangular sides. You can visualize a prism by imagining a 2D shape (like a triangle or hexagon) being 'extruded' or stretched upwards into the third dimension. The cross-section remains the same size and shape throughout its height.

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A cylinder is similar to a prism but features two congruent circular bases connected by a curved surface. If you were to 'unroll' the curved side of a cylinder, it would form a flat rectangle. The length of this rectangle is equal to the circumference of the circular base (2πr2\pi r), and the width is the height (hh) of the cylinder.

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Volume represents the total amount of 3D space occupied by an object. For any uniform prism or cylinder, the volume is found by calculating the area of the base shape and multiplying it by the height (V=Abase×hV = A_{base} \times h). Imagine filling the shape layer by layer with 1-unit cubes.

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The 'Net' of a 3D shape is a 2D diagram that shows what the shape would look like if it were unfolded and laid flat. For a rectangular prism, the net consists of six rectangles. For a cylinder, the net consists of two circles and one large rectangle.

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Surface Area is the total area of all the exterior faces of a 3D shape. To calculate it, you sum the areas of the two bases and the lateral (side) area. For prisms, the lateral area is the perimeter of the base multiplied by the height. For cylinders, the lateral area is the circumference of the circle multiplied by the height.

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Standard units of measurement are essential: Area is always measured in square units (e.g., cm2cm^2 or m2m^2), whereas Volume is always measured in cubic units (e.g., cm3cm^3 or m3m^3). Always ensure all dimensions are in the same unit before starting calculations.

📐Formulae

Volume of a General Prism: V=Abase×hV = A_{base} \times h

Volume of a Cylinder: V=πr2hV = \pi r^2 h

Total Surface Area of a General Prism: SA=(2×Abase)+(Pbase×h)SA = (2 \times A_{base}) + (P_{base} \times h) (where PP is perimeter)

Curved Surface Area (Lateral Area) of a Cylinder: CSA=2πrhCSA = 2 \pi r h

Total Surface Area of a Cylinder: TSA=2πr2+2πrhTSA = 2 \pi r^2 + 2 \pi r h

Area of a Triangle (for triangular prisms): A=12bhA = \frac{1}{2} b h

Circumference of a Circle: C=2πrC = 2 \pi r

💡Examples

Problem 1:

Calculate the volume and total surface area of a triangular prism with a base triangle of base 6 cm6\text{ cm} and height 4 cm4\text{ cm}. The length (height) of the prism is 12 cm12\text{ cm}, and the other two sides of the triangle are 5 cm5\text{ cm} each.

Solution:

  1. Base Area (AbaseA_{base}): A=12×6×4=12 cm2A = \frac{1}{2} \times 6 \times 4 = 12\text{ cm}^2 \n2. Volume: V=Abase×h=12×12=144 cm3V = A_{base} \times h = 12 \times 12 = 144\text{ cm}^3 \n3. Base Perimeter (PbaseP_{base}): P=6+5+5=16 cmP = 6 + 5 + 5 = 16\text{ cm} \n4. Surface Area: SA=(2×Abase)+(Pbase×h)=(2×12)+(16×12)=24+192=216 cm2SA = (2 \times A_{base}) + (P_{base} \times h) = (2 \times 12) + (16 \times 12) = 24 + 192 = 216\text{ cm}^2

Explanation:

To find the volume, we first find the area of the triangular cross-section and multiply by the prism's length. For the surface area, we add the areas of the two triangular ends to the area of the three rectangular sides (calculated using Perimeter ×\times Height).

Problem 2:

A cylinder has a radius of 3 cm3\text{ cm} and a height of 10 cm10\text{ cm}. Find its volume and total surface area. (Use π≈3.14\pi \approx 3.14)

Solution:

  1. Volume: V=πr2h=3.14×32×10=3.14×9×10=282.6 cm3V = \pi r^2 h = 3.14 \times 3^2 \times 10 = 3.14 \times 9 \times 10 = 282.6\text{ cm}^3 \n2. Surface Area: TSA=2πr2+2πrhTSA = 2\pi r^2 + 2\pi r h \nTSA=(2×3.14×32)+(2×3.14×3×10)TSA = (2 \times 3.14 \times 3^2) + (2 \times 3.14 \times 3 \times 10) \nTSA=(2×3.14×9)+(188.4)TSA = (2 \times 3.14 \times 9) + (188.4) \nTSA=56.52+188.4=244.92 cm2TSA = 56.52 + 188.4 = 244.92\text{ cm}^2

Explanation:

For volume, we calculate the area of the circular base (πr2\pi r^2) and multiply by the height. For the surface area, we calculate the area of the two circular lids (2πr22\pi r^2) and add the area of the curved side, which is a rectangle when flattened (2πr×h2\pi r \times h).